international research articles

Microbiome research in Africa must be based on equitable partnerships

• Ovokeraye H. Oduaran   ORCID: orcid.org/0000-0002-3033-7873 1 ,
• Moréniké Oluwátóyìn Foláyan 2 , 3 ,
• Arox W. Kamng’ona 4 ,
• Annettee Nakimuli   ORCID: orcid.org/0000-0003-2806-0243 5 ,
• Lamech M. Mwapagha   ORCID: orcid.org/0000-0003-1048-1787 6 ,
• Mathabatha E. Setati   ORCID: orcid.org/0000-0002-5450-009X 7 ,
• Michael Owusu 8 ,
• Nicola Mulder   ORCID: orcid.org/0000-0003-4905-0941 9 ,
• Thulani P. Makhalanyane   ORCID: orcid.org/0000-0002-8173-1678 10 , 11 &
• Soumaya Kouidhi 12  

Nature Medicine ( 2024 ) Cite this article

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• Developing world

A framework is presented for equitable and effective microbiome research partnerships between African researchers, international partners, healthcare professionals, policymakers and stakeholders.

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Acknowledgements

The authors thank all the members of the H3Africa Microbiome Task Force and the African Microbiome Special Interest Group for the feedback received in the writing process.

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Sydney Brenner Institute for Molecular Bioscience, University of the Witwatersrand, Johannesburg, South Africa

Ovokeraye H. Oduaran

Department of Child Dental Health, Obafemi Awolowo University, Ile-Ife, Nigeria

Moréniké Oluwátóyìn Foláyan

Oral Health Initiative, Center for Reproductive and Population Health Studies, Nigerian Institute of Medical Research, Yaba, Lagos, Nigeria

Kamuzu University of Health Sciences, School of Life Sciences and Allied Health Professions, Blantyre, Malawi

Arox W. Kamng’ona

Department of Obstetrics and Gynaecology, School of Medicine, Makerere University College of Health Sciences, Kampala, Uganda

Annettee Nakimuli

Department of Biology, Chemistry and Physics, Faculty of Health, Natural Resources and Applied Sciences, Namibia University of Science and Technology, Windhoek, Namibia

Lamech M. Mwapagha

African Microbiome Institute, Division of Molecular Biology and Human Genetics, Department of Biomedical Sciences, Stellenbosch University, Stellenbosch, South Africa

Mathabatha E. Setati

Department of Medical Diagnostics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

Michael Owusu

Computational Biology Division Department of Integrative Biomedical Sciences, IDM, Faculty of Health Sciences, University of Cape Town, Cape Town, South Africa

Nicola Mulder

Department of Microbiology, Faculty of Science, Stellenbosch University, Stellenbosch, South Africa

Thulani P. Makhalanyane

School for Data Science and Computational Thinking, Stellenbosch University, Stellenbosch, South Africa

Laboratory of Biotechnology and Valorisation of Bio-Geo Resources (LR11ES31), Higher Institute of Biotechnology of Sidi Thabet, BiotechPole of Sidi Thabet, University of Manouba, Ariana, Tunisia

Soumaya Kouidhi

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Oduaran, O.H., Foláyan, M.O., Kamng’ona, A.W. et al. Microbiome research in Africa must be based on equitable partnerships. Nat Med (2024). https://doi.org/10.1038/s41591-024-03026-2

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International Marketing Research: A State-of-the-Art Review and the Way Forward

• First Online: 21 October 2017

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• Leonidas C. Leonidou 5 ,
• Constantine S. Katsikeas 6 ,
• Saeed Samiee 7 &
• Bilge Aykol 8  

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The chapter provides a systematic review of 1722 international marketing articles published in the top six international business journals during the period 1995–2015. The analysis focuses on five major areas: profiles of authors involved in international marketing research; major contributors of international marketing articles based on their productivity and impact; structural characteristics of articles published on international marketing; articles with the greatest impact in the international marketing field; and specific themes of international marketing research covered and their evolution over time. The analysis revealed that the international marketing field has experienced a significant proliferation during the investigation period as demonstrated by its continuous refinement, improved quality, and extensive topical coverage. Fruitful directions for future research on the subject are discussed.

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Constantine S. Katsikeas

Collins College of Business, University of Tulsa, 800 South Tucker Drive, Tulsa, OK, 74104-9700, USA

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Leonidou, L.C., Katsikeas, C.S., Samiee, S., Aykol, B. (2018). International Marketing Research: A State-of-the-Art Review and the Way Forward. In: Leonidou, L., Katsikeas, C., Samiee, S., Aykol, B. (eds) Advances in Global Marketing. Springer, Cham. https://doi.org/10.1007/978-3-319-61385-7_1

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The benefits and challenges of international research collaboration

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Working with international colleagues can provide new insights and even a career boost, but it takes patience and planning.

Academic research is becoming ever more international. Whether it is to gain access to specialized equipment, develop new ideas or tap into new sources of funding, researchers are reaching out to their colleagues around the world, and their work is better for it.

“For me, it’s transparent that science is an international, global endeavour,” says Alejandro Adem, a mathematician at the University of British Columbia and chief executive officer of the non-profit research and training organization Mitacs. “Ideas transcend borders, no country controls the marketplace of ideas.”

For some fields of research, such as particle physics, working internationally is not a choice but a “way of life,” says Michael Roney, a professor of physics at the University of Victoria and director of its Institute of Particle Physics. Projects in particle physics often require huge, expensive infrastructure that no single country can afford on its own – not everybody can have their own Large Hadron Collider. “We need to collaborate to afford the science,” says Dr. Roney.

That means the field has developed a unique culture of collaboration, with researchers following their interests all over the world. Dr. Roney has worked extensively at the European Organization for Nuclear Research , known as CERN, the SLAC National Accelerator Laboratory in the United States and the KEK laboratory in Japan. This leads to “an extremely fertile cross-pollination of ideas,” he says. “New ideas come when you interact with people from diverse backgrounds. You think about things in a different way and suddenly see connections you never thought of before.”

Lori Beaman, who holds the Canada Research Chair in Religious Diversity and Social Change at the University of Ottawa, says her experiences working with international colleagues has provided her with an invaluable depth of perspective. In her studies of the decline of Christianity in North America, for example, she has found that many churches are closing and selling off their buildings. But, when working with colleagues in Sweden, she learned that in Scandinavia the state provides support to churches even when attendance is declining, so the buildings themselves remain.

“You might assume we were all handling an issue in the same way, but that’s not the case at all,” she says. “We live in a global world, we want to be able to think more broadly. We can’t do that without international collaboration.”

For Heather Aldersey, a researcher in rehabilitation therapy at Queen’s University who works on disability inclusion in the developing world, having international partners is vital to ensuring that her research translates into real actions to build capacity in the overseas communities where she works – for example, a recent project to develop a new occupational therapy program in Ethiopia. Without leadership and direction from her Ethiopian colleagues at the University of Gondar to ensure the research is used effectively on the ground, there would be little point in the work. “There has to be active buy-in and collaboration with the people affected by our research,” she says. “We have stellar partners who understand the community.”

International collaborations, and especially working abroad, can also provide a real career boost. Vincent Larivière, who holds the Canada Research Chair on the Transformations of Scholarly Communication at Université de Montréal, has found that academic mobility, whether it is having a second affiliation in another country or moving permanently abroad, is associated with work that has higher scientific impact as measured by citation rates. “Research by internationally mobile academics is more visible and more cited,” he says. “This correlates with what we know about the importance of international collaboration.”

Dr. Larivière’s study of the benefits of mobility, published in the influential journal Nature , is a case in point. It was one of the projects he worked on while a visiting professor at Leiden University in the Netherlands and involved researchers from Europe, the United States and Canada. “The project became an example of what we were studying,” he says.

Finding collaborators is an organic process, says Dr. Beaman. The timing needs to be right and an opportunity to work together might not present itself for years. Three years ago, for example, she met a Finnish academic after a conference in the Netherlands when they took the same train to the airport. They kept in touch and now that researcher will be a visiting scholar at U of Ottawa next year. Another colleague from Brazil will be visiting at the same time. Dr. Beaman says she’s excited about the possibilities for collaboration that could arise. “It’s an interesting configuration of people. I don’t know what will come of it,” she says.

The digital age is making international collaboration even easier. “I have international collaborators I’ve never met,” says Dr. Larivière. “Someone can just send an email to share ideas and data, and we can write a paper without ever having to meet each other in person.”

But international work has its share of challenges. Some are minor inconveniences that are easy to adjust to, while others can turn the project into a difficult grind.

Sometimes it can be something as simple as getting used to the work culture in another country. In Canada, Dr. Larivière was used to having access to his office around the clock and organizing his work schedule as he wanted. In the Netherlands, however, the university closes at 7 p.m. “I needed to adapt and change my working schedule,” he says.

Since the working language of science around the world tends to be English, Canadian academics don’t usually have too many problems with communication. “Luckily for us, the lingua franca is English,” says Dr. Roney.

But there can still be occasional hiccups with colleagues from countries where English isn’t a common second language. When visiting a collaborator’s lab in Brazil, Dr. Beaman says some in the group were able to provide an ad-hoc translation that worked fine. “It was very collegial, collaborative, lots of humour and people willing to go the extra mile to make sure that everybody understood what everybody else’s ideas were,” she says.

It’s important to keep in mind that international projects tend to move more slowly than usual. “Researchers often don’t realize how long it can take,” says Jennifer Morawiecki, international research manager in the office of research services at Dalhousie University. “They need to prepare for the long haul.”

Having collaborators scattered across the globe can make it tricky to set up meetings when dealing with multiple time zones. Researchers have to get used to having conference calls at odd times of the day. Even so, “some meetings it’s just not feasible to attend in person or remotely,” says Dr. Beaman.

This gets my vote for Best Poster here at #INORMS2018 . Mad props to my counterparts at @unimelb and @karolinskainst . Love it! pic.twitter.com/Yr1QPZ8e78 — Jennifer Morawiecki (@jmorawiecki) June 6, 2018

Arranging visas and permits for international work can also slow down the process, especially when working in the developing world. And funding timelines and application procedures don’t always align well, so getting agencies in different countries to work together can be slow and cumbersome. “Research administration [in other countries] is not always as well-developed as it is in Canada,” says Dr. Morawiecki. “The processes can run the gamut from wildly complicated to disturbingly simple.”

Politics can also intrude. Sometimes the security or political situation will change, which may shift a government’s funding priorities or even make it too dangerous to continue the project. Even in developed countries, politics can get in the way – the U.K.’s vote to leave the European Union has disrupted academia in the country , potentially making European collaborations more difficult.

Researchers will perhaps not be surprised to learn that the biggest challenge when it comes to international research is getting the money to do it. In a 2014 survey by Universities Canada on internationalization, 83 percent of universities cited the lack of research funding opportunities as the most significant barrier to international collaboration. “There are few dedicated funding mechanisms for international collaboration in Canada,” says Dr. Morawiecki. “There used to be more, but a lot have been cut in the past five to eight years.”

That means researchers sometimes have to get creative when it comes to finding funding. Many of the projects that Dr. Aldersey at Queen’s is involved with are funded by international development bodies, who are often more interested in concrete interventions than research questions. Her Ethiopia project, for example, funded by the Mastercard Foundation, was focused on scholarships and talent development, bringing Ethiopian students to Canada to study occupational therapy. She therefore had to find ways to embed research questions in the project both to evaluate it in a meaningful way and to add to the scientific body of knowledge.

One way she found to do this was to include a study of the current situation in Ethiopia. “When developing the first occupational therapy project in a country, we need to first know what OT looks like there now,” she says. “We look for multiple ways to benefit from one experience.”

For most researchers, international collaborations will involve trying to cobble together funding from multiple sources, as each member of the team applies independently to their home country’s funder in the hope that it all comes together at the right time. But it doesn’t always work. “If the pieces don’t all come together, the project can fall apart,” says U of Ottawa’s Dr. Beaman.

The research councils are trying to make it easier by working with their counterparts overseas to support cross-border research projects, says Brent Herbert-Copley, executive vice-president of the Social Sciences and Humanities Research Council. SSHRC is a member of the Trans-Atlantic Platform , or TAP, a collection of 18 social science funders from 12 countries in North America, South America and Europe that are working to align policies and offer funding to improve international links. “We’re trying to make it easier for collaboration to take place,” says Dr. Herbert-Copley.

TAP grew out of a European Union-funded initiative called Digging into Data , which supported data-intensive social science research collaborations between North America and Europe. The platform is now preparing to offer its second joint call for proposals, on the theme of social innovation. The call will offer a single application process for international teams, but funding for each collaborator will come from their own national funder. Dr. Herbert-Copley says TAP is a work in progress, but he says he’s excited about where it is heading. “It’s a learning curve, the TAP is the first time we’ve taken the time to work out how agencies could collaborate,” he says. “It will get easier the second and third times around.”

Canada’s research councils are also members of the Global Research Council , an organization that brings together the heads of national research councils from around the world to facilitate interaction and share best practices. Mario Pinto, who until recently was president of the Natural Sciences and Engineering Research Council, just completed a term as chairman of the organization. (Dr. Pinto stepped down as head of NSERC in September, after he was interviewed for this article.) He says that while the GRC does not fund research directly, it can help funders from different countries put together joint calls when the resources are available.

Canada’s funding agencies are consulting now on how to use the new $275-million Tri-Council Research Fund , announced in the 2018 federal budget to support international, interdisciplinary and high-risk research. Dr. Pinto says the GRC will provide useful contacts when setting up international projects. “The advantage of the GRC is that we have good relationships, so it is easy to forge partnerships,” he says.

Europe’s massive, multi-year research funding program Horizon 2020 , with a budget of €80 billion ($122 billion CAD), is a tempting target for researchers looking for international partners. “Our motto is ‘open to the world,’” says Viktoria Bodnarova, who runs the North American EURAXESS office in Washington, D.C. “We welcome international collaboration and we’re trying our best to attract international collaborators.”

The complex bureaucracy and red tape surrounding the program can be a deterrent, but the EU does provide guidance on how to get involved through its network of EURAXESS offices. Ms. Bodnarova says there are two main ways for Canadian researchers to work with their European colleagues through Horizon 2020.

The first is to join one of the consortia that compete for the large, collaborative grants. But that requires the researcher’s institution to be a partner in the project, and Canadian participants may still have to bring their own funding to the group. The other way is to apply for one of the individual grants, either through the Marie Curie fellowships or the European Research Council . These allow foreign researchers to pursue a project in any scientific field at a host institution in a European country, and Canadians have been remarkably successful at obtaining them, says Ms. Bodnarova.

For students, Mitacs offers a mobility program that allows graduate students to move between Canada and other countries for short periods. Dr. Adem says these visits can help build relationships between labs. “In a world of global mobility, it’s important for collaborators in different countries to have students going between them,” he says. “Students are agents for change and can teach the professors new techniques.” Dr. Adem says he hopes Mitacs will be able to expand some of its international mobility programs to include faculty as well.

For any researcher considering working internationally, the first stop should be their university’s research office, counsels Dalhousie’s Dr. Morawiecki. Most offices offer a free orientation on international research and will reach out to early career researchers to make sure that they’re aware of the opportunities available. “We want you to come talk to us,” she says.

While the challenges of international work can seem daunting, the rewards make it all worthwhile, says Dr. Morawiecki. And research offices, she says, are there to make the process as easy as possible. “It’s something researchers naturally want to do, but it takes time and effort,” she says. “You have to hustle a bit more to be successful.”

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Thank you for this article. In particular, We found the points raised regarding the benefits and challenges of engaging in international research collaboration to be very helpful. The “fertile cross-pollination of ideas” is truly a unique quality of the culture of international collaboration and can serve as a significant resource to producing a more valid evidence base for our scientific community. In our recently published paper (Guler et al., 2018), we developed a case study focused on the facilitators and barriers to team performance within the World Health Organization’s (WHO) Field Studies Coordination Group. We found similar findings regarding the complexity of global mental health collaboration.

If interested in reading: https://www.sciencedirect.com/science/article/pii/S1697260018300565

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Article Contents

1 introduction, 2 preliminaries, 3 entropy of noncommutative poisson boundaries, 4 trivial boundaries of amenable von neumann algebras, 5 choquet-deny property of von neumann algebras, acknowledgments.

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Noncommutative Poisson Boundaries, Ultraproducts, and Entropy

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Shuoxing Zhou, Noncommutative Poisson Boundaries, Ultraproducts, and Entropy, International Mathematics Research Notices , Volume 2024, Issue 10, May 2024, Pages 8794–8818, https://doi.org/10.1093/imrn/rnae022

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We construct the noncommutative Poisson boundaries of tracial von Neumann algebras through the ultraproducts of von Neumann algebras. As an application of this result, we complete the proof of Kaimanovich-Vershik’s fundamental theorems regarding noncommutative entropy. We also prove the Amenability-Trivial Boundary equivalence and Choquet-Deny-Type I equivalence for tracial von Neumann algebras.

Recently, Das-Peterson [ 10 ] extended the notion of Poisson boundary to the noncommutative setting: Let |$(M,\tau )$| be a tracial von Neumann algebra with separable predual. As a generalization of admissible measures on discrete groups, a normal regular strongly generating hyperstate on |$B(L^{2}(M))$| is a normal state |$\varphi \in B(L^{2}(M))_{*}$| with the standard form |$\varphi (T)=\sum _{n=1}^{\infty }\langle T \hat{z}_{n}^{*},\hat{z}_{n}^{*}\rangle $|⁠ , where |$\{z_{n}\}\subset M$| satisfying |$\sum _{n=1}^{\infty }z_{n}^{*}z_{n}=\sum _{n=1}^{\infty } z_{n}z_{n}^{*}=1$| and |$\{z_{n}\}$| generates |$M$| as a weakly closed unital algebra. The normal u.c.p. |$M$| -bimodular map associated to |$\varphi $| is defined as |${\mathcal{P}}_{\varphi }:T\in B(L^{2}(M))\mapsto \sum _{n=1}^{\infty } (Jz_{n}^{*}J)T(Jz_{n}J)\in B(L^{2}(M))$|⁠ . As a generalization of classical Poisson boundary, the noncommutative Poisson boundary of |$\varphi $| is the unique von Neumann algebra |${\mathcal{B}}_{\varphi }$| that is isomorphic to |$\textrm{Har}({\mathcal{P}}_{\varphi })=\{ T\in B(L^{2}(M))\mid \mathcal{P}_{\varphi }(T)=T\}$| as operator system. The isomorphism |${\mathcal{P}}:{\mathcal{B}}_{\varphi }\to \textrm{Har}({\mathcal{P}}_{\varphi })$| is called the Poisson transform. Note that since |$M\subset \textrm{Har}(\mathcal{P}_{\varphi })$|⁠ , |$M$| can always be embedded into |${\mathcal{B}}_{\varphi }$| as a subalgebra. And |$\zeta :=\varphi \circ{\mathcal{P}}$| is the canonical hyperstate on |${\mathcal{B}}_{\varphi }$|⁠ .

The following example [ 16 , Theorm 4.1] shows that the noncommutative Poisson boundary is a generalization of the classical one: Let |$(M,\tau )=(L(\Gamma ),\tau )$| be the group von Neumann algebra of a countable discrete group |$\Gamma $|⁠ , and |$\varphi (T)=\sum _{\gamma \in \Gamma }\mu (\gamma )\langle T\delta _{\gamma ^{-1}},\delta _{\gamma ^{-1}}\rangle $| be the hyperstate associated to |$\mu \in \textrm{Prob}(\Gamma )$|⁠ . Then the |$\varphi $| -Poisson boundary |${\mathcal{B}}_{\varphi }$| is exactly the group measure space construction |$L(\Gamma \curvearrowright B)$|⁠ , where |$(B,\nu _{B})$| is the |$(\Gamma ,\mu )$| -Poisson boundary.

Das-Peterson [ 10 ] already proved several analogues of the classical Poisson boundary theory, for examples, the injectivity, double ergodicity and rigidity of |${\mathcal{B}}_{\varphi }$|⁠ . The main goal of this paper is to further explore the analogues between classical and noncommutative Poisson boundaries.

Das-Peterson [ 10 ] also initiated entropy theory of noncommutative boundaries, which we refer to 2.3 for a brief introduction of the definition. Moreover, Das-Peterson [ 10 ] proved the analogues of part of Kaimanovich-Vershik’s fundamental theorems regarding entropy [ 18 ], while the rest are left open. In Section 3, inspired by [ 23 ], we develop a new way to construct |${\mathcal{B}}_{\varphi }$| with the ultraproducts of von Neumann algebras, which helps completing the proof of Kaimanovich-Vershik’s fundamental theorems:

(i) |$h_{\varphi }({\mathcal{B}}_{\varphi },\zeta )=h_{\varphi }({\mathcal{A}}^{\omega }, \varphi ^{\omega })=\lim _{\omega }\limits h_{\varphi }({\mathcal{A}}_{n}, \varphi _{n})=h(\varphi )$|⁠ .

(ii) A |$\varphi $| -boundary |$({\mathcal{B}}_{0},\zeta _{0})$| (i.e., von Neumann subalgebra of |$({\mathcal{B}}_{\varphi },\zeta )$| that contains |$(M,\tau )$|⁠ ) satisfies |$h_{\varphi }({\mathcal{B}}_{0},\zeta _{0})= h_{\varphi }({\mathcal{B}}_{\varphi },\zeta )$| if and only if |$({\mathcal{B}}_{0},\zeta _{0})=({\mathcal{B}}_{\varphi },\zeta )$|⁠ .

(iii) |$({\mathcal{B}}_{\varphi },\zeta )$| is trivial (i.e., |$({\mathcal{B}}_{\varphi },\zeta )=(M,\tau )$|⁠ ) if and only if |$h(\varphi )=0$|⁠ .

In [ 18 ], Kaimanovich-Vershik provided an important characterization of amenable groups involving classical Poisson boundary: A countable discrete group |$\Gamma $| is amenable if and only if there exists an admissible measure |$\mu \in \textrm{Prob}(\Gamma )$| such that the Poisson boundary of |$(\Gamma ,\mu )$| is trivial. In Section 4, we prove an analogue of Kaimanovich-Vershik’s amenability-trivial boundary equivalence:

The tracial von Neumann algebra |$(M,\tau )$| is amenable if and only if there exists a measure |$\mu \in \textrm{Prob}(\mathcal{U}(M))$| such that the associated hyperstate |$\varphi _{\mu }(T)=\int _{\mathcal{U}(M)} \langle T u^{*}\hat{1},u^{*}\hat{1}\rangle \textrm{d} \mu (u)$| has trivial Poisson boundary.

The Choquet-Deny property of locally compact groups is a property stronger than amenability: A locally compact group |$G$| is Choquet-Deny if for any admissible |$\mu \in \textrm{Prob}(G)$|⁠ , the |$\mu $| -Poisson boundary is trivial. Since the normal regular strongly generating hyperstate is a generalization of admissible measure, it’s natural to define the Choquet-Deny property of tracial von Neumann algebras to be that any normal regular strongly generating hyperstate has trivial Poisson boundary.

The classical Choquet-Deny theorem [ 8 ] states that any abelian countable discrete group is Choquet-Deny, of which Das-Peterson [ 10 ] proved an analogue, telling the same for abelian tracial von Neumann algebras. In Section 5, inspired by the latest progress regarding Choquet-Deny property of groups [ 12 , 17 ], we prove a stronger result, providing a more comprehensive understanding of the Choquet-Deny property of tracial von Neumann algebras:

Let |$(M,\tau )$| be a tracial von Neumann algebra with separable predual. Then |$(M,\tau )$| is Choquet-Deny if and only if |$M$| is of type |$\textrm{I}$|⁠ .

Following [ 16 , Theorm 4.1], for a countable discrete group |$\Gamma $|⁠ , the Choquet-Deny property of |$L(\Gamma )$| can always induce the Choquet-Deny property of |$\Gamma $|⁠ . Note that when |$\Gamma $| is finitely generated, |$\Gamma $| is Choquet-Deny if and only if |$\Gamma $| is virtually nilpotent (see [ 17 ] and [ 12 ]). And |$L(\Gamma )$| is of type |$\textrm{I}$| if and only if |$\Gamma $| is virtually abelian (see [ 26 ]). The Heisenberg group |$H_{3}(\mathbb{Z})$| is exactly such a finitely generated group that is virtually nilpotent but not virtually abelian. Hence for a countable discrete group |$\Gamma $|⁠ , the Choquet-Deny property of |$L(\Gamma )$| is not equivalent to, but strictly stronger than the Choquet-Deny property of |$\Gamma $|⁠ . That’s because the construction of the noncommutative Poisson boundary is more flexible than the construction of the classical Poisson boundary.

2.1 Hyperstates and u.c.p. maps

For |$\psi \in \mathcal{S}_{\tau }({\mathcal{A}})$| and |$\varphi \in \mathcal{S}_{\tau }(B(L^{2}(M)))$|⁠ , the convolution |$\varphi \ast \psi \in \mathcal{S}_{\tau }({\mathcal{A}})$| is defined to be the hyperstate associated to the |$M$| -bimodular u.c.p. map |$\mathcal{P}_{\varphi }\circ \mathcal{P}_{\psi }$|⁠ . And |$\psi $| is said to be |$\varphi $| -stationary if |$\varphi \ast \psi =\psi $|⁠ .

2.2 Noncommutative Poisson boundaries

regular , if |$\sum _{n=1}^{\infty }\limits z_{n}^{*}z_{n}=\sum _{n=1}^{\infty }\limits z_{n}z_{n}^{*}=1$|⁠ ;

strongly generating , if the unital algebra (rather than the unital |$\ast $| -algebra) generated by |$\{z_{n}\}$| is weakly dense in |$M$|⁠ .

According to [ 10 ], when |$\varphi $| is a normal regular strongly generating hyperstate, |$\zeta :=\varphi \circ{\mathcal{P}}\in \mathcal{S}_{\tau }({\mathcal{B}}_{\varphi })$| is a normal faithful hyperstate on |${\mathcal{B}}_{\varphi }$|⁠ .

In this paper, we will focus on the Poisson boundaries of normal regular strongly generating hyperstates, which can be viewed as a generalization of admissible measures on countable discrete groups.

2.3 Entropy

2.4 ultraproducts of von neumann algebras.

Ando-Haagerup [ 1 ] provided us with the relation between the two ultraproducts above: Let |$({\mathcal{A}}_{n},\varphi _{n})$| be as above and |$N:=\prod ^{\omega }({\mathcal{A}}_{n},L^{2}({\mathcal{A}}_{n},\varphi _{n}))$| be the Groh-Raynaud ultraproduct. Let |$\xi _{\varphi _{n}}\in L^{2}({\mathcal{A}}_{n},\varphi _{n})$| be the cyclic vector and |$\xi _{\omega }=(\xi _{\varphi _{n}})_{\omega }\in (L^{2}({\mathcal{A}}_{n},\varphi _{n}))_{\omega }$|⁠ . Let |$p\in N$| be the orthogonal projection onto |$\overline{N^{\prime} \xi _{\omega }}$| and define the normal state |$\varphi _{\omega }$| on |$N$| by |$\varphi _{\omega }(x)=\langle x \xi _{\omega },\xi _{\omega }\rangle $|⁠ .

In classical Poisson boundary theory, for a countable discrete group |$\Gamma $| and an admissible |$\mu \in \textrm{Prob}(\Gamma )$| (i.e., |$\cup _{n\in \mathbb{N}}(\textrm{supp} \, \mu )^{n}=\Gamma$|⁠ ) with |$H(\mu )<+\infty $|⁠ , the following statements are Kaimanovich-Vershik’s fundamental theorems regarding entropy (see [ 18 ]):

(a) For any |$(\Gamma ,\mu )$| -space |$(X,\nu _{X})$|⁠ , one has |$h_{\mu }(X,\nu _{X})\leq h(\mu )$|⁠ .

(b) For the |$\mu $| -Poisson boundary |$(B,\nu _{B})$|⁠ , the equality |$h_{\mu }(B,\nu _{B})= h(\mu )$| holds.

(c) A |$\mu $| -boundary |$(B_{0},\nu _{0})$| satisfies |$h_{\mu }(B_{0},\nu _{0})= h_{\mu }(B,\nu _{B})$| if and only if |$(B_{0},\nu _{0})$| is measurably isomorphic to |$(B,\nu _{B})$|⁠ .

(d) The |$\mu $| -Poisson boundary |$(B,\nu _{B})$| is trivial (i.e., |$B=\{*\}$| is a single point) if and only if |$h(\mu )=0$|⁠ .

In [ 10 ], Das-Peterson proved the noncommutative analogues of (a), a weaker version of (d) with |$h(\mu )$| replaced by |$h_{\mu }(B,\nu _{B})$|⁠ , and the “if” direction of (d) as a direct corollary.

In this section, we will prove the noncommutative analogues of (b)-(d). The method is inspired by [ 23 ], where Sayag-Shalom construct the classical Poisson boundaries through the ultraproduct of abelian |$\textrm{C}^{*}$| -algebras and apply it to the proofs of (b) and some other results. For the Tomita-Takesaki theory involved in this section, we refer to [ 25 , Chapter VI-IX] for details.

  Theorem 3.1. Let |$\varphi \in \mathcal{S}_{\tau }(B(L^{2}(M,\tau ))$| be a normal regular strongly generating hyperstate, and |$({\mathcal{B}}_{\varphi },\zeta )$| be the Poisson boundary. Take an increasing sequence |$\{a_{n}\}\subset \mathbb{R}$| satisfying |$\frac{1}{2}\leq a_{n}< 1 $| and |$\lim _{n}\limits a_{n}=1$|⁠ , and an ultrafilter |$\omega \in \beta \mathbb{N}\setminus \mathbb{N}$|⁠ . Since |$\varphi $| is strongly generating, we can take a sequence of von Neumann algebras with faithful normal hyperstates by $$\begin{align*} &({\mathcal{A}}_n,\varphi_n)=(B(L^2(M)),(1-a_n)\sum_{k=0}^{\infty}a_n^k \varphi^{*k}),\end{align*}$$ where |$\varphi ^{*0}(\,\cdot \,)=\langle \,\cdot \, \hat{1},\hat{1}\rangle $|⁠ . Let |$({\mathcal{A}}^{\omega }, \varphi ^{\omega })$| be the Ocneanu ultraproduct of |$\{({\mathcal{A}}_{n},\varphi _{n})\}_{n}$| along |$\omega $|⁠ . Then |$({\mathcal{B}}_{\varphi },\zeta )$| can be embedded into |$({\mathcal{A}}^{\omega }, \varphi ^{\omega })$| as a von Neumann subalgebra with normal conditional expectation and $$\begin{align*} &h_\varphi({\mathcal{B}}_\varphi,\zeta)=h_\varphi({\mathcal{A}}^\omega, \varphi^\omega).\end{align*}$$

Let |$l^{\infty }(\mathbb{N},{\mathcal{A}}_{n})$|⁠ , |$\mathcal{L}_{\omega }$|⁠ , |$\mathcal{I}_{\omega }$| and |$\mathcal{M}^{\omega }$| be as in ( 2.1 ). Then |$\mathcal{L}_{\omega }$| is a left ideal of |$l^{\infty }(\mathbb{N},{\mathcal{A}}_{n})$| and |$({\mathcal{A}}^{\omega }, \varphi ^{\omega })=\mathcal{M}^{\omega }/\mathcal{I}_{\omega }$|⁠ . For |$(x_{n})_{n} \in \mathcal{M}^{\omega }$|⁠ , denote by |$(x_{n})^{\omega }\in{\mathcal{A}}^{\omega }$| the image of |$(x_{n})_{n} $|⁠ .

Let |$\Delta : B(L^{2}(M))\to l^{\infty }(\mathbb{N},{\mathcal{A}}_{n})$| be the diagonal embedding. First, let’s prove that |$\mathcal{L}_{\omega }\Delta (M)\subset \mathcal{L}_{\omega }$|⁠ .

Assume that |$\varphi =\sum _{k=1}^{\infty }\langle \, \cdot \, \hat{z}_{k}^{*},\hat{z}_{k}^{*}\rangle $| where |$\sum _{k=1}^{\infty }z_{k}z_{k}^{*}=\sum _{k=1}^{\infty }z_{k}^{*}z_{k}=1$| and |$\{z_{k}\}\subset M$| generates |$M$| as a weakly closed unital subalgebra.

Since |$\varphi $| is strongly generating, |$\mathbb{C}[\{z^{*}_{k}\}]$| is weakly dense in |$M$|⁠ , inducing that it is also strongly dense in |$M$| (for a unital subalgebra, self-adjointness of its weak closure is enough to prove the von Neumann bicommutant theorem). So ( 3.2 ) also holds for any |$z\in M$|⁠ . Hence, we have |$\mathcal{L}_{\omega }\Delta (M)\subset \mathcal{L}_{\omega }$|⁠ .

Since |$\mathcal{I}_{\omega }=\mathcal{L}_{\omega } \cap \mathcal{L}^{*}_{\omega }$| and |$\mathcal{L}_{\omega }\Delta (M)\subset \mathcal{L}_{\omega }$|⁠ , we have |$\Delta (M)\subset \mathcal{M}^{\omega }$|⁠ . Hence, we can embed |$(M,\tau )$| into |$({\mathcal{A}}^{\omega },\varphi ^{\omega })$| through the diagonal map: |$x\in M\mapsto \Delta (x)+\mathcal{I}_{\omega }\in{\mathcal{A}}^{\omega }$|⁠ . And by ( 3.1 ), we know that |$\varphi ^{\omega }$| is |$\varphi $| -stationary.

Let’s prove that |${\mathcal{P}}_{\varphi ^{\omega }}(\bar{\Delta }(T))=T$| for any |$T\in \textrm{Har}({\mathcal{P}}_{\varphi })$|⁠ . Only need to prove for any |$a,b\in M$|⁠ , we have |$\varphi ^{\omega }(\bar{\Delta }(a)\bar{\Delta }(T)\bar{\Delta }(b))=\langle aTb\hat{1},\hat{1}\rangle$|⁠ .

Let |$({\mathcal{B}},\varphi ^{\omega }):=(\bar{\Delta }(\textrm{Har}({\mathcal{P}}_{\varphi })),\varphi ^{\omega })\subset ({\mathcal{A}}^{\omega },\varphi ^{\omega })$|⁠ . Now let’s prove that |$({\mathcal{B}},\varphi ^{\omega })$| is a von Neumann subalgebra isomorphic to |$({\mathcal{B}}_{\varphi },\zeta )$|⁠ .

Let |$\mathcal{P}:({\mathcal{B}}_{\varphi },\zeta )\to (\textrm{Har}({\mathcal{P}}_{\varphi }),\varphi )$| be the Poisson transform. Define |$\Phi :({\mathcal{B}}_{\varphi },\zeta )\to ({\mathcal{B}},\varphi ^{\omega })$| by |$\Phi =\bar{\Delta }\circ \mathcal{P}=({\mathcal{P}}_{\varphi ^{\omega }}|_{\mathcal{B}})^{-1}\circ \mathcal{P}$|⁠ . Following [ 7 , Remark 3.4], a bijective u.c.p. map between |$\textrm{C}^{*}$| -algebras with a u.c.p. inverse is necessarily a |$\textrm{C}^{*}$| -isomorphism. Hence |$\Phi :{\mathcal{B}}_{\varphi }\to{\mathcal{B}}$| is a |$\textrm{C}^{*}$| -isomorphism. Since both |$\bar{\Delta }$| and |${\mathcal{P}}$| are state preserving, we know that |$\Phi :({\mathcal{B}}_{\varphi },\zeta )\to ({\mathcal{A}}^{\omega },\varphi ^{\omega })$| preserves the normal faithful state |$\zeta $|⁠ . Hence, |$\Phi :({\mathcal{B}}_{\varphi },\zeta )\to ({\mathcal{A}}^{\omega },\varphi ^{\omega })$| is normal, |${\mathcal{B}}=\Phi ({\mathcal{B}}_{\varphi })$| is a von Neumann subalgebra of |${\mathcal{A}}^{\omega }$| and |$\Phi :{\mathcal{B}}_{\varphi }\to{\mathcal{B}}$| is a |$\textrm{W}^{*}$| -isomorphism that embeds |$({\mathcal{B}}_{\varphi },\zeta )$| into |$({\mathcal{A}}^{\omega },\varphi ^{\omega })$|⁠ . And by [ 25 , Theorem IX.4.2], the conditional expectation |$E:{\mathcal{A}}^{\omega }\to{\mathcal{B}}$| is normal.

Now let’s prove that |$h_{\varphi }({\mathcal{B}}_{\varphi },\zeta )=h_{\varphi }({\mathcal{A}}^{\omega }, \varphi ^{\omega })$|⁠ . By the isomorphism, we only need to prove that |$h_{\varphi }({\mathcal{B}},\varphi ^{\omega })=h_{\varphi }({\mathcal{A}}^{\omega }, \varphi ^{\omega })$|⁠ .

By ( 3.4 ) we only need to prove |$e_{M}\Delta _{({\mathcal{B}},\varphi ^{\omega })}^{it}e_{M}=e_{M}\Delta _{({\mathcal{A}}^{\omega },\varphi ^{\omega })}^{it}e_{M}$|⁠ , where |$e_{M}$| is the orthogonal projection from |$L^{2}({\mathcal{A}}^{\omega },\varphi ^{\omega })$| to |$L^{2}(M,\tau )$|⁠ .

Recall that |$E=\bar{\Delta }\circ{\mathcal{P}}_{\varphi ^{\omega }}:{\mathcal{A}}^{\omega }\to{\mathcal{B}}$| is a normal conditional expectation. According to [ 25 , Theorem IX.4.2], we have |$\Delta _{({\mathcal{B}},\varphi ^{\omega })}^{it}=\Delta _{({\mathcal{A}}^{\omega },\varphi ^{\omega })}^{it}|_{L^{2}({\mathcal{B}},\varphi ^{\omega })}$|⁠ .

  Lemma 3.2. With the same notation as above, when |$n=\omega $|⁠ , define |$({\mathcal{A}}_{n},\varphi _{n})$| to be |$({\mathcal{A}}^{\omega },\varphi ^{\omega })$|⁠ . For |$n\in \mathbb{N}\cup \{\omega \}$|⁠ , let |$\Delta _{\varphi _{n}}$| be the modular operator of |$({\mathcal{A}}_{n},\varphi _{n})$|⁠ , |$e_{n}$| be the orthogonal projection from |$L^{2}({\mathcal{A}}_{n}, \varphi _{n})$| to |$L^{2}(M,\tau )$| and |$A_{\varphi } \in B(L^{2}(M))$| be the positive trace class operator associated to |${\varphi }$|⁠ . Note that |$B(L^{2}(M))= e_{n} B(L^{2}({\mathcal{A}}_{n},\varphi _{n}))e_{n}$|⁠ . Then for any |$n\in \mathbb{N}\cup \{\omega \}$| and |$t>0$|⁠ , within |$B(L^{2}(M))$|⁠ , we have $$\begin{align*} &(1+t)^{-1} \leq e_n(\Delta_{\varphi_n}+t)^{-1}e_n\leq (\frac{1}{2}A_{\varphi}+t)^{-1}.\end{align*}$$

Now let’s prove |$e_{n}(\Delta _{\varphi _{n}}+t)^{-1}e_{n}\leq (\frac{1}{2}A_{\varphi }+t)^{-1}$|⁠ . First, we prove that |$\Delta _{\varphi _{n}}\geq \frac{1}{2}T^{*}A_{\varphi } T $| on |$D(\Delta _{\varphi _{n}})$|⁠ , where |$T:L^{2}({\mathcal{A}}_{n},\varphi _{n})\to L^{2}(M,\tau )$| is defined by |$T(a \xi _{\varphi _{n}})=\mathcal{P}_{\varphi _{n}}(a)\hat{1} \ (a\in{\mathcal{A}}_{n})$| and |$\xi _{\varphi _{n}}$| is the cyclic vector of |$L^{2}({\mathcal{A}}_{n},\varphi _{n})$|⁠ .

When |$n=\omega $|⁠ , since |$\varphi ^{\omega }$| is |$\varphi $| -stationary, according to the proof of [ 10 , Lemma 5.12], we have |$\Delta _{\varphi _{n}}\geq T^{*}A_{\varphi } T\geq \frac{1}{2}T^{*}A_{\varphi } T$| on |$D(\Delta _{\varphi _{n}})$|⁠ .

Now we have |$\Delta _{\varphi _{n}}\geq \frac{1}{2}T^{*}A_{\varphi } T $| for |$n\in \mathbb{N}\cup \{\omega \}$|⁠ , which induces |$(\Delta _{\varphi _{n}}+t)^{-1}\leq (\frac{1}{2}T^{*}A_{\varphi }T+t)^{-1}$| within |$B(L^{2}({\mathcal{A}}_{n},\varphi _{n}))$|⁠ .

Now we are ready to prove the analogue of the fact that for the |$(\Gamma ,\mu )$| -Poisson boundary |$(B,\nu _{B})$|⁠ , |$h_{\mu }(B,\nu _{B})=h(\mu )$|⁠ .

  Theorem 3.3. With the same notation as above, assume that |$H(\varphi )<+\infty $|⁠ . Then we have $$\begin{align*} h_\varphi({\mathcal{B}}_\varphi,\zeta)=h_\varphi({\mathcal{A}}^\omega, \varphi^\omega)=\lim_\omega\limits h_\varphi({\mathcal{A}}_n, \varphi_n)=h(\varphi).\end{align*}$$

Now we have |$F_{n}$| converge to |$F_{\omega }$| pointwise when |$n\to \omega $| and they are uniformly dominated by |$G\in L^{1}((0,+\infty ))$|⁠ . But Lebesgue’s dominated convergence theorem doesn’t work for nets, we need further discussion to prove |$\int _{0}^{+\infty } F_{\omega }(t) \, \textrm{d} t=\lim _{\omega }\limits \int _{0}^{+\infty } F_{n}(t) \, \textrm{d} t$|⁠ .

Recall that in the classical theory, for a quasi-invariant |$\Gamma $| -space |$(X,\nu _{X})$|⁠ , the Radon-Nikodym factor of |$(X,\nu _{X})$| is a quasi-invariant |$\Gamma $| -space |$(X_{\textrm{RN}},\Sigma _{\textrm{RN}},\nu _{X_{\textrm{RN}}})$|⁠ , where |$\Sigma _{\textrm{RN}}$| is generated by |$\{\frac{\textrm{d} \gamma \nu _{X}}{\textrm{d}\nu _{X}}\}_{\gamma \in \Gamma }$|⁠ , |$X_{\textrm{RN}}$| is the topological model of |$\Sigma _{\textrm{RN}}$| and |$\nu _{X_{\textrm{RN}}}=\nu _{X}|_{\Sigma _{\textrm{RN}}}$|⁠ . The following defines a generalization of Radon-Nikodym factor.

Let |$(\mathcal{A},\nu )$| be a von Neumann algebra with normal faithful state such that |$(M,\tau )\subset (\mathcal{A},\nu )$|⁠ . We define the noncommutative Radon-Nikodym factor of |$(\mathcal{A},\nu )$| to be |$(\mathcal{A},\nu )_{\textrm{RN}}:=(\mathcal{A}_{\textrm{RN}},\nu |_{\mathcal{A}_{\textrm{RN}}})$|⁠ , where |$\mathcal{A}_{\textrm{RN}}$| is the von Neumann subalgebra of |$\mathcal{A}$| generated by |$\{\sigma ^{\nu }_{t}(x)|x\in M,t\in \mathbb{R}\}$| and |$\{\sigma ^{\nu }_{t}(x)\}_{t\in \mathbb{R}}$| is the modular automorphism group of |$(\mathcal{A},\nu )$|⁠ .

The following example shows that the noncommutative Radon-Nikodym factor is a generalization of the classical one.

  Example 3.5. Let |$\Gamma $| be a countable discrete group and |$(X,\nu _{X})$| be a quasi-invariant |$\Gamma $| -space. Let |$(L(\Gamma ),\tau )$| be the group von Neumann algebra of |$\Gamma $| with canonical trace. Define the |$\tau $| -hyperstate |$\nu $| on |$L(\Gamma \curvearrowright X)$| by |$\nu (\sum _{\gamma \in \Gamma }a_{\gamma } u_{\gamma })=\int a_{e} \textrm{d} \nu _{X}$|⁠ . Then following [ 25 , Theorem X.1.17], the modular operator |$\Delta _{\nu }$| is $$\begin{align*} &\Delta_\nu(\gamma,x)=\frac{\textrm{d} \gamma^{-1}\nu_X}{\textrm{d}\nu_X}(x).\end{align*}$$ And the modular automorphism group satisfies $$\begin{align*} &\sigma_t^\nu(u_\gamma)=\left(\frac{\textrm{d} \gamma^{-1}\nu_X}{\textrm{d}\nu_X}\right)^{it}u_\gamma.\end{align*}$$ Hence, the |$(L(\Gamma ),\tau )$| -Radon-Nikodym factor of |$(L(\Gamma \curvearrowright X),\nu )$| is generated by |$\left \{\left (\frac{\textrm{d} \gamma ^{-1}\nu _{X}}{\textrm{d}\nu _{X}}\right )^{it}u_{\gamma }\right \}_{\gamma \in \Gamma ,t\in \mathbb{R}}$|⁠ , which is exactly the group measure space construction |$L(\Gamma \curvearrowright X_{\textrm{RN}})$|⁠ .

Let |$\varphi \in \mathcal{S}_{\tau }(B(L^{2}(M,\tau ))$| be a normal regular strongly generating hyperstate. Let |$(\mathcal{A},\nu )$| be as in Definition 3.4 such that |$h_{\varphi }(\mathcal{A},\nu )<+\infty $|⁠ . Let |$(\mathcal{A}_{0},\nu _{0})$| be a von Neumann subalgebra of |$(\mathcal{A},\nu )$| such that |$(M,\tau )\subset (\mathcal{A}_{0},\nu _{0})$|⁠ . Then |$h_{\varphi }(\mathcal{A}_{0},\nu _{0})=h_{\varphi }(\mathcal{A},\nu )$| if and only if |$(\mathcal{A}_{0},\nu _{0})_{\textrm{RN}}=(\mathcal{A},\nu )_{\textrm{RN}}$|⁠ .

Denote by |$\Delta _{0}$| and |$\Delta $| the modular operators of |$(\mathcal{A}_{0},\nu _{0})$| and |$(\mathcal{A},\nu )$|⁠ ; denote by |$\{\sigma _{t}^{0}\}_{t\in \mathbb{R}}$| and |$\{\sigma _{t}\}_{t\in \mathbb{R}}$| the modular automorphism groups of |$(\mathcal{A}_{0},\nu _{0}) $| and |$(\mathcal{A},\nu )$|⁠ . Then |$\Delta _{0}=e_{0}\Delta e_{0}$|⁠ , where |$e_{0}$| is the orthogonal projection from |$L^{2}(\mathcal{A},\nu )$| onto |$L^{2}(\mathcal{A}_{0},\nu _{0})$|⁠ .

Assume that |$(\mathcal{A}_{0},\nu _{0})_{\textrm{RN}}=(\mathcal{A},\nu )_{\textrm{RN}}$|⁠ . Denote by |$\{\sigma _{t}^{\textrm{RN}}\}_{t\in \mathbb{R}}$| the modular automorphism group of |$(\mathcal{A},\nu )_{\textrm{RN}}$|⁠ . Since |$\mathcal{A}_{\textrm{RN}}$| is invariant under the action of |$\{\sigma _{t}\}_{t\in \mathbb{R}}$|⁠ , by Tomita-Takesaki theory [ 25 , Theorem IX.4.2 and Theorem IX.4.18], we have |$\sigma ^{\textrm{RN}}_{t}=\sigma _{t}|_{\mathcal{A}_{\textrm{RN}}}$| for any |$t\in \mathbb{R}$|⁠ . By ( 3.4 ), we have |$h_{\varphi }(\mathcal{A}_{\textrm{RN}},\nu |_{\mathcal{A}_{\textrm{RN}}})=h_{\varphi }(\mathcal{A},\nu )$|⁠ . Since |$(\mathcal{A}_{0})_{\textrm{RN}}=\mathcal{A}_{\textrm{RN}}$|⁠ , for the same reason, |$h_{\varphi }(\mathcal{A}_{\textrm{RN}},\nu |_{\mathcal{A}_{\textrm{RN}}})=h_{\varphi }(\mathcal{A}_{0},\nu _{0})$|⁠ . Therefore, |$h_{\varphi }(\mathcal{A}_{0},\nu _{0})=h_{\varphi }(\mathcal{A},\nu )$|⁠ .

Assume that |$h_{\varphi }(\mathcal{A}_{0},\nu _{0})=h_{\varphi }(\mathcal{A},\nu )$|⁠ . First, let’s prove that |$\sigma _{t}|_{M}=\sigma _{t}^{0}|_{M}$|⁠ . The following proof is inspired by the proof of [ 10 , Lemma 5.14].

Since |$\{z_{k}\}$| generates |$M$|⁠ , we have |$\sigma _{t}|_{M}= \sigma _{t}^{0}|_{M}$| for any |$t\in \mathbb{R}$|⁠ . Therefore, according to Definition 3.4 , we have |$(\mathcal{A}_{0},\nu _{0})_{\textrm{RN}}=(\mathcal{A},\nu )_{\textrm{RN}}$|⁠ .

Recall that in classical theory, a |$(\Gamma ,\mu )$| -boundary is a |$(\Gamma ,\mu )$| -space |$(B_{0},\nu _{0})$| such that there exists a factor map from the |$\mu $| -Poisson boundary |$(B,\nu )$| onto |$(B_{0},\nu _{0})$|⁠ . The following defines a generalization of |$(\Gamma ,\mu )$| -boundary.

Up to state preserving isomorphisms, a |$\varphi $| -boundary |$({\mathcal{B}}_{0},\zeta _{0})$| is a von Neumann subalgebra of |$(B_{\varphi },\zeta )$| satisfying |$(M,\tau )\subset ({\mathcal{B}}_{0},\zeta _{0})\subset (\mathcal{B}_{\varphi },\zeta )$|⁠ .

The following theorem is the analogue of the fact that the Poisson boundary is the unique boundary with maximal entropy.

Let |$\varphi \in \mathcal{S}_{\tau }(B(L^{2}(M,\tau ))$| be a normal regular strongly generating hyperstate such that |$H(\varphi )<+\infty $|⁠ . Let |$({\mathcal{B}}_{\varphi },\zeta )$| be the Poisson boundary. Then for any |$\varphi $| -boundary |$({\mathcal{B}}_{0},\zeta _{0}) \subset (\mathcal{B}_{\varphi },\zeta )$|⁠ , |$h_{\varphi }({\mathcal{B}}_{0},\zeta _{0})=h_{\varphi }({\mathcal{B}}_{\varphi },\zeta )$| if and only if |$({\mathcal{B}}_{0},\zeta _{0})=({\mathcal{B}}_{\varphi },\zeta )$|⁠ .

Assume that |$h_{\varphi }({\mathcal{B}}_{0},\zeta _{0})=h_{\varphi }({\mathcal{B}}_{\varphi },\zeta )$|⁠ . Then by Theorem 3.6 , we have |$({\mathcal{B}}_{0},\zeta _{0})_{\textrm{RN}}=({\mathcal{B}}_{\varphi },\zeta )_{\textrm{RN}}$|⁠ .

Take |$({\mathcal{B}}_{0},\zeta _{0})$| to be |$(M,\tau )$| in Theorem 3.8 , we have the following corollary, which is an analogue of the fact that the |$(\Gamma ,\mu )$| -Poisson boundary is trivial if and only if |$h(\mu )=0$|⁠ . See also [ 10 , Corollary 5.16] for the “if” direction.

Let |$\varphi \in \mathcal{S}_{\tau }(B(L^{2}(M,\tau ))$| be a normal regular strongly generating hyperstate such that |$H(\varphi )<+\infty $|⁠ . Then |${\mathcal{B}}_{\varphi }=M$| if and only if |$h(\varphi )=0$|⁠ .

According to [ 18 ], a countable discrete group |$\Gamma $| is amenable if and only if there exists an admissible measure |$\mu \in \textrm{Prob}(\Gamma )$| (i.e., |$\bigcup _{n\in \mathbb{N}}(\textrm{supp} \, \mu )^{n}=\Gamma$|⁠ ) such that the Poisson boundary of |$(\Gamma ,\mu )$| is trivial. In this section, we will prove an analogue for tracial von Neumann algebras.

  Definition 4.1. For |$\mu \in \textrm{Prob}(\mathcal{U}(M))$|⁠ , we say that |$\mu $| is an atomic measure if $$\begin{align*} &\mu=\sum_{\mu(u)>0} \mu(u)\cdot \delta_{u}. \end{align*}$$ In this case, we simply denote |$\{u\in \mathcal{U}(M)\mid \mu (u)>0\}$| by |$\textrm{supp}_{\textrm{ess}}\, \mu $|⁠ , which is a countable |$\mu $| -conull (hence dense) subset of |$\textrm{supp} \, \mu$|⁠ .

  Definition 4.2. For (atomic) |$\mu \in \textrm{Prob}(\mathcal{U}(M))$|⁠ , define the (normal) hyperstate |$\varphi _{\mu } \in \mathcal{S}_{\tau }(B(L^{2}(M))$| to be $$\begin{align*} &\varphi_\mu(T)=\int_{\mathcal{U}(M)} \langle T u^*\hat{1},u^*\hat{1}\rangle \textrm{d} \mu (u), \ T\in B(L^2(M)).\end{align*}$$

According to [ 10 , Theorem 2.10], to find a normal hyperstate with trivial boundary, we only need to find a nice hyperstate |$\varphi $|⁠ , such that |$\varphi ^{*n}$| tends to be |$M$| -central when |$n \to \infty $|⁠ . Before constructing the |$\varphi $| we need, we also need the following lemma to make sure such |$\varphi $| can be obtained through a measure on |$\mathcal{U}(M)$|⁠ .

  Lemma 4.3. Let |$(M, \tau )$| be an amenable tracial von Neumann algebra with separable predual and $$\begin{align*} &\mathcal{S}_0=\{\varphi_\mu \in \mathcal{S}_\tau(B(L^2(M))\mid \mu \in \mathrm{Prob}(\mathcal{U}(M)) \mbox{ is of finite support}\}.\end{align*}$$ Then there exists a hypertrace in the weak* closure of |$\mathcal{S}_{0}$|⁠ .

So for any |$\epsilon>0$|⁠ , there exist a |$\varphi _{\mu _{n}^{\prime}}\in \mathcal{S}_{0}$| such that |$\|\varphi _{\mu _{n}}-\varphi _{\mu _{n}^{\prime}}\|\leq 2 \epsilon $|⁠ . Hence |$\varphi _{\mu _{n}}$| is in the weak* closure of |$\mathcal{S}_{0}$|⁠ , and so is |$\varphi _{0}$|

The following theorem is inspired by [ 18 , Theorem 4.3].

  Theorem 4.4. Let |$(M, \tau )$| be an amenable tracial von Neumann algebra with separable predual. Then there exists an atomic measure |$\mu \in \textrm{Prob}(\mathcal{U}(M))$|⁠ , such that |$\bigcup _{n\in \mathbb{N}}(\textrm{supp}_{\textrm{ess}}\,\mu )^{n}$| is weakly dense in |$\mathcal{U}(M)$| and the normal hyperstate |$\varphi =\varphi _{\mu }$| satisfies |$\varphi \geq \frac{1}{2}\langle \, \cdot \, \hat{1}, \hat{1} \rangle $|⁠ , and for any |$x \in M$|⁠ , $$\begin{align*} &\lim_{n \to \infty}\limits \|\varphi^{*n} (x \, \cdot \, )- \varphi^{*n} ( \, \cdot \, x) \|=0,\end{align*}$$ which is equivalent to $$\begin{align*} &\lim_{n \to \infty}\limits \|xA_n- A_n x \|_{1,Tr}=0,\end{align*}$$ where |$A_{n}$| is the trace-class operator associated to |$\varphi ^{*n}$|⁠ .

We construct a sequence |$\{\varphi _{n}\}\subset \mathcal{S}_{0}$| as follows: Let |$\{1\}=K_{1}\subset K_{2} \subset ...$| be an increasing sequence of finite subsets of |$(M)_{1}$|⁠ , such that |$K:=\bigcup _{n=1}^{\infty }K_{n}$| is strongly dense in |$(M)_{1}$|⁠ . Let |$\{1\}=U_{1}\subset U_{2} \subset ...$| be an increasing sequence of finite subsets of |$\mathcal{U}(M)$|⁠ , such that |$U:=\bigcup _{n=1}^{\infty }U_{n}$| is weakly dense in |$\mathcal{U}(M)$|⁠ . Take |$\frac{1}{2}=t_{1}>t_{2}>...$| to be a decreasing series of positive real number, such that |$\sum _{n=1}^{\infty }\limits t_{n}=1$|⁠ .

First of all, |$U=\bigcup _{n} U_{n} \subset \bigcup _{n} \textrm{supp} \, \mu _{n}=\textrm{supp}_{\textrm{ess}}\, \mu$|⁠ . Since |$U$| is weakly dense in |$\textrm{Prob}(\mathcal{U}(M))$|⁠ , we have that |$\bigcup _{n\in \mathbb{N}}(\textrm{supp}_{\textrm{ess}}\,\mu )^{n}$| is weakly dense in |$\textrm{Prob}(\mathcal{U}(M))$|⁠ .

For a fixed |$x \in K$|⁠ , there exists a |$n_{0} \in \mathbb{N}$| such that |$x\in K_{n}$| for any |$n\geq n_{0}$|⁠ . For any |$n\geq n_{0}$|⁠ , we consider |$\|\varphi ^{*m_{n}} (x \, \cdot \, )- \varphi ^{*m_{n}} ( \, \cdot \, x) \|$|⁠ .

So |$\varphi $| satisfies all the conditions we want.

Now we can prove the following main theorem of this section.

(i) There exists an atomic measure |$\mu \in \textrm{Prob}(\mathcal{U}(M))$|⁠ , such that |$\bigcup _{n\in \mathbb{N}}(\textrm{supp}_{\textrm{ess}}\,\mu )^{n}$| is weakly dense in |$\textrm{Prob}(\mathcal{U}(M))$| and the normal hyperstate |$\varphi =\varphi _{\mu }$| has trivial Poisson boundary;

(ii) There exists a hyperstate |$\varphi \in \mathcal{S}_{\tau }(B(L^{2}(M)))$| with trivial Poisson boundary;

(iii) |$(M, \tau )$| is amenable.

(i) |$\Rightarrow $| (ii) is clear.

(ii) |$\Rightarrow $| (iii): Assume that the hyperstate |$\varphi $| has trivial Poisson boundary. Then by the injectivity of Poisson boundaries [ 10 , Proposition 2.4], |$M=\textrm{Har}({\mathcal{P}}_{\varphi })$| is injective (amenable).

(iii) |$\Rightarrow $| (i) is a direct corollary of [ 10 , Theorem 2.10] and Theorem 4.4 .

Recall that a locally compact group |$G$| is called Choquet-Deny if for any admissible |$\mu \in \textrm{Prob}(G)$|⁠ , the Poisson boundary of |$\mu $| is trivial. In this section, we will define the Choquet-Deny property for tracial von Neumann algebras and prove the equivalence between Choquet-Deny property and type |$\textrm{I}$|⁠ , which is inspired by the fact that a finitely generated countable discrete group |$\Gamma $| is Choquet-Deny if and only if |$\Gamma $| is virtually nilpotent (see [ 17 ] and [ 12 ]), and |$\Gamma $| is virtually abelian if and only if |$L(\Gamma )$| is of type |$\textrm{I}$| (see [ 26 ]).

Let |$(M, \tau )$| be a tracial von Neumann algebra with separable predual. We say that |$(M,\tau )$| is Choquet-Deny if for any normal regular strongly generating hyperstate |$\varphi \in \mathcal{S}_{\tau }(B(L^{2}(M)))$|⁠ , one has |${\mathcal{B}}_{\varphi }=M$|⁠ .

We only consider weakly separable |$(M,\tau )$|⁠ , because it is the only way that |$M$| admits a normal strongly generating hyperstate.

According to [ 10 , Corollary 3.2], any abelian |$(M,\tau )$| with separable predual is Choquet-Deny. Let’s see more examples of Choquet-Deny and non-Choquet-Deny von Neumann algebras.

For |$n\in \mathbb{N}_{+}$|⁠ , the type |$\textrm{I}_{n}$| factor |$(M_{n}(\mathbb{C}), \frac{1}{n}\textrm{Tr})$| is Choquet-Deny.

Therefore, |$\textrm{Har}(\mathcal{P}_{\varphi })=M_{n}(\mathbb{C})$|⁠ . So |$M_{n}(\mathbb{C})$| is Choquet-Deny.

The unique amenable |$\textrm{II}_{1}$| factor |$L(S_{\infty })$| is not Choquet-Deny.

According to [ 17 ] and [ 12 ], a countable discrete group is Choquet-Deny if and only if it has no |$\textrm{ICC}$| quotients. But |$S_{\infty }$| itself is |$\textrm{ICC}$|⁠ , so |$S_{\infty }$| is not Choquet-Deny.

Therefore, there exists an admissible |$\mu \in \textrm{Prob}(S_{\infty })$|⁠ , that is, |$\textrm{supp} \, \mu$| generates |$S_{\infty }$| as semigroup, such that the |$\mu $| -Poisson boundary |$(B,\nu _{B})$| is not trivial.

Also view |$\mu $| as an element in |$\textrm{Prob}(\mathcal{U}(L(S_{\infty })))$|⁠ . Then |$\varphi _{\mu }$| is a normal regular strongly generating hyperstate. According to [ 16 , Theorm 4.1], the |$\varphi _{\mu }$| -Poisson boundary |${\mathcal{B}}_{\varphi _{\mu }}$| is |$L(S_{\infty }\curvearrowright B)$|⁠ .

Since |$(B,\nu _{B})$| is not trivial, |$L(S_{\infty }\curvearrowright B)\not =L(S_{\infty })$|⁠ . Hence, |$L(S_{\infty })$| is not Choquet-Deny.

Now we do some preparations for the proof of Choquet-Deny-type |$\textrm{I}$| equivalence.

Let |$(M,\tau )$| be a tracial von Neumann algebra, let |$\varphi $| be a normal regular strongly generating hyperstate. Then |$\textrm{Har}(\mathcal{P}_{\varphi })\subset Z(M)^{\prime}$|⁠ .

Therefore, for any |$u\in \mathcal{U}(Z(M))$| and |$x \in \mathcal{B}_{\varphi }$|⁠ , we have |$uxu^{*}=x$|⁠ . By Poisson transform, which is |$M$| -bimodular, we know that for any |$u\in \mathcal{U}(Z(M))$| and |$T \in \textrm{Har}(\mathcal{P}_{\varphi })$|⁠ , |$uTu^{*}=T$|⁠ . Therefore, |$\textrm{Har}(\mathcal{P}_{\varphi })\subset Z(M)^{\prime}$|⁠ .

For the next theorem, we refer to [ 24 , Chapter IV] for details on direct integral theory.

  Theorem 5.5. Let |$(M,\tau )$| be a tracial von Neumann algebra with direct integral decomposition on Borel probability measure space |$(X,\mu _{X})$|⁠ : $$\begin{align*} &(M,\tau)=\int_X^\oplus (M_x,\tau_x)\textrm{d}\mu_X(x).\end{align*}$$ Let |$\varphi $| be a normal regular strongly generating hyperstate on |$B(L^{2}(M))$|⁠ . By direct integral theory, |$\varphi $| admits the following decomposition while restricting to the decomposable operators: $$\begin{align*} &\varphi|_{\int_X^\oplus B(L^2(M_x))\textrm{d}\mu_X(x)}=\int_X^\oplus \varphi_x\textrm{d}\mu_X(x),\end{align*}$$ where for |$x\in X$|⁠ , |$\varphi _{x}$| is a normal |$\tau _{x}$| -hyperstate on |$B(L^{2}(M_{x}))$|⁠ . Then we have $$\begin{align*} &\textrm{Har}(\mathcal{P}_\varphi)=\int_X^\oplus \textrm{Har}(\mathcal{P}_{\varphi_x})\textrm{d}\mu_X(x)\end{align*}$$ and $$\begin{align*} &\mathcal{B}_\varphi=\int_X^\oplus \mathcal{B}_{\varphi_x}\textrm{d}\mu_X(x).\end{align*}$$

Hence, for |$T=\int _{X}^{\oplus } T_{x} \in \int _{X}^{\oplus } B(L^{2}(M_{x}))\textrm{d}\mu _{X}(x)$|⁠ , |$\mathcal{P}_{\varphi }(T)=T$| if and only if |$\mathcal{P}_{\varphi _{x}}(T_{x})=T_{x}$| for |$\mu _{X}$| -a.e. |$x\in X$|⁠ .

Therefore, |$\textrm{Har}(\mathcal{P}_{\varphi })=\int _{X}^{\oplus } \textrm{Har}(\mathcal{P}_{\varphi _{x}})\textrm{d}\mu _{X}(x)$| and |$\mathcal{B}_{\varphi }=\int _{X}^{\oplus } \mathcal{B}_{\varphi _{x}}\textrm{d}\mu _{X}(x)$|⁠ .

Theorem 5.5 shows that the noncommutative Poisson boundary is somehow additive, indicating that in some cases, the study of noncommutative Poisson boundaries can be simplified by focusing the Poisson boundaries of tracial factors.

Now we are ready to prove the following main theorem.

(i) |$M$| is Choquet-Deny;

(ii) For any atomic measure |$\mu \in \textrm{Prob}(\mathcal{U}(M))$| such that |$\textrm{supp}_{\textrm{ess}}\, \mu $| generates |$M$| as a weakly closed unital subalgebra, |$\varphi _{\mu }$| has a trivial Poisson boundary;

(iii) |$M$| is of type |$\textrm{I}$|⁠ .

(ii) |$\Rightarrow $| (iii): Assume that |$M$| satisfies (ii). First, let’s prove that for |$\mu _{X}$| -a.e. |$x\in X$|⁠ , |$(M_{x},\tau _{x})$| is amenable.

Take an atomic measure |$\mu _{1}\in \textrm{Prob}(\mathcal{U}(M_{1}))$| such that |$\textrm{supp}_{\textrm{ess}}\, \mu _{1}$| generates |$M_{1}$| as a weakly closed unital subalgebra. Take an atomic measure |$\mu _{2}\in \textrm{Prob}(\mathcal{U}(L^{\infty }(X_{2})))$| such that |$\textrm{supp}_{\textrm{ess}}\, \mu _{2}$| generates |$L^{\infty }(X_{2})$| as a weakly closed unital subalgebra.

Therefore, |$\mu _{X}(X_{2})=0$| and |$M=M_{1}$| is of type |$\textrm{I}$|⁠ .

(iii) |$\Rightarrow $| (i): Assume that |$M$| is of type |$\textrm{I}$|⁠ . Then for |$\mu _{X}$| -a.e. |$x\in X$|⁠ , |$(M_{x},\tau _{x})$| is a finite type |$\textrm{I}$| factor. Hence, after replacing |$X$| with a conull subset, we may assume that for every |$x\in X$|⁠ , |$(M_{x},\tau _{x})=(M_{n_{x}}(\mathbb{C}),\frac{1}{n_{x}}Tr)$| for some |$n_{x}\in \mathbb{N}_{+}$|⁠ .

Fix a normal regular strongly generating hyperstate |$\varphi $|⁠ , we want to prove that |$\textrm{Har}(\mathcal{P}_{\varphi })=M$|⁠ .

Therefore, |$\textrm{Har}(\mathcal{P}_{\varphi })=M$| for any normal regular strongly generating |$\varphi $|⁠ . Hence, |$M$| is Choquet-Deny.

As mentioned in the introduction part, for a countable discrete group |$\Gamma $|⁠ , the Choquet-Deny property of |$L(\Gamma )$| is not equivalent to, but strictly stronger than the Choquet-Deny property of |$\Gamma $|⁠ . Actually, there exists no such property (CD) for tracial von Neumann algebras satisfying that |$\Gamma $| is Choquet-Deny if and only if |$L(\Gamma )$| has the property (CD). This is because a Choquet-Deny group and a non-Choquet-Deny group can have a same group von Neumann algebra. Consider two groups: the Heisenberg group |$H_{3}(\mathbb{Z})$| and |$\mathbb{Z}\times S_{\infty }$|⁠ . Since |$H_{3}(\mathbb{Z})$| is amenable and of type |$\textrm{II}_{1}$| [ 2 , Example 7.D.5 and Theorem 8.F.4], we have |$L(H_{3}(\mathbb{Z}))=Z(L(H_{3}(\mathbb{Z})))\bar{\otimes }\mathcal{R}$|⁠ , where |$\mathcal{R}$| is the unique AFD |$\textrm{II}_{1}$| factor. Since |$Z(H_{3}(\mathbb{Z}))\cong \mathbb{Z}$|⁠ , |$Z(L(H_{3}(\mathbb{Z})))$| must be diffuse for having a diffuse subalgebra |$L(\mathbb{Z})$|⁠ . Therefore, |$Z(L(H_{3}(\mathbb{Z})))\cong L^{\infty }([0,1])$| and |$L(H_{3}(\mathbb{Z}))=L^{\infty }([0,1])\bar{\otimes }\mathcal{R}$|⁠ . We also have |$L(\mathbb{Z}\times S_{\infty })=L(\mathbb{Z})\bar{\otimes } L(S_{\infty })=L^{\infty }([0,1])\bar{\otimes }\mathcal{R}$|⁠ . Hence these two groups has the same group von Neumann algebra. However, following [ 17 ] and [ 12 ], since |$H_{3}(\mathbb{Z})$| is finitely generated and nilpotent, it is Choquet-Deny; while |$\mathbb{Z}\times S_{\infty }$| has the ICC quotient |$S_{\infty }$|⁠ , hence it is non-Choquet-Deny. Therefore, there exists no property for tracial von Neumann algebras that can perfectly match with the Choquet-Deny property of groups.

This paper was completed under the supervision of Professor Cyril Houdayer. Theorem 4.5 and Theorem 5.6 were initially introduced by Professor Houdayer as conjectures. The author would like to express sincere gratitude to Professor Cyril Houdayer for numerous insightful discussions and valuable comments on this paper. The author would also like to thank Professor Amine Marrakchi for useful comments regarding this paper.

Communicated by Prof. Dan-Virgil Voiculescu

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International planet hunters unveil massive catalog of strange worlds

NASA TESS-Keck Survey details mass, density of 126 planets

While thousands of planets have been discovered around other stars, relatively little is known about them. A NASA catalog featuring 126 exotic, newly discovered worlds includes detailed measurements that allow for comparisons with our own solar system. 

The catalog details a fascinating mix of planet types beyond our solar system, from rare worlds with extreme environments to ones that could possibly support life. 

The planets were analyzed by a large, international team of scientists using NASA’s Transiting Exoplanet Survey Satellite (TESS) in collaboration with the W.M. Keck Observatory on Maunakea, Hawai’i. They are described in today’s edition of The Astrophysical Journal Supplement. 

“Relatively few of the previously known exoplanets have a measurement of both the mass and the radius. The combination of these measurements tell us what the planets could be made of and how they formed,” said Stephen Kane, UC Riverside astrophysicist and principal investigator of the TESS-Keck Survey. 

“With this information, we can begin to answer questions about where our solar system fits in to the grand tapestry of other planetary systems,” Kane said. 

The research team spent three years developing the catalog. They analyzed more than 13,000 radial velocity (RV) measurements to calculate the masses of 120 confirmed planets, plus six candidate planets, spread out over the northern sky. 

Though the planets themselves aren’t visible, they do have a visible effect. As they orbit, the planets tug on their host stars, causing them to “wobble.” When the star moves toward a telescope, its visible light turns slightly bluer; when it moves away from us, the light shifts slightly redder. 

This is much like how sound behaves. Due to the Doppler effect, a fire truck’s siren gets higher-pitched as it travels closer and sounds lower-pitched as it drives farther away.

“These RV measurements let astronomers detect and learn the properties of these exoplanetary systems. When we see a star wobbling regularly back and forth, we can infer the presence of an orbiting planet and measure the planet’s mass,” said Ian Crossfield, University of Kansas astrophysicist and catalog co-author. 

Several planets in the TESS-Keck Survey stand out as touchstones for deepening astronomers’ understanding of the diverse ways planets form and evolve. 

A related survey paper authored by UCR graduate student Michelle Hill announces the discovery of two new planets orbiting a star like our sun. The first is a “sub-Saturn” planet with a mass and radius that are between those of Neptune and Saturn. 

“There is ongoing debate about whether sub-Saturn planets are truly rare, or if we are just bad at finding planets like these,” Hill said. “So, this planet, TOI-1386 b, is an important addition to this demographic of planets.”

TOI-1386 b only takes 26 days to orbit its star. Meanwhile its neighbor, a planet with a mass close to that of Saturn, takes 227-days to orbit the same star. 

Another survey paper authored by UCR graduate student Daria Pidhorodetska describes a planet about half the size of Neptune that takes a mere 19 days to orbit its star, which is much like our Sun. 

“Planets smaller than Neptune but larger than Earth are the most prevalent worlds in our galaxy, yet they are absent from our own Solar System. Each time a new one is discovered, we are reminded of how diverse our Universe is, and that our existence in the cosmos may be more unique than we can understand,” Pidhorodetska said. 

There are a lot of stars that are not similar to our sun. If scientists want to make apt comparisons between our world and others, they need to find stars of a similar age, size, and mass. “Then we can do apples-to-apples comparisons,” Kane said. “That’s the exciting part of the papers produced by Michelle and Daria, because they allow for this.”

Planets with even more extreme, ultra-short orbits around stars unlike our sun are also detailed in the catalog. One is so close to its orange dwarf star it completes orbit in less than 12 hours.   

“TOI-1798 c orbits its star so quickly that one year on this planet lasts less than half a day on Earth. Because of their proximity to their host stars, planets like this one are also ultra hot — receiving more than 3,000 times the radiation that Earth receives from the sun,” said Alex Polanski, University of Kansas physics and astronomy graduate student and lead author of the catalog paper. 

“Existing in this extreme environment means that this planet has likely lost any atmosphere that it initially formed,” Polanski said. 

Ultimately, this new catalog represents a major contribution both to NASA’s TESS mission, and toward answering the question of whether other planets are capable of hosting life as we know it. 

“Are we unusual? The jury is still out on that one, but our new mass catalog represents a major step toward answering that question,” Kane said.  

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McElroy-Heltzel’s research among International Journal of Psychology’s top cited articles

Stacey McElroy-Heltzel, University of Iowa assistant professor of counseling psychology and researcher in the field of psychology, has garnered recognition for her work on the impact of COVID-19 on mental health. Her article titled "Resource loss and mental health during COVID-19: Psychosocial protective factors among US older adults and those with chronic disease," published in the International Journal of Psychology , has been listed among the top 10 most-cited papers from January 1, 2022 through December 31, 2023.

McElroy-Heltzel's research delves into the intricate relationship between resource loss and mental health outcomes during the COVID-19 pandemic, with a particular focus on vulnerable populations such as older adults and individuals with chronic diseases. In the wake of unprecedented challenges posed by the pandemic, understanding the psychosocial protective factors that mitigate adverse mental health effects is of paramount importance.

“During the pandemic, the prevailing narrative was that we were a nation experiencing multiple crises, including a mental health crisis. While that was true in many ways, I think it's important that that's not the only narrative out there,” McElroy-Heltzel says. “We have a tendency to over pathologize normal responses to stressful events, and that's not a good message to send. It’s normal and appropriate to experience distress when bad things happen. While some people will develop significant mental health symptoms, many people experience both stress and resilience by drawing upon social support or having an optimistic outlook. It’s possible to recognize that while things are difficult at the moment, there is hope that things will get better.” 

The study conducted by McElroy-Heltzel and her team sheds light on the resilience exhibited by certain segments of the population amidst adversity. By examining factors such as social support, coping strategies, and access to essential resources, the research elucidates some protective factors that allow individuals to maintain psychological well-being in the face of resource constraints and heightened stressors.

The recognition of McElroy-Heltzel's work as one of the top 10 most-cited papers underscores its significance and impact within the scientific community. Her findings not only contribute to the academic discourse on mental health resilience but also have practical implications for policymakers, health care professionals, and community stakeholders striving to support vulnerable populations during public health crises.

In an era characterized by unprecedented global challenges, McElroy-Heltzel's research serves as a positive outlook, offering insights into the resilience of human spirit and the potential for collective action to foster positive mental health outcomes amidst adversity. As the world continues to navigate the complexities of the COVID-19 pandemic and its aftermath, the lessons gleaned from McElroy-Heltzel's work remain invaluable in shaping effective interventions and strategies to promote mental well-being for all.

“Many people find it difficult to reach out for social support during stressful times, especially if we believe everyone around us is distressed. We may fear being a burden. Yet, research has indicated that social support is one of the most consistent and robust correlates of psychological health. Therefore, we want to acknowledge that while many people are struggling, through connection, we can build resilience,” McElroy-Heltzel says. “Our findings support this idea. In this article, we were able to show how things like social support, resilience, and optimism mitigated some of the impacts of the pandemic related to factors on peoples’ mental distress, even for vulnerable populations.”

For McElroy-Heltzel, this recognition serves as a testament to the importance of interdisciplinary collaboration and rigorous scientific inquiry in addressing pressing societal issues. Moving forward, she remains committed to advancing research that not only informs policy and practice but also empowers individuals and communities to thrive in the face of adversity.

COE faculty recognized for excellence in research

Neal, Hoffman receive Collegiate Teaching Awards

Brown honored with the Michael J. Brody Award for Faculty Excellence in Service

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New research initiative tackles pressing global development issues

By alison fromme cornell sc johnson college of business.

Fundamental challenges in food insecurity, poverty, agriculture, health, education and markets form the focus of Collaboration for International Development Economics Research (CIDER) , a new initiative launched by the Office of the Provost, the SC Johnson College of Business, the College of Agriculture and Life Sciences, the College of Arts and Sciences and the Cornell Jeb E. Brooks School of Policy.

“CIDER builds on a long history of Cornell research and engagement in the economies of nations, particularly developing and emerging economies challenged by global economic forces,” said Provost Michael I. Kotlikoff. “This initiative expands the interdisciplinary focus of these efforts, bringing economists, social scientists, policy experts and agricultural experts together to pursue solutions to some of our most difficult global challenges.”

Hosted by the SC Johnson College, CIDER unites 24 faculty across campus and the world, along with students, staff, researchers and external partners, to create and share knowledge. CIDER’s activities will encompass research, workshops, seminars, internships, career mentoring and continuing-education coursework.

“We’re delighted to embark on this new collaborative effort in development economics,” said Andrew Karolyi, the Charles Field Knight Dean of the Cornell SC Johnson College of Business. “CIDER taps into existing expertise and a grand legacy of intellectual leadership at Cornell going back decades. I can’t wait to see the tangible impact CIDER makes on campus and around the world.”

CIDER’s inaugural faculty director is Chris Barrett , the Stephen B. and Janice G. Ashley Professor of Applied Economics and Management in the Charles H. Dyson School of Applied Economics and Management and professor in the Brooks School.

“We expect CIDER will further reinforce Cornell's already formidable reputation in this space,” Barrett said. The university’s impact in development economics was established over many decades and reinforced when standard measures of poverty and food insecurity were developed here in the 1980s. Now, CIDER provides a forum to collaborate on large-scale projects, advance policy-relevant research and train early career scholars and practitioners.

Through a workshop hosted by CIDER on May 10, the center is already encouraging new collaborations in East African dryland drought research, risk management and policy.

The World Bank, the African Development Bank, private organizations and multiple East African national governments are currently investing nearly $1 billion in the region to address drought, Barrett said.

“The efficacy and the impact of these massive investments can be directly influenced by research findings,” he said. “Indeed, research by Cornell and partners going back to the late 2000s underpins the initiative. We’re now working to produce timely policy-relevant, clearly communicated and rigorous research that can inform that effort.”

Among other presentations at the workshop, Karlijn Morsink , Utrecht University economist and CIDER-affiliated adjunct associate professor at the Dyson School, discussed her work leading the evaluation of World Bank programs in the region and share opportunities for Cornell faculty and student involvement.

“This workshop and the collaborations it represents offer just one example,” Barrett said. “We look to scale this type of effort across a range of different domains.”

CIDER will also support early career mentoring through two formal programs. Structural Transformation of Agriculture and Rural Spaces (STARS) , an existing Cornell program, previously paired early career researchers who earned degrees in Africa with mentors at Cornell and affiliated institutions. Now under CIDER’s umbrella, STARS is open to scholars across all low- and lower-middle-income countries.

Additionally, a predoctoral program for scholars who have not yet earned advanced degrees will provide one to three years of research experience and professional development training with core faculty. One predoctoral fellow already began work in January, and three more arrive this summer.

Building professional networks, increasing research capacity, disseminating best practices in the field and shaping early career researchers for the next generation are at the heart of CIDER’s mission, Barrett said. “This is a really exciting venture.”

Alison Fromme is a writer for the Cornell SC Johnson College of Business.

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College Honors ECE Graduate Student for Excellence in Research

• by Matt Marcure
• May 23, 2024

Zhengfeng (Jeff) Lai has received the 2024 Excellence in Graduate Student Research award from the College of Engineering, recognizing the work he completed throughout his Ph.D. program in the Department of Electrical and Computer Engineering. The honor is part of the annual College of Engineering Graduate Student Excellence awards that celebrate student contributions to research, service, safety and DEI.

Lai's research stands to advance the possibilities of computer vision and machine learning. Throughout his graduate program, he published 14 articles (11 first-authored), which have already received over 200 citations.

Notably, Lai won a Best Paper Award at the 2022 Conference on Computer Vision and Pattern Recognition for pioneering research on a semi-supervised learning framework that does not make assumptions about unlabeled data. While pursuing his doctorate, he also filed a patent application for a project where he successfully applied machine learning to detect congenital heart disease in newborns.

Lai graduated in December 2023 and currently works as a machine learning research scientist at Apple, where he has revolutionized the company's use of large-scale pre-training and multimodal modeling.

Meet all the awardees

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Demi Moore, Cher and more stars raise money for AIDS research at amfAR gala near Cannes

On the red carpet for the annual amfAR gala in Cannes, Cher said the organization is an important one to her. (May 23)

Demi Moore poses for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Scott A Garfitt/Invision/AP)

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Cher poses for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Vianney Le Caer/Invision/AP)

Tommy Hilfiger, left, and Dee Ocleppo pose for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Vianney Le Caer/Invision/AP)

Leni Olumi Klum, left, and Heidi Klum pose for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Scott A Garfitt/Invision/AP)

Alexander Edwards, from left, Cher, Demi Moore, and Nick Jonas pose for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Vianney Le Caer/Invision/AP)

Demi Moore, left, and Cher pose for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Scott A Garfitt/Invision/AP)

Paris Jackson, center, poses for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Scott A Garfitt/Invision/AP)

Winnie Harlow poses for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Vianney Le Caer/Invision/AP)

Alex Scott poses for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Vianney Le Caer/Invision/AP)

Kelly Rowland poses for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Vianney Le Caer/Invision/AP)

Nick Jonas poses for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Vianney Le Caer/Invision/AP)

Maria Borges poses for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Vianney Le Caer/Invision/AP)

Maria Bakalova poses for photographers upon arrival at the amfAR Cinema Against AIDS benefit at the Hotel du Cap-Eden-Roc, during the 77th Cannes international film festival, Cap d’Antibes, southern France, Thursday, May 23, 2024. (Photo by Vianney Le Caer/Invision/AP)

ANTIBES, France (AP) — Some of the biggest stars in the French Rivera for the Cannes Film Festival made appearances at the 30th annual amfAR gala to raise money for AIDS research.

Demi Moore, whose film “The Substance” caused a stir at Cannes, hosted this year’s gala, a role launched by Elizabeth Taylor in 1993.

The red carpet at the exclusive Hôtel du Cap, Eden Roc, was awash with models, actors, singers and fashion designers as well as plenty of festival movers and shakers, who paid thousands of euros for a table at the hottest event in town.

Michelle Yeoh, Heidi Klum, Kelly Rowland, Andie MacDowell, Diane Kruger, Colman Domingo, Michelle Rodriguez, Winnie Harlow, Robin Thicke, Diplo, Paris Jackson, Petra Nemcova, Karolina Kurkova, Natasha Poly and Sarah Ferguson, the Duchess of York, all attended the fundraiser.

Rowland addressed her Cannes controversy from earlier in the week where she appeared to have an altercation with security on the red carpet.

“The woman knows what happened. I know what happened. And, I have a boundary, and I stand by those boundaries, and that is it. And there were other women that attended that carpet who did not quite look like me, and they didn’t get scolded, or pushed off or told to get off,” Rowland told The Associated Press . “And, I stood my ground and she felt like she had to stand hers but I stood my ground and that was it.”

As always the night began with champagne and cocktails under the stars. The sweeping hotel walkway was transformed into a carpet of red glitter down to the ocean, although this year the flashy super yachts were not blocking the horizon.

Guests drank champagne, margaritas, palomas and vodka martinis and posed for multiple selfies and photos at the black tie charity event.

Many of the works of art and sculptures from the auction were on display around the grounds including an iconic hand signed Andy Warhol lithograph of Taylor, which later raised 350,000 euros ($378,289).

A specially commissioned hand drawn picture of the Queen with Swarovski crystals on an aluminum frame, from royal artist Chris Levine, reached 475,000 euros ($513,392) for the charity. Sarah Ferguson took to the podium to help auction this lot with a plea of “this was my mother-in-law” and “stop being so bored, Stop looking at your phone” to the diners.

Guests were served a starter of truffled zucchini flower with cream of mushroom to the live soundtrack of Jess Glynne’s opening performance, followed by beef brulee with a potato emulsion and a strawberry and elderflower profiterole dessert while auctioneers called for bids.

Part way through the dinner guests paused and took to their feet to line the temporary catwalk through the tables for the fashion runway show curated annually by Carine Roitfeld and auctioned off to the highest bidder. This year’s collection, entitled “Fairy Tales” included 26 looks from Chanel, Dior, Louis Vuitton, Saint Laurent, Vivienne Westwood and Prada to name a few. After the models gathered on stage to help the auctioneer up the bids, the collection finally went for 500,000 euros ($540,412).

As part of the experiences on offer, a walk on part in “Emily in Paris“ series 5, plus an invite to the season 4 LA premiere went for 250,000 euros ($270,262).

Nick Jonas brought brother Joe on stage for a medley that got the room dancing and then Cher closed the show with a tribute to Taylor and a high-octane performance of “Believe” complete with gyrating dancers. The afterparty carried on by the hotel’s swimming pool with Diplo on the decks and more drinks as guests danced outdoors until the early hours.

AmfAR, The Foundation for AIDS Research is dedicated to the support of AIDS research, HIV prevention, treatment education and advocacy, raising nearly $900 million in support of it’s programs since 1985.