# Descriptive Set Theory and Functional Analysis (12w5099)

Arriving in Banff, Alberta Sunday, June 17 and departing Friday June 22, 2012

## Organizers

Edward Effros (University of California, Los Angeles)

George Elliott (University of Toronto)

Ilijas Farah (York University)

Andrew Toms (Purdue University)

## Objectives

Objectives of the workshop

Recently, there has been great progress at the interface of operator algebras, descriptive set theory, and ergodic theory. Here we single out examples of four types, and explain how our workshop will advance research in each area. (We emphasize the first two types, as they exhibit particularly strong connections between these fields. And of course, the topics covered at the workshop will not be restricted to these four.)

I. Borel reducibility and the complexity of classification problems.

Mackey's opinion that Borel equivalence relations yielding non-standard Borel spaces are simply ``unclassifiable" has more recently been countered with a rich theory of cardinality for such relations. Given Polish spaces $X$ and $Y$ carrying Borel equivalence relations $E$ and $F$, respectively, one says that $(X,E)$ is {it Borel reducible} to $(Y,F)$ if there is a Borel map $Theta:X to Y$ with the property that

[

xEy Leftrightarrow Theta(x)F Theta(y).

]

In words, assigning invariants to $F$-classes is at least as difficult as assigning them to $E$-classes; one writes $E leq_B F$. There are infinitely many degrees of complexity in this picture, and the relationships between them remain murky in places. There are, however, some standout types: $E$ is {it classifiable by countable structures}, if, roughly, it is no more complex than the isomorphism relation on countable graphs; $E$ is {it turbulent} if there is a Borel reduction from the orbit equivalence (OE) relation of a turbulent group action into $E$; $E$ is {it below a group action} if there is a Borel reduction from $E$ into the OE relation of a Polish group action.

Several results regarding the Borel complexity of C$^*$- and W$^*$-algebras and group actions upon them have recently emerged. Sasyk-T"ornquist have established turbulence for isomorphism of von Neumann factors of all types, while Kerr-Li-Pichot have done the same for various types of group actions on standard probablity spaces and the hyperfinite $mathrm{II}_1$ factor. On the C$^*$-algebra side, Farah-Toms-T"ornquist have proved that the isomorphism relation for unital nuclear simple separable C$^*$-algebras (the primary object's in G. A. Elliott's $mathrm{K}$-theoretic classification program) is turbulent yet below a group action, and have established a similar result for metrizable Choquet simplices.

Our workshop will bring together these and other researchers to work on new questions in Borel reducibility, such as assessing the complexity of exact and non-exact C$^*$-algebras and non-commutative $L_p$ spaces (and perhaps finding therein an instance of the Kechris-Louveau conjecture concerning $E_1$ and group actions), and determining whether various classification functors in functional analysis have Borel computable inverses.

II. Measure preserving group actions.

Three steadily weaker notions of equivalence for free ergodic actions of countable groups on a standard probability space are conjugacy, orbit equivalence, and von Neumann equivalence (isomorphism of the von Neumann algebra crossed products associated to $Gamma curvearrowright X$ and $Lambda curvearrowright Y$). The last of these is very weak, as any two free ergodic actions of amenable groups are equivalent in this sense. Nevertheless Popa has since 2002 developed a remarkable deformation/rigidity theory which has allowed him and subsequently many others to establish orbit equivalence from von Neumann equivalence for a wide array of non-amenable groups. This has led to striking results in the descriptive theory of orbit equivalence relations, including the proof (due to Ioana and Epstein) that every countable non-amenable group admits continuum many orbit inequivalent actions, giving a strong converse to the Connes-Feldman-Weiss Theorem. More recently, Hjorth has used techniques derived from Epstein-Ioana to prove that there are continuum many treeable countable equivalence relations up to Borel reducibility (see I. above).

We will address several problems in this theory at our workshop, such as the question of whether any countable non-amenable group admits continuum many free Borel actions up to Borel reducibility, and whether the isomorphism relation for separable $mathrm{II}_1$ factors is the universal equivalence relation for unitary group actions.

III. Banach spaces.

The descriptive theory of Banach spaces is another active area under this proposal's umbrella. Rosendal, Ferenczi, Louveau and Todorcevic are among the prime actors. Recent results include the proof that the isomorphism problem for separable Banach spaces is equivalent to the maximally complicated analytic equivalence relation in the Borel hierarchy, and a partial classification of Banach spaces in terms of minimal subspaces. The second item is part of Gowers' program to classify Banach spaces by finding characteristic spaces present in every space. That program will be pursued further at our workshop, as will applications of infinite-dimensional Ramsey theory to Banach spaces.

IV. The structure of C$^*$-algebras.

C$^*$-algebra theory has seen many old problems solved lately using set theory as a fundamental tool. (It must be said that the set theory involved is not really descriptive, but we nevertheless have another important interaction between set theory and functional analysis.) These results include the proof by Farah and Phillips-Weaver that the question of whether all automorphisms of the Calkin algebra are inner is independent of ZFC, and the Akemann-Weaver proof of the consistency of a negative answer to Naimark's problem (``Must a C$^*$-algebra with only one irreducible representation up to unitary equivalence be isomorphic to the compact operators on some Hilbert space?"). Further questions to be addressed at our workshop include the possibility that a positive answer to Naimark's problem is consistent, and the question of whether the Calkin algebra admits a $mathrm{K}_1$-reversing automorphism.

Timeliness and relevance.

The progress described above has led to a tremendous amount of new collaboration and dialogue between functional analysts and descriptive set theorists, albeit through a multitude of largely independent projects. That is why a 5-day workshop at BIRS on Descriptive Set Theory and Functional Analysis will be especially effective: we will not only disseminate research and lay the groundwork for progress on major problems in the field--any BIRS workshop should do as much--but also give new coherence to this interdisciplinary field. Success in this last goal will prove particularly helpful to young researchers wanting to enter the field, as they will get a panoramic view of its research and be able to discuss their own research with a cast of faculty never before assembled at a single meeting.

The profile of interdisciplinary research in set theory and functional analysis has been rising steadily. For instance, Texas A&M University hosted a 5-day conference on the topic in August, 2010, and there have been two Appalachian Set Theory Workshops (a NSF funded series) by Kechris and Farah discussing several of the recent results described in this proposal. Our BIRS workshop, however, will be an order of magnitude more significant than these events, not least because of the quality of the participants. They include 10 ICM speakers with 13 ICM talks between them (3 of them plenary), and editors of {it Journal of the American Mathematical Society}, {it Fundamenta Mathematicae}, {it Journal of Symbolic Logic}, {it Bulletin of Symbolic Logic}, {it Canadian Journal of Mathematics}, {it Journal of Functional Analysis}, {it Pacific Journal of Mathematics}, {it Mathematical Research Letters}, and {it Journal of Operator Theory}.

As for timeliness, let us point out that in addition to the conference activity mentioned above, most of the significant results in this proposal have appeared in the last five years. Moreover, Popa's plenary ICM address on $mathrm{II}_1$ equivalence relations came in 2006, and Vaes gave his ICM lecture this year in Hyderabad.

Recently, there has been great progress at the interface of operator algebras, descriptive set theory, and ergodic theory. Here we single out examples of four types, and explain how our workshop will advance research in each area. (We emphasize the first two types, as they exhibit particularly strong connections between these fields. And of course, the topics covered at the workshop will not be restricted to these four.)

I. Borel reducibility and the complexity of classification problems.

Mackey's opinion that Borel equivalence relations yielding non-standard Borel spaces are simply ``unclassifiable" has more recently been countered with a rich theory of cardinality for such relations. Given Polish spaces $X$ and $Y$ carrying Borel equivalence relations $E$ and $F$, respectively, one says that $(X,E)$ is {it Borel reducible} to $(Y,F)$ if there is a Borel map $Theta:X to Y$ with the property that

[

xEy Leftrightarrow Theta(x)F Theta(y).

]

In words, assigning invariants to $F$-classes is at least as difficult as assigning them to $E$-classes; one writes $E leq_B F$. There are infinitely many degrees of complexity in this picture, and the relationships between them remain murky in places. There are, however, some standout types: $E$ is {it classifiable by countable structures}, if, roughly, it is no more complex than the isomorphism relation on countable graphs; $E$ is {it turbulent} if there is a Borel reduction from the orbit equivalence (OE) relation of a turbulent group action into $E$; $E$ is {it below a group action} if there is a Borel reduction from $E$ into the OE relation of a Polish group action.

Several results regarding the Borel complexity of C$^*$- and W$^*$-algebras and group actions upon them have recently emerged. Sasyk-T"ornquist have established turbulence for isomorphism of von Neumann factors of all types, while Kerr-Li-Pichot have done the same for various types of group actions on standard probablity spaces and the hyperfinite $mathrm{II}_1$ factor. On the C$^*$-algebra side, Farah-Toms-T"ornquist have proved that the isomorphism relation for unital nuclear simple separable C$^*$-algebras (the primary object's in G. A. Elliott's $mathrm{K}$-theoretic classification program) is turbulent yet below a group action, and have established a similar result for metrizable Choquet simplices.

Our workshop will bring together these and other researchers to work on new questions in Borel reducibility, such as assessing the complexity of exact and non-exact C$^*$-algebras and non-commutative $L_p$ spaces (and perhaps finding therein an instance of the Kechris-Louveau conjecture concerning $E_1$ and group actions), and determining whether various classification functors in functional analysis have Borel computable inverses.

II. Measure preserving group actions.

Three steadily weaker notions of equivalence for free ergodic actions of countable groups on a standard probability space are conjugacy, orbit equivalence, and von Neumann equivalence (isomorphism of the von Neumann algebra crossed products associated to $Gamma curvearrowright X$ and $Lambda curvearrowright Y$). The last of these is very weak, as any two free ergodic actions of amenable groups are equivalent in this sense. Nevertheless Popa has since 2002 developed a remarkable deformation/rigidity theory which has allowed him and subsequently many others to establish orbit equivalence from von Neumann equivalence for a wide array of non-amenable groups. This has led to striking results in the descriptive theory of orbit equivalence relations, including the proof (due to Ioana and Epstein) that every countable non-amenable group admits continuum many orbit inequivalent actions, giving a strong converse to the Connes-Feldman-Weiss Theorem. More recently, Hjorth has used techniques derived from Epstein-Ioana to prove that there are continuum many treeable countable equivalence relations up to Borel reducibility (see I. above).

We will address several problems in this theory at our workshop, such as the question of whether any countable non-amenable group admits continuum many free Borel actions up to Borel reducibility, and whether the isomorphism relation for separable $mathrm{II}_1$ factors is the universal equivalence relation for unitary group actions.

III. Banach spaces.

The descriptive theory of Banach spaces is another active area under this proposal's umbrella. Rosendal, Ferenczi, Louveau and Todorcevic are among the prime actors. Recent results include the proof that the isomorphism problem for separable Banach spaces is equivalent to the maximally complicated analytic equivalence relation in the Borel hierarchy, and a partial classification of Banach spaces in terms of minimal subspaces. The second item is part of Gowers' program to classify Banach spaces by finding characteristic spaces present in every space. That program will be pursued further at our workshop, as will applications of infinite-dimensional Ramsey theory to Banach spaces.

IV. The structure of C$^*$-algebras.

C$^*$-algebra theory has seen many old problems solved lately using set theory as a fundamental tool. (It must be said that the set theory involved is not really descriptive, but we nevertheless have another important interaction between set theory and functional analysis.) These results include the proof by Farah and Phillips-Weaver that the question of whether all automorphisms of the Calkin algebra are inner is independent of ZFC, and the Akemann-Weaver proof of the consistency of a negative answer to Naimark's problem (``Must a C$^*$-algebra with only one irreducible representation up to unitary equivalence be isomorphic to the compact operators on some Hilbert space?"). Further questions to be addressed at our workshop include the possibility that a positive answer to Naimark's problem is consistent, and the question of whether the Calkin algebra admits a $mathrm{K}_1$-reversing automorphism.

Timeliness and relevance.

The progress described above has led to a tremendous amount of new collaboration and dialogue between functional analysts and descriptive set theorists, albeit through a multitude of largely independent projects. That is why a 5-day workshop at BIRS on Descriptive Set Theory and Functional Analysis will be especially effective: we will not only disseminate research and lay the groundwork for progress on major problems in the field--any BIRS workshop should do as much--but also give new coherence to this interdisciplinary field. Success in this last goal will prove particularly helpful to young researchers wanting to enter the field, as they will get a panoramic view of its research and be able to discuss their own research with a cast of faculty never before assembled at a single meeting.

The profile of interdisciplinary research in set theory and functional analysis has been rising steadily. For instance, Texas A&M University hosted a 5-day conference on the topic in August, 2010, and there have been two Appalachian Set Theory Workshops (a NSF funded series) by Kechris and Farah discussing several of the recent results described in this proposal. Our BIRS workshop, however, will be an order of magnitude more significant than these events, not least because of the quality of the participants. They include 10 ICM speakers with 13 ICM talks between them (3 of them plenary), and editors of {it Journal of the American Mathematical Society}, {it Fundamenta Mathematicae}, {it Journal of Symbolic Logic}, {it Bulletin of Symbolic Logic}, {it Canadian Journal of Mathematics}, {it Journal of Functional Analysis}, {it Pacific Journal of Mathematics}, {it Mathematical Research Letters}, and {it Journal of Operator Theory}.

As for timeliness, let us point out that in addition to the conference activity mentioned above, most of the significant results in this proposal have appeared in the last five years. Moreover, Popa's plenary ICM address on $mathrm{II}_1$ equivalence relations came in 2006, and Vaes gave his ICM lecture this year in Hyderabad.