50 Open Problems: Ultraproduct II$_1$ factors
A collection of 50 open problems around the structure theory of ultraproducts of II$_1$ factors is presented, along with some annotations and references.
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arXiv subjects
Srivatsav Kunnawalkam Elayavalli
Publications and source records attributed to Srivatsav Kunnawalkam Elayavalli.
Strongly converging unitary representations for extensions by exact groups
We prove the existence of strongly converging unitary representations in various new settings of countable groups, in particular, arising as extensions by exact groups: semidirect products with amenable groups; generalized wreath products with abelian base; graph wreath products; various free-by-cyclic groups including Gersten's group; and general Bernoulli shift crossed products on $C^*$-algebras.
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Selfless C*-correspondences, operator valued C*-probability spaces and completely positive maps
We develop a general theory of selflessness for C*-correspondences, with several applications. Specializing this theory to completely positive maps, in particular to conditional expectations, gives rise to a novel notion of relative selflessness. Among applications, the machinery developed here yields new examples of selfless C*-algebras, for instance among minimal tensor products and reduced crossed products, new examples of MF C*-algebras arising as reduced amalgamated free products, a conceptually new proof of Kirchberg's $\mathcal{O}_{\infty}$-absorption theorem, and the lack of ``phantom'' traces on ultrapowers.
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Relative biexactness and mixing in von Neumann algebras
We develop a new technique to upgrade relative biexactness in general von Neumann algebras: suppose that $\{N_i\}_{i\in I}\subset M$ are mixing and biexact subalgebras of a separable von Neumann algebra with expectation, and if $M$ is biexact relative to $\{N_i\}_{i\in I}$, then $M$ is biexact. This result yields several new examples of biexact von Neumann algebras, notably including amalgamated free products. By generalizing the relative biexactness results of Hoshino to the von Neumann algebra setting and applying our result above along with certain bimodule computations, we in fact obtain, as an application, a new classification result for biexactness for graph products of finite dimensional von Neumann algebras. This yields significant extensions of prior works of Caspers-Borst, and Blufstein-Goldman-Oyakawa.
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Questions on the structure of random embeddings of $L(\mathbb{F}_2)$
Motivated by recent developments at the interface of operator algebras and random matrix theory, we propose new conjectures concerning the asymptotic structure of random matrix models of the countable free groups. The first conjecture predicts a random matrix analogue of the Akemann-Ostrand property for free groups, and reveals a succinct approach to recover the Peterson-Thom property for $L(\mathbb{F}_2)$. The second stronger conjecture is motivated by continuous model theory. It predicts that the \emph{random} embedding of the free group factor into a matrix ultraproduct is \emph{existential}. We discuss the interesting relationship between these conjectures.
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Selfless reduced amalgamated free products and HNN extensions
We find a general family of selfless inclusions in reduced amalgamated free products of C*-algebras. Apart from generalizing prior works due to McClanahan, Ivanov and Omland, our work yields a few other applications. We present a short new approach to construct HNN extensions of C*-algebras and find several new examples of selflessness using this. This generalizes results of Ueda, Ivanov and de la Harpe-Preaux. As another application our work yields a short proof of selflessness for arbitrary graph products of C*-algebras over graphs of more than 2 vertices and diameter greater than 3.
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Toeplitz exactness for strong convergence
We prove a new "Toeplitz exactness" theorem for strong convergence. This is a machine to upgrade strong convergence in the general setting of $C^\ast$-correspondences, and has several applications.
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On free components of Artin and Coxeter groups
The number of connected components can be remembered by the von Neumann algebra among Artin groups, the only possible exception being the case that corresponds to the free group factor problem. In the case of Coxeter groups, this result is obtained in the absence of relatively hyperbolicity. We also discuss a specific case of the analogous problem in measure equivalence where each factor group is a product of nonabelian free groups.
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A new source of purely finite matricial fields
A countable group $G$ is said to be \emph{matricial field} (MF) if it admits a strongly converging sequence of approximate homomorphisms into matrices; i.e, the norms of polynomials converge to those in the left regular representation. $G$ is \emph{purely MF} (PMF) if these maps are actual homomorphisms, and $G$ is further \emph{purely finite field} (PFF) if the image of each homomorphism is finite. By developing a new operator algebraic approach to these problems, we are able to prove the following result bringing several new examples into the fold. Suppose $G$ is a MF (resp., PMF, PFF) group and $H<G$ is separable (i.e., $H=\cap_{i\in \mathbb{N}}H_i$ where $H_i<G$ are finite index subgroups) and $K$ is a residually finite MF (resp., PMF, PFF) group. If either $G$ or $K$ is exact, then the amalgamated free product $G*_{H}(H\times K)$ is MF (resp., PMF, PFF). Our work has several applications, we list some below: 1. The Brown--Douglas--Fillmore semigroups of many new examples of reduced group $C^*$-algebras are shown to be not groups. 2. Arbitrary group doubles $G*_HG$ of MF (resp., PMF, PFF) over separable subgroups $H$ are MF (resp., PMF, PFF). Moreover, $G*H$ is PFF whenever $G,H$ are PFF, and either $G$ or $H$ is exact. 3. Arbitrary graph products of residually finite exact MF (resp., PMF, PFF) groups are MF (resp., PMF, PFF), yielding a significant generalization of the breakthrough work of M. Magee and J. Thomas. 4. The open problem of proving PFF for fundamental groups of closed hyperbolic 3-manifolds is resolved. This has geometric significance in the theory of minimal surfaces via A. Song's approach.
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Strongly 1-bounded inner amenable groups
It is shown that finitely presented icc inner amenable groups yield strongly 1-bounded II_1 factors.
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Conjugacy co-amenability
In this note we study a natural analytic property of inclusions of groups akin to co-amenability: the property of existence of a non-compactly supported invariant state for the conjugation action of a group $G$ on the von Neumann algebra generated by the characteristic functions $\{\mathbf{1}_{gHg^{-1}}\}_{g\in G}$ viewed inside $\ell^\infty(G)$. Some interesting settings and examples of this phenomena are proved. We also comment on a consideration related to proper proximality, which motivated this property.
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Non-isomorphism of reduced free group $C^\ast$-algebras
Using a new approach involving embedding spaces in II$_1$ factors with plenty of freely independent Haar unitaries, we prove that $C^\ast_r(\mathbb{F}_n)\ncong C^\ast_r(\mathbb{F}_m)$ for $n \neq m$. This recovers the seminal result of Pimsner and Voiculescu with a short new proof.
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Exotic full factors via weakly coarse bimodules
We are able to explicitly compute the bimodule structure of von Neumann algebra inclusions in handle constructions, which arise as inductive limits of iterated amalgamated free products not elementarily equivalent to $L(\mathbb{F}_2)$. Our computation is achieved via identifying delicate normal form decompositions in amalgamated free products built in an iterated fashion. Using these techniques, we are able to show that the handles constructions are always full, without any need to appeal to Property (T) phenomena which was essential in all previous works. Furthermore our bimodule machinery works in the setting of arbitrary von Neumann algebras equipped with faithful normal states, yielding examples of full $\mathrm{III}_1$ factors via handle constructions.
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Selfless Inclusions of C*-Algebras
We introduce and study a natural notion of selflessness for inclusions of C*-probability spaces, which in particular implies that all intermediate C*-algebras are selfless in the sense of Robert. We identify natural sources of selfless inclusions in the realms of Z-stable and free product C*-algebras. As an application of this, we prove selflessness for a new family of C*-probability spaces outside the regime of free products and group C*-algebras. These include the reduced free unitary compact quantum groups.
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Some remarks on decay in countable groups and amalgamated free products
In this note, we first study the notion of subexponential decay (SD) for countable groups with respect to a length function, which generalizes the well-known rapid decay (RD) property, first discovered by Haagerup in 1979. Several natural properties and examples are studied, especially including groups that have SD, but not RD. This consideration naturally has applications in $C^*$-algebras. We also consider in this setting a permanence theorem for decay in amalgamated free products (proved also recently by Chatterji--Gautero), and demonstrate that it is in a precise sense optimal.
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On conjugacy and perturbation of subalgebras
We study conjugacy orbits of certain types of subalgebras in tracial von Neumann algebras. For any separable II$_1$ factor $N_0$ we construct a highly indecomposable non Gamma II$_1$ factor $N$ such that $N_0 \subset N$ and moreover every von Neumann subalgebra of $N$ with Haagerup's property admits a unique embedding up to unitary conjugation. Such a factor necessarily has to be non separable, but we show that it can be taken of density character $2^{\aleph_0}$. On the other hand we are able to construct for any separable II$_1$ factor $M_0$, a separable II$_1$ factor $M$ containing $M_0$ such that every property (T) subfactor admits a unique embedding into $M$ up to uniformly approximate unitary equivalence; i.e., any pair of embeddings can be conjugated up to a small uniform $2$-norm perturbation.
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Strict comparison in reduced group $C^*$-algebras
We prove that for every $n\geq 2$, the reduced group $C^*$-algebras of the countable free groups $C^*_r(\mathbb{F}_n)$ have strict comparison. Our method works in a general setting: for $G$ in a large family of non-amenable groups, including hyperbolic groups, free products, mapping class groups, right-angled Artin groups etc., we have $C^*_r(G)$ have strict comparison. This work also has several applications in the theory of $C^*$-algebras including: resolving Leonel Robert's selflessness problem for $C^*_r(G)$; uniqueness of embeddings of the Jiang-Su algebra $\mathcal{Z}$ up to approximate unitary equivalence into $C^*_r(G)$; full computations of the Cuntz semigroup of $C^*_r(G)$ and future directions in the $C^*$-classification program.
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Soficity for group actions on sets and applications
In this article we develop a notion of soficity for actions of countable groups on sets. We show two equivalent perspectives, several natural properties and examples. Notable examples include arbitrary actions of both amenable groups and free groups, and actions of sofic groups with locally finite stabilizers. As applications we prove soficity for generalized wreath products (and amalgamated free generalized wreath products) of sofic groups where the underlying group action is sofic. This generalizes the result of Hayes and Sale [HS18], and proves soficity for many new families of groups.
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