Operator Algebras and Operator Theory Ical Atom

A C*-algebra is said to be K-stable if its nonstable K-groups are naturally isomorphic to the usual K-theory groups. In this talk, we shall study continuous C(X)-algebras, each of whose fibers are K-stable. We will show that such an algebra is itself K-stable under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension.

Rokhlin dimension is a regularity property for group actions on $C^*$-algebras. It was originally introduced for actions of the integers and finite groups, and later the definition was extended to other classes of groups. Rokhlin dimension comes in two flavors, commuting and non-commuting towers, which at least for finite group actions, turn out to be different. The main interest in Rokhlin dimension was as a tool to show that various regularity properties of a $C^*$-algebra pass to the crossed product. For those types of theorems, one only cares about whether this dimension is finite or infinite, and not the actual value. For actions of finite groups on simple $C^*$-algebras, the only known examples had dimensions 0,1,2 or infinity. Nuclear dimension, a related non-dynamical dimension for $C^*$-algebras, is known to only admit the values 0,1 or infinity on simple $C^*$-algebras, so it might seem plausible that Rokhlin dimension would exhibit similar behavior. In this talk, I’ll describe work in preparation which shows that arbitrarily large values can be achieved (though we don’t know how to achieve all known examples), as well as finer conclusions which can be deduced from the actual value, as opposed to merely whether the dimension is finite. This shows that the value Rokhlin dimension can in fact be seen as an interesting invariant of the group action. The tools required for proving it involve equivariant K-theory and the Atiyah-Segal completion theorem; I will not assume that the audience is familiar with those.

This is joint work with N. Christopher Phillips.

In this talk, I will describe joint work with Mike Jury and Rob Martin. The focus of this talk will be on noncommutative (NC) rational function, i.e., elements of the free skew field on d generators. Suppose such a function is bounded on all finite-dimensional row contractions. In that case, it admits an inner-outer factorization as elements of the free semigroup algebra (analogous to the classical factorization in function theory on the disc). Both the inner and outer functions are NC rational, as well. I will describe the theory behind this factorization and discuss how one obtains representations of the Cuntz algebra $\mathcal{O}_d$ from inner elements of the free semigroup algebra. I will show that from NC rational inners, one obtains the finitely-correlated representations introduced by Bratelli and Jorgensen. I will finish the discussion with some open questions.

I will discuss a free noncommutative analogue of the classical Christoffel–Darboux kernels, namely the reproducing kernel of the space of noncommutative polynomials up to a given degree with the scalar product induced by a tracial positive functional on the free $^*$-algebra on $n>1$ generators. Using operator space methods one can show that despite it being matrix valued, the Christoffel–Darboux kernel is — similarly to the classical case — the solution of an extremal problem. If there is time left, I will introduce the analogues of the Siciak extremal function with the eventual goal to study the asymptotic behaviour of the kernel as the degree goes to infinity.

The talk is based on joint work with Serban Belinschi and Victor Magron.

In this talk, I will discuss a noncommutative (nc) analogue of the Gleason problem and its application in the “NC Cowen-Douglas” class. The Gleason problem was first studied by Andrew Gleason in studying the maximal ideals of a commutative Banach algebra. In particular, he showed that if the maximal ideal consisting of functions in the Banach algebra $\mathcal{A} ( \mathbb{B} ( 0, 1 ) )$ vanishing at the origin is finitely generated then it has to be generated by the coordinate functions where $\mathcal{A} ( \mathbb{B} ( 0, 1 ) )$ is the Banach algebra of holomorphic functions on the open unit ball $\mathbb{B} ( 0, 1 )$ at $0$ in $\mathbb{C}^n$ which can be continuously extended up to the boundary. The question – whether the maximal ideals in algebras of holomorphic functions are generated by the coordinate functions – has been named the Gleason problem. It turns out that the existence of a local solution of the Gleason problem in a reproducing kernel Hilbert space provides a sufficient condition for the membership of the tuple of adjoint of multiplication operators by coordinate functions in the Cowen-Douglas class.

After briefly discussing these classical aspects of the Gleason problem, I will introduce its nc counterpart for uniformly analytic nc functions and show that such a problem in the nc category is always locally uniquely solvable unlike the classical case. As an application one obtains a characterization of nc reproducing Hilbert spaces of uniformly analytic nc functions on a nc domain in $\mathbb{C}^d_{ \text{nc} }$ so that the adjoint of the $d$ - tuple of left multiplication operators by the nc coordinate functions are in the nc Cowen-Douglas class. Along the way, I will recall necessary materials from nc function theory.

This is a part of my ongoing work jointly with Professor Vinnikov on the nc Cowen-Douglas class.

(Part 2 of the talk from last week.)

In this talk, I will discuss a noncommutative (nc) analogue of the Gleason problem and its application in the “NC Cowen-Douglas” class. The Gleason problem was first studied by Andrew Gleason in studying the maximal ideals of a commutative Banach algebra. In particular, he showed that if the maximal ideal consisting of functions in the Banach algebra $\mathcal{A} ( \mathbb{B} ( 0, 1 ) )$ vanishing at the origin is finitely generated then it has to be generated by the coordinate functions where $\mathcal{A} ( \mathbb{B} ( 0, 1 ) )$ is the Banach algebra of holomorphic functions on the open unit ball $\mathbb{B} ( 0, 1 )$ at $0$ in $\mathbb{C}^n$ which can be continuously extended up to the boundary. The question – whether the maximal ideals in algebras of holomorphic functions are generated by the coordinate functions – has been named the Gleason problem. It turns out that the existence of a local solution of the Gleason problem in a reproducing kernel Hilbert space provides a sufficient condition for the membership of the tuple of adjoint of multiplication operators by coordinate functions in the Cowen-Douglas class.

After briefly discussing these classical aspects of the Gleason problem, I will introduce its nc counterpart for uniformly analytic nc functions and show that such a problem in the nc category is always locally uniquely solvable unlike the classical case. As an application one obtains a characterization of nc reproducing Hilbert spaces of uniformly analytic nc functions on a nc domain in $\mathbb{C}^d_{ \text{nc} }$ so that the adjoint of the $d$ - tuple of left multiplication operators by the nc coordinate functions are in the nc Cowen-Douglas class. Along the way, I will recall necessary materials from nc function theory.

This is a part of my ongoing work jointly with Professor Vinnikov on the nc Cowen-Douglas class.

When studying quotients of C-algebras generated by creation and annihilation operators on analogues of Fock space, the question of the existence of a co-universal quotient plays an important role in answering fundamental questions in the theory. The study of co-universal quotients goes back to works of Cuntz, and Cuntz and Krieger, on uniqueness theorems for C-algebras arising from symbolic dynamics, and by now co-universal quotients have been shown to exist in several broad classes of examples of Toeplitz C*-algebras.

When associating Toeplitz C-algebras to random walks on a group $G$, new notions of *ratio-limit space and boundary emerge from searching for their co-universal quotients, and the existence of these co-universal quotients becomes intimately related to the group dynamics on the ratio-limit boundary.

In this talk I will exlain how we extended results of Woess to show that there is co-universal quotient for a large class of symmetric random walks on relatively hyperbolic groups. This sheds light on some questions of Woess on ratio-limits for random walks on relatively hyperbolic groups, and extends a result mine on the existence of co-universal quotients for Toeplitz C*-algebras for random walks.

*This talk is based on joint work with Ilya Gekhtman.

Recently, Ofek, Pandey, and Shalit have defined a version of Banach-Mazur distances on the space of isomorphism classes of finite-dimensional complete Pick spaces. By the universality theorem of Agler and McCarthy, every finite-dimensional complete Pick space on n points is equivalent to a subspace of the Drury-Arveson space spanned by n kernels at points of the unit ball of some C^d. We propose to study the space of projections on finite-dimensional multiplier coinvariant subspaces of the Drury-Arveson space. The metric on this space is induced by the norm. We show that if we restrict ourselves to the subspace of projection on spaces spanned by distinct n kernels, then this space is homeomorphic to the symmetrized polyball. It then follows that the invariant distance obtained induces the same topology on the space of isomorphism classes of complete Pick space as the Banach-Mazur distance of Ofek, Pandey, and Shalit. Time permuting we will show a potential application of this idea

Free nc function theory is an extension of the theory of holomorphic functions of several complex variables to the theory of functions on matrix tuples $Z=(Z_1,\cdots,Z_d)$ where $Z_i\in M_n(\mathbb{C})$ and $n$ is allowed to vary.

An nc function is a function defined on such tuples $Z$ and takes values in $\cup_{n\in \mathbb{N}} M_n(\mathbb{C})$ which is graded and respects direct sums and similarity (equivalently, respects intertwiners).

The classical correspondence between positive kernels and Hilbert spaces of functions has been recently extended by Ball, Marx and Vinnikov to nc completely positive kernels and Hilbert spaces of nc functions. In a previous, unpulished work, we have developed a similar theory for matricial functions where $\mathbb{C}$ is replaced by a von Neumann algebra $M$, $\cup_{n\in \mathbb{N}} M_n(\mathbb{C})$ is replaced by a suitable disjoint union of correspondences over $M$ and the ``index set” $\mathbb{N}$ is replaced by the set of representations of $M$.

Thus, in both situations we deal with structures that are fibred. In each settings there are situations where we need to move among fibres. This led us to consider actions of categories on fibred sets and we study here functions and kernels that are invariant under certain actions of categories.

This is a joint work (in progress) with Paul Muhly.