Abstracts

Cartan embeddings and intermediate algebras
Adam Fuller

A Cartan subalgebra of a von Neumann algebra is a suitably large abelian subalgebra. Cartan subalgebras have played an important role in the study of von Neumann algebras. Analogues have been given in C*-algebras and (purely algebraic) associative algebras. In all three of these settings, Cartan inclusions are uniquely determined by some underlying structure. For von Neumann algebras, this underlying structure is usually described in terms of measured equivalence relations; for C*-algebras and algebras the description is given in terms of groupoids.

Let $D$ be a Cartan subalgebra of a von Neumann algebra/C*-algebra/algebra $M$. If $C$ is another algebra such that $D \subseteq C \subseteq M$, is $D$ a Cartan subalgebra of $M$? To what extent does the equivalence relation/groupoid structure given by $D \subseteq M$, determine the intermediate algebras $C$? In this talk, we will give a brief survey of these questions, highlighting the parallels and differences between the 3 notions of a Cartan inclusion.

C*-covers : RFD and non-RFD (plus Ken Davidson by example)
Adam Humeniuk

A C*-cover of a (non-selfadjoint) operator algebra $A$ is a C*-algebra generated by a completely isometric copy of $A$. Centrally important is the unique minimum C*-cover: The C*-envelope. The envelope is an important invariant, but very far from a complete invariant. In recent joint work with Chris Ramsey and Elias Katsoulis, and forthcoming work with additional coauthors, we argue that studying the whole collection of C*-covers (which form a complete lattice) and not just the envelope, yields important information about $A$. For instance, $A$ is residually finite dimensional (RFD) if and only if it has RFD C*-covers. However, counterexamples of Clouâtre-Ramsey and Hartz show that neither the minimal nor maximal C*-cover may be RFD. Building in this direction, I will explain why the RFD C*-covers do not form a sub-lattice of the whole C*-cover lattice, even when $A$ is the disk algebra. In the spirit of Saturday afternoon, we will keep this talk light and focused on examples. I will alternate math-talk with a selection of stories, reflections, and maybe a bad joke or two about Ken's outsized influence on me, mathematics, and the world.

From dilation theory to quantum computers (and back again)

David Kribs

Ken’s energy, brilliance and determination were inspiring to witness as his student, and a few small comments and bits of advice he gave me in subsequent years had an outsized influence on my career. In this (somewhat self-indulgent) talk, I’ll give a summary of a research journey that started with Ken and my doctoral work under his supervision on dilation theory for non-commuting n-tuples of operators. During my postdoc years the journey moved into graph algebras and the theory of quantum channels. This was ultimately followed by a full-on jump into quantum error correction and related topics for several years, and then surprisingly (certainly to me anyway) to recent work with Xanadu Quantum Technologies, a Canadian company that is building (actual) quantum computers. The self-indulgent part: I’ll use a series of my papers as signposts for the discussion. 

A journey in dilation theory
Boyu Li

Dilation theory began with Sz. Nagy’s dilations of contractive operators, and has since found many applications. In this talk, I will share my personal journey through the world of dilation theory, highlighting several results on regular dilations and their applications, and reflecting on the invaluable guidance and support I’ve received from Ken along the way.

40+ years of math with Ken
Vern Paulsen

Since the 1980's, Ken and I collaborated on 4 papers in Operator Algebras and 3 papers in QI. In this talk, I will give an overview of the first 4 papers, to give some overview of Ken's interests through the ages. I will try to show some of the reasons that these papers wouldn't have existed without Ken's strengths.

Exotic ideals in free transformation group C*-algebras
David Pitts

In this talk, I will describe some results obtained jointly with R. Exel and V. Zarikian. Let (A, B) be a pair of unital C*-algebras with B ⊆ A and I_A ∈ B. An exotic ideal for (A, B) is a non-zero ideal J ⊆ A such that J ∩ B = {0}. Consider a C*-dynamical system (C(X), Γ, τ), where Γ is a countable discrete group, X is a compact Hausdorff space, and τ is an action of Γ on C(X) such that the associated action of Γ on X is free. Let η be a C*-norm on the convolution algebra C_c(Γ, C(X)); its completion, denoted C(X) ⋊_η Γ, is called a represented free transformation group C*-algebra. Using the theory of exotic crossed product functors, we showed that when Γ is nonamenable and the dynamical system admits an invariant state, one may choose η so that exotic ideals for (C(X) ⋊_η Γ, C(X)) exist in abundance and the associated conditional expectation E_η: C(X) ⋊_η Γ → C(X) is not faithful. These facts give answers to questions raised by K. Thomsen. Using the Koopman representation, a theorem of Elek, and a technique of Chou, Lao, and Rosenblatt, we showed that when Γ has property (T), and X is the Cantor set, there exists a free and minimal action of Γ on X and a C*-norm η such that the compact operators are an exotic ideal for (C(X) ⋊_η Γ, C(X)). As a consequence, when B ⊆ B(H) is a non-atomic MASA and A is the norm-closed span of the normalizers for B, then K(H) is an exotic ideal for (A, B); this answers a question of Katavolos and Paulsen.

Noncommutative entire functions
Eli Shamovich

Noncommutative analytic functions appeared in J. L. Taylor's attempt to build a general functional calculus for arbitrary tuples of operators. In this talk, I will discuss the most natural algebra of noncommutative entire functions. This algebra is interesting both as an algebraic object and an analytic one. I will explain why this algebra behaves in many ways, like the algebra of entire functions on the complex plane. In particular, using techniques developed by Davidson and Pitts for the free semigroup algebra, we can show that the algebra of noncommutative entires is a semi-free ideal ring (every finitely generated right ideal is free as a module) and that every finitely generated ideal is closed. These results lead to an extension of Haagerup and Thorbjornsen's theorem on almost sure convergence of random matrices to the setting of noncommutative meromorphic functions.

Negative resolution of the C*-algebraic Tarski problem

Christopher Schafhauser

TBA