Seminar in operator theory and operator algebras (MATH 9310)
Fall 2013

The seminar is organized by David Sherman. We meet Tuesdays 3:30-4:30 in Kerchof 326.

### September 10

#### Operator algebras for analytic varieties

To an analytic variety V one can consider the restriction of the multiplier algebra of Drury-Arveson space to V. The goal of these two talks is to show that two such operator algebras are completely isometrically isomorphic if and only if there is an automorphism of the ball taking one variety onto the other.

### September 17

#### Operator algebras for analytic varieties

To an analytic variety V one can consider the restriction of the multiplier algebra of Drury-Arveson space to V. The goal of these two talks is to show that two such operator algebras are completely isometrically isomorphic if and only if there is an automorphism of the ball taking one variety onto the other.

### September 24

#### Basic notions of free probability (Part 1 of 2)

In Part 1, we will first set up the basic framework of non-commutative probability and discuss some basic notions of this topic with examples. The notion of free independence will be introduced and contrasted with classical (tensor) independence. To illustrate this contrast, we will exhibit combinatorial proofs of the Central Limit Theorem when either of these two notions of independence is considered.

Looking ahead to Part 2, there are a number of possible topics we may discuss, including but not limited to: computations using the free central limit theorem, the surprising result of Wigner's theorem for large random matrices, the relation between free independence and non-crossing partitions via free cumulants, free convolution, and the R-transform.

### October 1

#### Basic notions of free probability (Part 2 of 2)

We will begin by completing the proof of the Central Limit Theorems in this algebraic context. As we will see in this proof, non-crossing partitions of n arise naturally from free independence. We will go on to examine this relationship between non-crossing partitions and free independence by looking at the concept of free cumulants (related to moments via the moment-cumulant formula). These free cumulants will let us study free convolution and the related R-transform (this is the free analog of the logarithm of the Fourier transform). We will use these ideas to discuss the free convolution of two projections and the free analog of the Poisson distribution.

### October 8

#### Department reception for Barbara and Tom in the lounge (3:30 pm)

October 12 is VOTCAM, in honor of Barbara and Tom, at UVa.

### October 22

#### Characters and II1 Factor Representations of Thompson's group

A character of an infinite group G is defined as a positive-definite class function. In view of the Gelfand-Naimark-Segal construction, characters are in one-to-one correspondence with finite (in the sense of Murray-von Neumann) von Neumann algebra representations of the group. Characters of a group form a Choquet simplex with extreme characters being in one-to-one correspondence with type II_1 factor representations of the group in question.

Studying the representation theory of the infinite symmetric group S(∞), Vershik noticed that many of its characters come from ergodic actions of S(∞) on measure spaces (X,μ) by the formula f(g) = μ(FixedPoints(g)). The question we are trying to address is for which infinite groups all of the characters can be given a similar dynamical interpretation.

We will reexamine some older results on classification of characters of such groups as Special Linear Group of Infinite Matrices over a finite field and Groups of Rational Permutation of the unit interval. We will then classify characters of locally finite groups determined by Bratteli diagrams and the Higman-Thompson groups. In the case of (the commutator of) a Higman-Thompson group, based on the fact that its natural action on the unit interval has no invariant measures, we will show that this group has no non-trivial characters. As a result, the only non-trivial II1 factor representation of it is the group von Neumann algebra. The absence of non-trivial characters has some implications in the theory of random subgroups.

### October 29

#### Equivalent operator categories

Leaving rigorous definitions to the talk, operator categories are natural classes that include C*-algebras, operator systems, hereditary manifolds, operator algebras, Jordan operator algebras, etc. I will show how to associate the following features to any such category: an operator topology, a representation theory, and a convexity/dilation theory. It turns out that if one of these features agrees for a pair of categories, then all three do, in which case the categories are called equivalent. I will discuss some equivalences, along the way obtaining new observations about Arveson's hyperrigidity and maybe even triangles.

### November 5

#### Every matrix is a product of Toeplitz matrices

Recently Ye and Lim showed that every matrix can be written as a product of Toeplitz matrices. In this talk, I will outline a proof of this result. I will also discuss the maximal algebras within the Toeplitz matrices and show how some alterations of the Ye and Lim proof will show that the n by n matrices can be written as a product of maximal algebras of Toeplitz matrices.

### November 12

#### Inverse continuity of the numerical range map

The numerical range of a matrix has many applications in linear algebra, physics, and engineering. The numerical range is the image of the complex unit n-sphere under the quadratic map [x maps to x*Ax]. Recently there has been interest in the pre-images of this numerical range map. I will discuss the continuity properties of the multivalued inverse of the numerical range map. Along the way, we will revisit some of the classical results concerning the numerical range.

### November 19

#### Forcing Axioms and Baire Category Theorem

The Baire Category Theorem (BCT) states that in certain topological spaces, the intersection of countably many open dense sets is dense. "Forcing Axioms" assert that for a wide class of Stone Spaces, the "countably many" requirement in the BCT can be relaxed. Forcing Axioms can neither be proved nor disproved in classical mathematics, but have many interesting consequences; e.g. Farah proved that Forcing Axioms imply there are no outer automorphisms of the Calkin algebra. I will discuss Forcing Axioms and their enemy, Jensen's Diamond Principle.

### November 26

#### Naimark's problem: a primer

We know that every irreducible representation of the compact operators on a Hilbert space is unitarily equivalent to the identity representation. In 1951, Naimark asked if the converse was true, and this came to be known as Naimark's problem. We will discuss the history of Naimark's problem, including results of Rosenberg and Glimm. We will discuss a criterion for unique extensions of pure states due to Anderson and then introduce an important crossed product construction utilizing the homogeneity of the state space of separable C*-algebras (Futamura, Kataoka, and Kishimoto 2001, Kishimoto, Ozawa, and Sakai 2003). In a subsequent talk we will present the 2004 construction of a counterexample to Naimark's problem due to Akemann and Weaver.

### December 122 PMKerchof 128

#### Naimark's problem: the counterexample

In a previous talk we discussed the history of Naimark's problem. In this talk we will work through the construction of a counterexample to Naimark's problem: a C*-algebra with only one irreducible representation up to unitary equivalence which is not *-isomorphic to the compact operators on a Hilbert space. This construction is due to Akemann and Weaver in 2004 and utilizes Jensen's diamond principle, which is known to be independent of ZFC.