Seminar in operator theory and operator algebras (MATH 9310)
Fall 2011

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

### August 30

#### Factorizing collections of weakly compact operators

A fundamental result of Davis, Figiel, Johnson and Pelczynski from 1974, states that for each weakly compact operator T : X to Y (X and Y are Banach spaces) there is a reflexive space Z and operators S:X to Z, R:Z to Y such that T=RS. In other words, every weakly compact operator factors through a reflexive space. Using recent results on embedding classes of reflexive Banach spaces, we outline the proof that, under certain assumption on X and Y, if A is an Borel collection of weakly compact operators in the strong operator topology, then there is a single reflexive space Z such that every operator T in A factors through Z. As a consequence, we get the known result, for any two Banach spaces X and Y, there is a reflexive space Z such that every compact operator factors through Z.

### September 20

#### Nonexpansive maps and the horofunction boundary

The horofunction boundary is a tool from metric geometry that has been useful in studying the dynamics of iterated nonexpansive maps. In this talk, I will introduce the horofunction boundary and discuss some recent applications to nonexpansive maps in finite dimensional normed spaces. I will present some new results and also show how horofunctions can be used to give elegant proofs of some well known theorems about nonexpansive maps.

### September 27

#### Some thoughts on Livsic's theorem I

A 1946 theorem of Livsic determined when two symmetric operators are unitarily equivalent by means of a characteristic function. That is to say, with each symmetric operator T with (1, 1) deficiency indices, there is a certain function wT, called the characteristic function for T, such that the following theorem holds: If T1 and T2 are symmetric operators with (1, 1) deficiency indices, then T1 is unitarily equivalent to T2 if and only if wT1 is a unimodular multiple of wT2. There is a version of this result for symmetric operators with (n, n) deficiency indices where the characteristic functions here are nxn matrix-valued functions and the condition is T1 is unitarily equivalent to T2 if and only if there are unitary matrices u and v such that wT1 = u wT2 v. In this joint work with Alexandru Aleman, we have a similar type of result but our characteristic function is, in certain circumstances, more computable than Livsic's.

### October 4

#### Some thoughts on Livsic's theorem II

Using our more computable Livsic-type characteristic function from the first talk, we examine when two symmetric Toeplitz operators are unitarily equivalent.

### October 18

#### When is the C*-algebra generated by a C*-algebra and a group of unitary operators isomorphic to a crossed product?

If A is a C*-algebra of bounded operators on a Hilbert space, G is a discrete, amenable group of unitary operators on the same space, and UAU*=A for all U in G, then one can easily construct a C*-dynamical system related to C*(A, G), and there exists a *-homomorphism from the associated crossed product onto C*(A, G). In this talk, we will consider the question "Under what conditions is this map a *-isomorphism?" This question has been extensively studied in the literature on singular integral operators. We will review the known results and also discuss how the crossed product structure can be used to investigate the invertibility of elements in C*(A, G) via the trajectorial approach.

### October 25

#### Introduction to the Cuntz semigroup

The Cuntz semigroup has become an important tool in the classification of simple C*-algebras. For example, it was recently used to distinguish two separable simple nuclear unital C*-algebras with the same classification data (in the sense of Elliott). In this talk, we will discuss some elementary properties of the Cuntz semigroup and other basic notions from the classification program.

### November 1

#### Choquet boundaries, from classical to noncommutative

The Choquet boundary of a unital function space was defined in 1959 by Bishop and de Leeuw. Both its existence and applications rely on convexity results like the Krein-Milman theorem (1940) and Choquet's theorem (1956).

In 1969 Arveson imitated this construction for a unital operator space S, which should be thought of as a "noncommutative unital function space." But there were no noncommutative convexity results to rely on, and no one could show that the "noncommutative Choquet boundary" was robust enough to determine S.

Arveson's analogue for a point in the classical Choquet boundary is a certain kind of irreducible representation. Dropping the irreducibility requirement, the theory has more limited scope... but at least it was shown to work(!) by Dritschel-McCullough (2005), based on a noncommutative convexity result of Agler (1988). Then Arveson (2008) used disintegration techniques to prove that a separable S is determined by its noncommutative Choquet boundary.

Still the analogues of many classical theorems are not known. My talk includes background for Craig Kleski's VOTCAM lecture this Saturday, in which he will explain his 2011 result that (in the separable setting) the noncommutative Choquet boundary is a norm-attaining set of representations of S.

### November 8

#### M-ideals in Banach spaces: application to lifting problems

We are going to discuss relations in C*-algebras and their lifting properties. We will show that some techniques from geometry of Banach spaces might be useful for solving lifting problems. In particular we are going to discuss the question about lifting of nilpotents posed by Loring and Pedersen.

### November 15

#### Dynamics of linear operators

We will introduce the class of linear operators that have n-weakly dense orbits or scaled orbits. We will discuss some examples both in finite and infinite dimensions, and state some open questions.

### November 29

#### Amenable groups and quasidiagonal C*-algebras

Halmos introduced the notion of a quasidiagonal operator on a Hilbert space, which is a block diagonal operator plus a compact operator. Halmos's definition transfers naturally to the C*-algebra setting. Despite the simple definition of quasidiagonal C*-algebras, the class is far from well-understood. In this talk, we'll discuss the relationship between the amenability of a discrete group and the quasidiagonality of its reduced C*-algebra. In particular we'll talk about some quantitative non-quasidiagonality results for the free group as well as some amenable groups whose reduced C*-algebras fail to be strongly quasidiagonal.