Linear Analysis Seminar

I'll discuss a strengthening of the completely positive approximation property for nuclear C*-algebras and how it relates to notions of topological covering dimension and near inclusions of nuclear C*-algebras. This is joint work with Ilan Hirshberg and Eberhard Kirchberg.

A distribution of a random variable is a probability measure, which in the bounded case can be identified with a positive functional on polynomials. Let B be a C*-algebra. A distribution of a B-valued random variable is a completely positive B-bimodule map μ from the algebra of non-commutative polynomials with B coefficients to B. Just like in the scalar case, there is an operation of free convolution on such distributions, and in particular one can talk about free convolution powers μ^t for real positive t. We show that, in the operator-valued case, one can more generally define μ^η for a c.p. map η from B to B. If μ^t exists for all positive t, then μ^η automatically exist for all c.p. η. For general μ, the convolution powers exists whenever (η - 1) is c.p.

This is joint work with Serban Belinschi, Maxime Fevrier, and Andu Nica. I will start by defining scalar version of all the objects, so only minimal background should be required.

In 1972 Kadison and Kastler introduced a notion of closeness of operator algebras in terms of the Hausdorff distance between their unit balls. They raised the question of whether two close algebras are isomorphic, or even unitarily equvalent. In the context of von Neumann algebras, we say that M is KK-stable if it is isomorphic to any suitably close algebra. KK-stability is known for all amenable von Neumann algebras, a result due to Christensen. In this talk I will describe the first known classes of nonamenable KK-stable von Neumann algebras. These arise as crossed products by certain nonamenable groups. The talk will be aimed at a general audience. This is joint work with Jan Cameron, Erik Christensen, Allan Sinclair, Stuart White and Alan Wiggins.

The Cuntz semigroup of a C*-algebra has been successfully used in various questions on the structure and classification of C*-algebras. In spite of this, the Cuntz semigroup has some limitations as a tool to study C*-algebras. For example, it is trivial for all purely infinite simple C*-algebras. The root of the problem lies on the Cuntz comparison of positive elements, which is used to define the Cuntz semigroup. I will discuss some variations on the Cuntz comparison relation aimed at expanding its usefulness.

We examine the amalgamated free product of hyperfinite von Neumann algebras over multimatrix subalgebras. Using the concept of standard embeddings we are able to show these are composed of direct sums interpolated free group factors and hyperfinite algebras, and we show that this class is closed under this type of amalgamated free product. We also show that the free dimension satisfies the equation fdim(A∗_DB)=fdim(A)+fdim(B)-fdim(D).

We define a Riesz type interpolation property for the Cuntz semigroup of a C*-algebra and prove it is satisfied by the Cuntz semigroup of every C*-algebra with the ideal property. Related to this, we obtain two characterizations of the ideal property in terms of the Cuntz semigroup of the C*-algebra. Some additional characterizations are proved in the special case of the stable, purely infinite C*-algebras, and two of them are expressed in language of the Cuntz semigroup. We introduce a notion of comparison of positive elements for every unital C*-algebra that has (normalized) quasitraces. We prove that large classes of C*-algebras (including large classes of AH algebras) with the ideal property have this comparison property. This is joint work with Francesc Perera.