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.

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.