Operator Algebras Ical Atom

Given a path of self-adjoint Fredholm operators, one can count the net number of eigenvalues which cross from negative to positive as one moves along the path. This number (an integer) is called the spectral flow. The idea of spectral flow can be generalized to semifinite von Neumann algebras by using the trace on the algebra to measure the change in the spectrum, and obtain a spectral flow which is a real number. In a different direction, the same idea can be applied to some paths of unbounded self-adjoint operators. The goal of this talk is to explain the formal definition of spectral flow, and give an overview of its connections to other ideas in mathematics.

The goal of this talk is to explain the connection of spectral flow to K-theory/K-homology, and to introduce the p-summable integral formula for spectral flow. Recall that spectral flow (introduced in the talk on Nov. 1st) measures, for a path of Breuer-Fredholm self-adjoint operators, the net amount of spectrum which crosses zero in the positive direction as you move along the path. In a specific context, the spectral flow can be used to calculate the index pairing between K-theory and K-homology. I will start with a bird’s eye view of K-theory and K-homology, leading up to the main result of the talk, which is the p-summable integral formula for spectral flow.

During the last two weeks, we discussed the definition of spectral flow and its connection to noncommutative geometry. This week, we will go over a proof of the integral formula for spectral flow which calculates the index pairing between (the equivalence classes of) a unitary and a p-summable semifinite Fredholm module.

In this talk I will describe joint work with Chris Phillips and Qingyun Wang. The weak tracial Rokhlin property for actions of discrete amenable groups on simple unital C-algebras is defined by Qingyun Wang [https://arxiv.org/abs/1410.8170]. We show that the class of simple separable unital exact C-algebras with strict comparison and almost divisible Cuntz semigroup is closed under taking crossed products by such actions. We use this to show that the class of simple separable unital nuclear $\mathcal{Z}$-stable C*-algebras is also preserved.

Examples include the non-commutative Bernoulli shift of any discrete amenable group $\Gamma$ on $\bigotimes_{\Gamma} \mathcal{Z} \cong \mathcal{Z}$ and others.

In this talk I will describe joint work with Chris Phillips and Qingyun Wang. The weak tracial Rokhlin property for actions of discrete amenable groups on simple unital C-algebras is defined by Qingyun Wang [https://arxiv.org/abs/1410.8170]. We show that the class of simple separable unital exact C-algebras with strict comparison and almost divisible Cuntz semigroup is closed under taking crossed products by such actions. We use this to show that the class of simple separable unital nuclear $\mathcal{Z}$-stable C*-algebras is also preserved.

Examples include the non-commutative Bernoulli shift of any discrete amenable group $\Gamma$ on $\bigotimes_{\Gamma} \mathcal{Z} \cong \mathcal{Z}$ and others.

An operator space is a complex vector space V together with a sequence of (complete) norms on square matrices of all sizes over V satisfying certain compatibility conditions. These conditions are due to Ruan who showed that they are necessary and sufficient for the sequence of matrix norms to be induced from a linear embedding of V as a closed subspace into the space of bounded linear operators on a Hilbert space. There are notions of completely bounded maps and complete isometries between operator spaces that correspond to bounded maps and isometries between Banach spaces. There is also a notion of the dual operator space.

There are (infinitely) many ways to extend the given norm on a Banach space to a sequence of matrix norms to obtain an operator space. In particular, there is a variety of natural operator space structures on a Hilbert space H, none of which turns out to be self dual. Pisier showed that there is a unique operator space structure on H that is self dual, i.e., such that the canonical isometry from H to the conjugate Hilbert space is a complete isometry; he called this operator space the (corresponding) operator Hilbert space, or OH.

There are two constructions of OH, a rather abstract one using complex interpolation for operator spaces and a more concrete one, using a noncommutative version of the Cauchy–Schwartz inequality that is due to Haagerup. In this talk, I will review some operator space basics, and then present a variation of the second construction that is motivated by the recent theory of completely positive noncommutative kernels (see Ball–Marx–Vinnikov, arXiv:1602.00760).

Perhaps the simplest dilation result in operator theory is the dilation of an isometry to a unitary. However, when one generalizes an isometry to a Toeplitz-Cuntz-Krieger family of a directed graph, things become much more complicated.

The analogue of a unitary operator in this case is a (full) Cuntz-Krieger family, and a result of Skalski and Zacharias on C-correspondences supplies us with a such a dilation when the graph is row-finite and sourceless. We apply Arveson’s non-commutative Choquet theory to answer this question for arbitrary graphs. We compute the non-commutative Choquet boundary of graph tensor algebras and are able to recover a result of Katsoulis and Kribs on the computation of the C-envelope of these algebras.

However, as the non-commutative Choquet boundary of the operator algebra is a more delicate information than the C*-envelope, we are able to dilate any TCK family to a (full) CK family. In fact, we are able to make progress on a decade old problem of Skalski and Zacharias, that asks for the multivariable analogue, generalizing a result of Ito’s dilation theorem. More precisely, we are able to show that TCK families of graphs $G_1,…,G_d$ that commute according to a higher rank row-finite sourceless directed graph have CK dilations that still commute according to the higher rank graph structure.

By a flow I mean a one-parameter point-norm continuous group of automorphisms of a C*-algebra. In 1996, Kishimoto introduced a concept of the Rokhlin property for flows, which is analogous to the Rokhlin property for a single automorphism. I’ll discuss a generalization of this, Rokhlin dimension. The results parallel to a great extent results previously obtained in the discrete settings for actions of Z, Z^n, finite groups and certain residually finite groups.

The main results are that crossed products by flows with finite Rokhlin dimension preserve finite nuclear dimension and D-absorption (the latter with an additional technical assumption), crossed products by flows with finite Rokhlin dimension are stable, and any free flow on a commutative C*-algebra with a finite dimensional spectrum has finite Rokhlin dimension. In particular, this shows that crossed products of commutative algebras with finite dimensional spectrum by minimal flows fall under the Elliott’s classification program, provided the have non-zero projections (which follows, e.g., if the flow has a transversal).

This is joint work with Szabo, Winter and Wu, to appear in Comm. Math. Phys.

The context for this series of talks is the topic of an action $\alpha$ of a locally compact abelian group $G$ on a C*-algebra $A$, and the resulting properties of the crossed product of $A$ by $G$. In the first two talks, we will lead up to the definition of the strong Connes spectrum of the action $\alpha$, and discuss elements of the multiplier algebra of the crossed product which are integrable with respect to the dual action.

The goal for the final talk on this topic is to discuss the proof of the following result:

Theorem (Kishimoto, 1980): Suppose $(A, G, \alpha)$ is a C*-dynamical system, where $G$ is a locally compact abelian group. Then the cross product $A \rtimes_\alpha G$ is simple if and only if $A$ is $\alpha$-simple and the strong Connes spectrum of $\alpha$ is equal to the dual group of $G$.

The main references are the following two articles:

• Olesen & Pedersen - Applications of the Connes spectrum to C*-dynamical systems (1978)
• Kishimoto - Simple crossed products of C*-algebras by locally compact abelian groups (1980).