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.