In the early '90s, Elliott conjectured that separable simple nuclear C*-algebras are classified up to isomorphism by their K-theory groups and traces, in analogy with the Connes--Haagerup classification of separably acting injective factors by their type and flow of weights. In the last few years, Elliott's classification program has been completed under the two additional hypotheses: Z-stability and the UCT. In the von Neumann algebraic setting, the classification of injective factors was followed by a delicate analysis of the symmetries of such factors. For example, Oceanu proved every amenable group admits a outer action on the hyperfinite II_1 factor R, which is unique up to cocycle conjugacy, and Popa and Takesaki proved the automorphism group of R is contractable. In the spirit of the latter result, I will discuss recent joint work with Jamie Gabe on the homotopy type of automorphism groups of ``classifiable'' C*-algebras.
Homotopy groups of automorphism groups of classifiable C*-algebras. Sponsored by the Meyer Fund
Oct. 27, 2022 3:35pm (MATH 3…
Probability
Benedek Valko (University of Wisconsin Madison)
X
If one removes the first row and column of an n by n random Haar unitary matrix then the eigenvalues of the resulting matrix are inside the unit disk. Zyczkowski and Sommers (2000) derived the joint pdf of these eigenvalues, and showed that they form a determinantal process. Killip and Kozhan (2017) extended this result to circular beta ensembles (with beta=2 corresponding to the unitary case), describing the eigenvalues of the truncated model via a random recursion. (For general beta the resulting ensemble is not determinantal.) We derive the point process limit of the truncated ensemble together with the scaling limit of the normalized characteristic polynomials. The limiting objects are closely connected to the random analytic function appearing as the limit of the normalized characteristic polynomials of the (full) circular beta ensemble.
Joint with Yun Li (Tsinghua University, Beijing)
Scaling limits of the truncated circular beta ensemble