Purdue OA Seminar

Title: The Rank Ratio Function and the Radius of Comparison Function

Abstract: In this talk, for any given Cuntz semigroup, we introduce an associated function, called the Rank Ratio Function. This function plays a very important role in computing the relative radii of comparison of Cuntz semigroups. Time allowing, we will discuss both the C*-algebraic and algebraic aspects of the relative radius of comparison, and introduce another function called the Radius of Comparison Function.

This function provides a method for constructing non-classifiable C*-algebras with distinct relative radii of comparison. 

Title: C*-algebras over the circle: some classification and limitation

Abstract: We first review the traditional Elliott classification program for AT-algebras, i.e. C*-algebras that are obtained as inductive limits of building blocks over C(T). After recalling earliest satisfactory results in both the simple and real rank zero case, we turn to more recent developments involving the Cuntz semigroup and its refined versions. 

In particular, we discuss the classification of *-homomorphisms from C(T) into various codomains, and highlight current limitation to complete the classification of these C*-algebras.

Title: Projective representations of almost unimodular groups

Abstract: Given a locally compact group $G$ with a 2-cocycle $\omega: G\times G\to \mathbb{T}$, Colin Sutherland showed that any left Haar measure uniquely determines a faithful normal semifinite weight on the associated twisted group von Neumann algebra. This weight, which we call the twisted Plancherel weight, is tracial if and only if $G$ is unimodular, and for countable discrete groups it is the usual tracial state. In the setting of non-unimodular groups, the modular automorphism group of the necessarily non-tracial twisted Plancherel weight is explicitly determined by the so-called modular function of $G$. The twisted group von Neumann algebra is generated by the left regular $\omega$-projective representation of $G$. In 1958, George Mackey showed that $\omega$-projective representations of $G$ are connected to the representations of the central extension of $G$ by $\mathbb{T}$, when $G$ is second countable. In this talk, we will introduce the class of "almost unimodular groups" for which the twisted Plancherel weight is almost periodic, in the sense of Connes from 1972.  We will also give some examples of such groups admitting a 2-cocycle such that the group von Neumann algebras are purely infinite and not factors, but the twisted group von Neumann algebras are semifinite factors. 

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