Venue
Online (Google Meet)
Speaker
Anshu
Affiliation
IISER Bhopal
Title
Homotopical Stable Ranks of $\mathrm{C}^*$-algebras
Abstract- The goal of this talk is to discuss the connected stable rank and general stable rank, which are collectively known as homotopical stable ranks of a $\mathrm{C}^*$-algebra. The notion of homotopical stable ranks was introduced by Marc Rieffel in the pursuit of understanding the stability properties of $\mathrm{C}^*$-algebras. After Rieffel's foundational work, many ranks have been associated with $\mathrm{C}^*$-algebras, real rank, tracial rank, and decomposition rank, to name a few. The study of these and many more ranks associated with a $\mathrm{C}^*$-algebra is referred to as "noncommutative dimension theory".
In the first part of the talk, we will define the connected stable rank and general stable rank. We will then discuss some basic examples and properties of these ranks. We will also discuss the relationship between the homotopical stable ranks of a $\mathrm{C}^*$-algebra $A$ and its $K$-theory. Then we will move on to give brief descriptions of a $C(X)$-algebra and a crossed product $\mathrm{C}^*$-algebra by a finite group. In the later part of the talk, we will provide estimates of the connected stable rank for upper semicontinuous $C(X)$-algebras and crossed product $\mathrm{C}^*$-algebras by finite groups. We will talk about a condition on the finite group actions, which is called the Rokhlin property. We will also prove that if $A$ has any of the homotopical stable rank one, then the crossed product $\mathrm{C}^*$-algebra by an action with the Rokhlin property also has the corresponding homotopical stable rank one.
In the first part of the talk, we will define the connected stable rank and general stable rank. We will then discuss some basic examples and properties of these ranks. We will also discuss the relationship between the homotopical stable ranks of a $\mathrm{C}^*$-algebra $A$ and its $K$-theory. Then we will move on to give brief descriptions of a $C(X)$-algebra and a crossed product $\mathrm{C}^*$-algebra by a finite group. In the later part of the talk, we will provide estimates of the connected stable rank for upper semicontinuous $C(X)$-algebras and crossed product $\mathrm{C}^*$-algebras by finite groups. We will talk about a condition on the finite group actions, which is called the Rokhlin property. We will also prove that if $A$ has any of the homotopical stable rank one, then the crossed product $\mathrm{C}^*$-algebra by an action with the Rokhlin property also has the corresponding homotopical stable rank one.