2024/2025

K-Theory of C*-Algebras

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 3, 4 module

Open to: students of all HSE University campuses

Instructors: Alexei Yu. Pirkovskii

Language: English

ECTS credits: 3

Course Syllabus

Abstract

$K$-theory of $C^*$-algebras appeared in the 1970iesas a noncommutative counterpart of Atiyah-Hirzebruch topological $K$-theory.In some sense, this theory may be viewed as ``algebraic topology for $C^*$-algebras''.$K$-theory naturally associates two abelian groups, $K_0(A)$ and $K_1(A)$, to every$C^*$-algebra $A$. These groups are quite important invariants of $A$.On the one hand, theycontain much information about $A$, and on the other hand, there are powerful toolsto explicitly calculate them.If $A=C(X)$, the algebra of continuous functions on a compactHausdorff topological space $X$, then $K_0(A)$ and $K_1(A)$ are just the topological $K$-groups$K^0(X)$ and $K^1(X)$, respectively. Thus topological $K$-theory is fully embeddedinto $K$-theory of $C^*$-algebras. A number of fundamental results in topological $K$-theory,including the Bott periodicity, have natural extensions to $C^*$-algebras.At the same time, $K$-theory of $C^*$-algebras has some interesting``purely noncommutative'' properties, which do not have classical prototypes.In this course we define $K$-theory for $C^*$-algebras,prove its basic properties (including the Bott periodicity theorem), and calculatethe $K$-groups in some important cases.