K-Theory of C*-Algebras

2024/2025

Academic Year

ENG

Instruction in English

3

ECTS credits

Delivered at:

Faculty of Mathematics

Course type:

Optional course (faculty)

When:

3, 4 module

Instructor

Pirkovskii, Alexei Y.

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.