K-Theory of C*-Algebras

2024/2025

Учебный год

ENG

Обучение ведется на английском языке

3

Кредиты

Кто читает:

Факультет математики

Статус:

Дисциплина общефакультетского пула

Когда читается:

3, 4 модуль

Преподаватель

Пирковский Алексей Юльевич

Course Syllabus

Abstract

$K$-theory of $C^*$-algebras appeared in the 1970ies as 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, they contain much information about $A$, and on the other hand, there are powerful tools to explicitly calculate them. If $A=C(X)$, the algebra of continuous functions on a compact Hausdorff 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 embedded into $K$-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 calculate the $K$-groups in some important cases.