K-Theory of C*-Algebras

2024/2025

Academic Year

ENG

Instruction in English

3

ECTS credits

Delivered by:

Faculty of Mathematics

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.

Learning Objectives

• -

Expected Learning Outcomes

• ---

Course Contents

• Basic facts on 𝐶∗-algebras (a survey).
• Equivalence relations for projections. The group 𝐾0(𝐴). Remarks on the commutative case (vector bundles, the Serre – Swan theorem, topological 𝐾-theory).
• Homotopy invariance, half-exactness, and stability of 𝐾0.
• Equivalence of unitaries. The group 𝐾1(𝐴).
• The index map in 𝐾-theory. A relation to the Fredholm index. The exact sequence of 𝐾-groups induced by a 𝐶∗-algebra extension.
• The Toeplitz algebra. The Bott periodicity.
• Inductive limits of 𝐶∗-algebras. The continuity of 𝐾0. The order structure on 𝐾0(𝐴). AF-algebras and their Bratteli diagrams. Elliott’s classification of AF-algebras in terms of 𝐾-theory.

Assessment Elements

• Midterm
• Exam

Interim Assessment

• 2024/2025 4th module
  Final grade = 0.3 × (midterm grade) + 0.7 × (exam grade). Both the midterm and the final exam will have the form of written take-home individual assignments. You will have appr. a week for preparing your solutions.

Bibliography

Recommended Core Bibliography

• Algebraic K-theory. Vol.1: Higher K-theories, , 1973
• Algebraic K-theory. Vol.2: "Classical" algebraic K-theory, and connections with arithmetic, , 1973
• Algebraic K-theory. Vol.3: Hermitian K-theory and geometric applications, , 1973

Recommended Additional Bibliography

• K-theory : lectures, Atiyah, M. F., 1989

Authors

• Иконописцева Юлия Вахтаногвна
• PIRKOVSKIY ALEKSEY YULEVICH