Homotopy theory of C*-algebras

TL;DR

This paper develops a comprehensive homotopy theory framework for C*-algebras, introducing model structures, stabilization techniques, and new invariants, bridging classical homotopy and noncommutative geometry.

Contribution

It constructs both unstable and stable homotopy theories for C*-algebras, incorporating homotopy techniques and defining new K-theory for these algebras.

Findings

Established a model structure for C*-algebra homotopy

Developed stable homotopy categories and spectra for C*-algebras

Introduced new invariants such as stable homotopy groups and noncommutative motives

Abstract

In this work we construct from ground up a homotopy theory of C*-algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C*-algebras. The spaces in C*-homotopy theory are certain hybrids of functors represented by C*-algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C*-algebra circle of complex-valued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory, we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C*-homotopy theory. The stable homotopy category of C*-algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We work out examples related to the emerging subject of noncommutative motives and zeta functions of C*-algebras. In addition, we employ homotopy theory to define a new type of K-theory of C*-algebras.