Gysin sequences and SU(2)-symmetries of C*-algebras

TL;DR

This paper introduces SU(2)-equivariant subproduct systems of Hilbert spaces to analyze symmetries in C*-algebras, revealing their topological invariants and KK-equivalence properties, with implications for noncommutative topology.

Contribution

It develops the notion of SU(2)-equivariant subproduct systems and analyzes their associated Toeplitz and Cuntz-Pimsner algebras, providing new insights into their K-theoretic invariants.

Findings

Toeplitz algebra is KK-equivalent to complex numbers for irreducible SU(2) representations

Established a six-term exact sequence of K-groups with a noncommutative Euler class

Connected symmetries of C*-algebras with topological invariants via Kasparov theory

Abstract

Motivated by the study of symmetries of C*-algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz-Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K-theory. In particular, starting from an irreducible representation of SU(2), we show that the corresponding Toeplitz algebra is equivariantly KK-equivalent to the algebra of complex numbers. In this way, we obtain a six term exact sequence of K-groups containing a noncommutative analogue of the Euler class.