Lifting units in self-injective rings and an index theory for Rickart C^∗-algebras

Pere Menal and Juame Moncasi

Vol. 126 (1987), No. 2, 295–329

DOI: 10.2140/pjm.1987.126.295

Abstract

In this paper we study the following question: If R is a right self-injective ring and I an ideal of R, when can the units of R∕I be lifted to units of R?

We answer this question in terms of K₀(I). For a purely infinite regular right self-injective ring R we obtain an isomorphism between K₁(R∕I) and K₀(I) which can be viewed as an analogue of the index map for Fredholm operators.

By giving a purely algebraic description of the connecting map K₁(A∕I) → K₀(I) in the case where A is a Rickart C^∗-algebra, we are able to extend the classical index theory to Rickart C^∗-algebras in a way which also includes Breuer’s theory for W^∗-algebras.

Mathematical Subject Classification 2000

Primary: 18F25

Secondary: 19A99, 19K56, 46C05, 16A54

Milestones

Received: 29 November 1984

Revised: 10 December 1985

Published: 1 February 1987