Sur la simplicité essentielle du groupe des inversibles et du groupe unitaire dans une \(C^*\)-algèbre simple. (French) Zbl 0573.46033

De la Harpe, Pierre; Skandalis, Georges

J. Funct. Anal. 62, 354-378 (1985).

Let A be a simple approximately finite-dimensional \(C^*\)-algebra with unit, let \(GL_ 1(A)\) be the group of invertible elements and let \(U_ 1(A)\) be that of unitaries in A. In a previous work the abelianized groups \(GL_ 1(A)/DGL_ 1(A)\) and \(U_ 1(A)/DU_ 1(A)\) have been described with Grothendieck’s group \(K_ 0(A)\) viewed as a dimension group for A [See Ann. Inst. Fourier 34, No.1, 241-260 and No.4, 169-202 (1984; Zbl 0534.46036 and Zbl 0548.46044)]. Here, it is shown that the groups \(DGL_ 1(A)\) and \(DU_ 1(A)\) are simple up to their centres. There are K-theoretic corollaries.

MSC:

46L05	General theory of \(C^*\)-algebras
46L40	Automorphisms of selfadjoint operator algebras
46M20	Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)

Keywords:

approximately finite-dimensional \(C^*\)-algebra with unit; group of invertible elements; Grothendieck’s group; dimension group

Citations:

Zbl 0534.46036; Zbl 0548.46044

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