On two methods for investigation of invertibility of operators of \(C^*\)-algebras, generated by dynamical systems. (Russian) Zbl 0566.47026

Let B be a \(C^*\)-algebra, G a group, \(T_ g\) a unitary representation of G in B and A a \(C^*\)-subalgebra of B. The algebra B is called of type \(C^*(A,G,T_ g)\) if \(T_ gAT_ g^{-1}=A\) for \(g\in G\) and the set of all finite sums \(\sum a_ gT_ g\) \((a_ g\in A)\) is dense in B. The author studies the case when A is isomorphic to the \(C^*\)-algebra HOM(E) of homomorphisms of an n-dimensional complex vector-space bundle E over a compact space M. let \(B^ i\) be of type \(C^*(A^ i,G,T^ i_ g).\) He proves that under some conditions, an isomorphism \(\phi: A^ 1\to A^ 2\) admits an extension to an isomorphism of \(B^ 1\) onto \(B^ 2\). He also characterizes the invertibility of operators \(a_ 0+a_ 1T\) in algebras of type \(C^*(A,T)\) by means of hyperbolic linear extensions.
Reviewer: D.Petz


47C15 Linear operators in \(C^*\)- or von Neumann algebras
46L55 Noncommutative dynamical systems
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