## 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

### MSC:

 47C15 Linear operators in $$C^*$$- or von Neumann algebras 46L55 Noncommutative dynamical systems
Full Text: