## Solution of the similarity problem for cyclic representations of $$C^*- algebras$$.(English)Zbl 0543.46033

This is a quite deep and engrossing paper on the famous similarity problem for representations of $$C^*$$-algebras. This problem can be stated as follows:
Is any bounded, nonself-adjoint representation $$\pi$$ of a $$C^*$$-algebra A on a Hilbert space H similar to a *-representation?
As a solution to this problem the author proves that every bounded, cyclic representation of a $$C^*$$-algebra A on a Hilbert space H is similar to a *-representation. He also further shows that if A is a $$C^*$$-algebra without tracial states then the above result also holds for non-cyclic representations. It is also established that when $$\pi$$ is a cyclic, bounded representation of a $$C^*$$-algebra A on a Hilbert space H, then the bicommutant of $$\pi$$ (A) is equal to the $$\sigma$$-weak closure of $$\pi$$ (A).
Reviewer: N.K.Thakare

### MSC:

 46L05 General theory of $$C^*$$-algebras 46K10 Representations of topological algebras with involution 46L30 States of selfadjoint operator algebras
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