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

Haagerup, Uffe

Ann. Math. (2) 118, 215-240 (1983).

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

Keywords:

bounded *-representation; non-degenerate representation; finite cyclic set; similarity problem for representations of \(C^*\)-algebras; cyclic representation; \(C^*\)-algebra without tracial states; bicommutant