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**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).

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 |