\(C^*\)-algebras generated by subnormal operators. (English) Zbl 0665.47022
Let H be a separable complex Hilbert space, and, as usually, let B(H) be the algebra of all bounded linear operators on H.
The author considers the \(C^*\)-algebra \(C^*(S)\) generated by an irreducible, essentially normal, subnormal operator \(S\in B(H)\), and he denotes a minimal normal extension of S by N. In the first section the author characterizes the \(C^*\)-algebra \(C^*(S)\) and determines the generators of \(C^*(S)\). Using the theory of L. G. Brown, R. G. Douglas and P. A. Fillmore [Lect. Notes Math. 345, 58-128 (1973; Zbl 0277.46053)] the author states a necessary and sufficient condition on an operator \(A\in B(H)\) in order that the \(C^*\)-algebra \(C^*(A)\) generated by A is spatially \({}^*\)-isomorphic to \(C^*(S)\), i.e. \(C^*(A)\cong C^*(S)\), this is the case if and only if the following conditions on \(A\in B(H)\) holds
i) A is an irreducible, essentially normal operator on H;
ii) for every continuous function f on the spectrum \(\sigma\) (N), i.e. \(f\in C^ 0(\sigma (N))\), such that the restriction of f to the essential spectrum \(\sigma_ e(S)\) of S is one-to-one, one has (a) \(\sigma_ e(A)=f(\sigma_ e(S))\), and (b) index (A-\(\lambda)=index (f(S)-\lambda)\) for every \(\lambda \in C\setminus \sigma_ e(A).\)
The authors’ next step is the computation of the essential spectrum \(\sigma_ e(S)\) and the index of f(S) for every \(f\in C^ 0(\sigma (N))\) with \(f|_{\sigma_ e(S)}\) is one-to-one. Some enlighting examples of irreducible, subnormal operators S with a prescribed spectral picture and the \(C^*\)-algebras \(C^*(S)\) generated by them are given. These examples leads the author to a classification of the \(C^*\)- algebras \(C^*(A)\) generated by irreducible, essentially normal operators A with essential spectrum \(\sigma_ e(A)\) equal to a finite union of disjoint Jordan curves and index (A-\(\lambda)\)\(\neq 0\) \((\lambda \in C\setminus \sigma_ e(A)\) suitable).
Finally the author considers two operators \(A,B\in B(H)\) which are similar (resp. quasi-similar) to each other. It is well-known, that in general the similarity (resp. the quasi-similarity) do not lead to any relation between the \(C^*\)-algebras \(C^*(A)\) generated by A and the \(C^*\)-algebra \(C^*(B)\) generated by B, but in an earlier work by J. B. Conway and the author [(*), Trans. Am. Math. Soc. 284, 153-161 (1984; Zbl 0583.47027)] some surprising results contrary to the facts last mentioned are obtained for hyponormal or subnormal operators A, B. These results are extended to quasi-similar cyclic subnormal operators A, B, in this case one has \(C^*(A)\cong C^*(B)!\) Last but not least a conjecture of C. R. Putnam [J. Oper. Theory 11, 243-254 (1984; Zbl 0538.47016)] on hyponormal operators with rank 1 self-commutators, which was proved in (*), is extended and also proved.
The author considers the \(C^*\)-algebra \(C^*(S)\) generated by an irreducible, essentially normal, subnormal operator \(S\in B(H)\), and he denotes a minimal normal extension of S by N. In the first section the author characterizes the \(C^*\)-algebra \(C^*(S)\) and determines the generators of \(C^*(S)\). Using the theory of L. G. Brown, R. G. Douglas and P. A. Fillmore [Lect. Notes Math. 345, 58-128 (1973; Zbl 0277.46053)] the author states a necessary and sufficient condition on an operator \(A\in B(H)\) in order that the \(C^*\)-algebra \(C^*(A)\) generated by A is spatially \({}^*\)-isomorphic to \(C^*(S)\), i.e. \(C^*(A)\cong C^*(S)\), this is the case if and only if the following conditions on \(A\in B(H)\) holds
i) A is an irreducible, essentially normal operator on H;
ii) for every continuous function f on the spectrum \(\sigma\) (N), i.e. \(f\in C^ 0(\sigma (N))\), such that the restriction of f to the essential spectrum \(\sigma_ e(S)\) of S is one-to-one, one has (a) \(\sigma_ e(A)=f(\sigma_ e(S))\), and (b) index (A-\(\lambda)=index (f(S)-\lambda)\) for every \(\lambda \in C\setminus \sigma_ e(A).\)
The authors’ next step is the computation of the essential spectrum \(\sigma_ e(S)\) and the index of f(S) for every \(f\in C^ 0(\sigma (N))\) with \(f|_{\sigma_ e(S)}\) is one-to-one. Some enlighting examples of irreducible, subnormal operators S with a prescribed spectral picture and the \(C^*\)-algebras \(C^*(S)\) generated by them are given. These examples leads the author to a classification of the \(C^*\)- algebras \(C^*(A)\) generated by irreducible, essentially normal operators A with essential spectrum \(\sigma_ e(A)\) equal to a finite union of disjoint Jordan curves and index (A-\(\lambda)\)\(\neq 0\) \((\lambda \in C\setminus \sigma_ e(A)\) suitable).
Finally the author considers two operators \(A,B\in B(H)\) which are similar (resp. quasi-similar) to each other. It is well-known, that in general the similarity (resp. the quasi-similarity) do not lead to any relation between the \(C^*\)-algebras \(C^*(A)\) generated by A and the \(C^*\)-algebra \(C^*(B)\) generated by B, but in an earlier work by J. B. Conway and the author [(*), Trans. Am. Math. Soc. 284, 153-161 (1984; Zbl 0583.47027)] some surprising results contrary to the facts last mentioned are obtained for hyponormal or subnormal operators A, B. These results are extended to quasi-similar cyclic subnormal operators A, B, in this case one has \(C^*(A)\cong C^*(B)!\) Last but not least a conjecture of C. R. Putnam [J. Oper. Theory 11, 243-254 (1984; Zbl 0538.47016)] on hyponormal operators with rank 1 self-commutators, which was proved in (*), is extended and also proved.
Reviewer: U.Grimmer
Keywords:
\(C^*\)-algebra generated by an irreducible, essentially normal, subnormal operator; algebra of all bounded linear operators; minimal normal extension; essentially normal operator; prescribed spectral picture; hyponormal operators with rank 1 self-commutatorsReferences:
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