tailieunhanh - Báo cáo toán học: "Stone-Weierstrass theorem for separable C*-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Stone-Weierstrass theorem for separable C*-algebras . | J. OPERATOR THEORY 6 1981 375-405 Copyright by INCREST 1981 FINITELY CONNECTED DOMAINS G REPRESENTATIONS OF Hm G AND INVARIANT SUBSPACES B. CHEVREAU c. M. PEARCY and A. L. SHIELDS 1. INTRODUCTION Let Z1 be a separable infinite dimensional complex Hilbert space and let eetye denote the algebra of all bounded linear operators on ye. Recall that if .J is a subalgebra of s ye and Ji is a subspace of ye such that 0 Ji ye and BJt c Ji for every B in J then JI is a nontrivial invariant subspace for J. If J is the algebra of all polynomials in a fixed operator A then Jể is a nontrivial invariant subspace for A and if J is the commutant of A then Jể is a nontrivial hyperinvariant subspace for A. The question whether every operator in e ye has a nontrivial invariant subspace is of course one of the most important unsolved problems in operator theory. As a result there is considerable interest in whether all operators of a given sort in have invariant subspaces. A big breakthrough in this area was made about three years ago by Scott Brown who showed in 12 that every subnormal operator in ye has invariant subspaces and simultaneously originated a technique for constructing invariant subspaces that could be applied to a much wider class of operators. Brown s pioneering work was rapidly followed by a sequence of papers exploiting this breakthrough see the attached bibliography . The present paper is another in that sequence although herein we take a somewhat different point of view. In any case this paper should be regarded as a sequel to 13 In that paper it was shown that if A is a contraction in ỉeịye with the property that the intersection of the spectrum a A of A and the open unit disc D in c is a dominating subset of D see 2 for definitions then A has invariant subspaces. One of the purposes of this paper is to generalize that theorem. We replace D by an arbitrary bounded domain G in c and we study operators A for which G is an Af-spectral set for A and ơ A n G is a .