Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các sản phẩm của phép đo quang phổ. | J. OPERATOR THEORY 6 1981 363-374 Copyright by INCREST 1981 STONE-WEIERSTRASS THEOREMS FOR SEPARABLE C -ALGEBRAS JOEL ANDERSON and JOHN w. BUNCE 1. INTRODUCTION Suppose A is a c -algebra and B is a c -subalgebra of A. In this context the classical commutative Stone-Weierstrass theorem asserts that if B separates the pure states of A then B A. The main results of this paper are as follows. Assume A is separable and unital. 1 If B separates the pure states of A and for each state on B tĩị B contains a regular maximal abelian subalgebra then B A. 2 If B separates the factor states of A and each factor state on B extends to a factor state on A then B A. See 1 4 7 9 10 and 12 for other Stone-Weierstrass theorems for non-commutative c -algebras. If A is a nonunital c -algebra let A denote the c -algebra obtained by adjoining an identity to A. As noted in 12 if B separates the pure states of A and zero then C B 1 separates the pure states of A. We may thus assume that A has a unit. Moreover if B separates the pure states of A then B contains the unit of A 12 Lemma 1 . We therefore assume throughout that A is unital and 1e B A. The paper is organized as follows. In Section 2 we collect some results that will be needed in the sequel. Section 3 contains the main results and in Section 4 we present some miscellaneous related facts. If A is a c -algebra then we use S 4 P A and F A to denote the states on A the pure states on A and the factor states on A respectively. If f e S l then Ttf ye J and 1 y denote the cyclic representation arising from the Hilbert space on which Ttf A acts and the cannonical cyclic vector. If s is a set of operators on a Hilbert space then WZ S denotes the von Neumann algebra generated by S and if 7 is a vector then Sty denotes the closed subspace generated by elements of s acting on t . Finally maximal abelian subalgebras of von Neumann algebras are always assumed to be self-adjoint. 364 JOEL ANDERSON and JOHN w. BUNCE 2. PRELIMINARY RESULTS Probably
