TAILIEUCHUNG - Báo cáo toán học: "Non-self-adjoint crossed products. III: Infinite algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Không tự liên hợp vượt qua các sản phẩm. III: Infinite đại số. | J. OPERATOR THEORY 12 1984 3 22 Copyright by INCREST 1984 NON-SELF-ADJOINT CROSSED PRODUCTS. Ill INFINITE ALGEBRAS MICHAEL McASEY PAUL s. MUHLY and KICHI-SUKE SAITO 1. INTRODUCTION Let M be a von Neumann algebra let a be a -automorphism of M and let M xa z be the von Neumann algebra crossed product determined by M and a. In this work which is a continuation of 16 and 17 we investigate the structure of a certain non-self-adjoint subalgebra of iWxaZ which we call a non-self-adjoint crossed product and which we denote by AfxaZ . Precise definitions are given in the next section. In 16 and 17 we assumed that M was finite and that a preserves a faithful normal trace. Here M may be arbitrary and we assume only that a fixes a faithful normal state on M. Our primary objective is to describe the invariant subspace structure of MxxZ and to exploit some of its consequences. We view MxxZ in standard form we identify the underlying Hilbert space with Haagerup s ZAspace see 8 and 27 L2 and we identify M Xx z with a von Neumann algebra of operators acting on the left of L2. The commutant of M Xx z then is identified with a von Neumann algebra 91 of operators acting on the right of L2. In this identification M Xx z is denoted by fl . There is a special subspace H2 of L2 which stands in the same relation to L2 as the classical Hardy space on the unit disc stands in relation to Ư of the circle. Our main result Theorem asserts that if fixes the finite central projections of M elementwise then every subspace 90Ĩ of L2 that is invariant under and contains no reducing subspace for may be written SƠĨ 7 PH2 where Rv is a partial isometry in 91. Conversely if each subspace of L2 that is invariant for and contains no reducing subspace for has the form 7 H2 for a suitable partial isometry in 9Ĩ then a fixes the finite central projections in M elementwise. Thus an exact analogue of Beurling s theorem as extended by Lax and Halmos is true for M Xx z precisely when a fixes each finite .

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