The slice map problem for $\sigma $-weakly closed subspaces of von Neumann algebras

Abstract

A a-weakly closed subspace ` of B(' X) is said to have Propett S_ if for any a-weakly closed subspace \5 of a von Neumann algebra -I, {.x E 5 0 A: R-(,(.x) (= ' for all rp E B(0.)) = X ' l, where R , is the right slice map associated with T. It is shown that semidiscrete von Neumann algebras have Property Sa, and various stability properties of the class of a-weakly closed subspaces with Property S0 are established. It is also shown that if (?111, G, a) is a W*-dynamical system such that 'k has Property S0 and C is compact abelian, then all of the spectral subspaces associated with a have Property S0. Some applications of these results to the study of tensor products of spectral subspaces and tensor products of reflexive algebras are given. In particular, it is shown that if L , is a commutative subspace lattice with totally atomic core, and L, is an arbitrary subspace lattice, then alg(i 0 E2) algC1 0 algL?2. Introduction. Slice maps, introduced by Tomiyama, have proved to be a powerful tool in studying tensor products of C*-algebras and von Neumann algebras (see, e.g., [23,24,26-281). In this paper we use slice maps to study tensor products of a-weakly closed subspaces of von Neumann algebras, particularly tensor products of spectral subspaces, and tensor products of reflexive algebras. If ')T and 6. are von Neumann algebras, then for each qp in X*, the predual of .X, there is a unique a-weakly continuous linear map R., from 9DZ 0 9 to 6 such that R,W,(a X b) = (a, q) b (a E 91R, b E DL)l called the right slice map associated with p. Similarly, for each 4 E 9L, there is a left slice map LI, from 9DTh0 6 to 9DT. If 5 C 9Th and Y C % are a-weakly closed subspaces, and 5 8 '5 denotes the a-weak closure of the algebraic tensor product of 5 and C, it is clear that if x E 0 0 then R (x) C 9 for all p E 9<X*. If (x E 5 0 : R ,(x) Efor all qg E ' } = 5 Received by the editors February 19, 1982 and, in revised form, July 20, 1982. 1980 Mathematics Subject Classification. Primary 46L10, 46L55, 46M05, 47D25.