tailieunhanh - Báo cáo toán học: "Orthogonal pairs of *-subalgebras in finite von Neumann algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: *- Subalgebras trực giao của các đồng nghiệp trong giới hạn von Neumann đại số. | J. OPERATOR THEORY 9 1983 253 268 Copyright by INCREST 1983- ORTHOGONAL PAIRS OF -SUBALGEBRAS IN FINITE VON NEUMANN ALGEBRAS SORIN POPA 1. INTRODUCTION Let M be a finite von Neumann algebra with a finite faithful normal trace T t 1 1. Let Blt B c M be von Neumann subalgebras of M. Bỵ is orthogonal to B2 if 0 for all bỵ G Bị b2 G B2 T bỵ b2 0. Starting from this notion we present in this paper a method of computing the von Neumann algebra generated by the normalizer of certain subalgebras in some type IIX factors. Let B c M be a von Neumann subalgebra B its normalizer in M. In order to compute B it is enough to compute its orthogonal in M that is the set of all XE M such that r x ố 0 for b G B . To this end we first show the following technical result if B has no atoms and u is a unitary element in M such that uBu is orthogonal to B then u is orthogonal to B . Consequently to determine B it is sufficient to find enough such unitary elements in M. This method is particularly useful in the case when M is the von Neumann algebra L G associated to the left regular representation of a discrete group G. Most of the paper contains applications of this method. We show that the hyperfinite IIX factor R has a proper subfactor Ro such that any maximal abelian -subalgebra abbreviated as MASA of Ro is maximal abelian in R. This gives a negative answer to the type IIX case of a problem of Kadison cf. 8 Problem 12 for the case when M is of type III see 16 . Let s be an arbitrary nonempty set and 5 0 c s a nonempty subset of S . Denote by Fs the free group with generators in S . Then L Fs0 is naturally embedded as a subalgebra in FS . We show that if B is an arbitrary completely nonatomic subalgebra of L F 0 then the normalizer of B in L FS is contained in L F 0 . In particular this shows that any MASA of L F is maximal abelian in L FS . It also shows that the commutant in L FS of any completely nonatomic von Neumann subalgebra is separable. Thus if s is uncountable then L FS is .