On transitive algebras containing a standard finite von Neumann subalgebra

Abstract

Let M be a finite von Neumann algebra acting on a Hilbert space H and A be a transitive algebra containing M . In this paper we prove that if A is 2-fold transitive, then A is strongly dense in B(H). This implies that if a transitive algebra containing a standard finite von Neumann algebra (in the sense of [U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975) 271-283]) is 2-fold transitive, then A is strongly dense in B(H). Non-selfadjoint algebras related to free products of finite von Neumann algebras, e.g., LF n and (M 2 (C), 1 2 Tr) * (M 2 (C), 1 2 Tr), are studied. Brown measures of certain operators in (M 2 (C), 1 2 Tr) * (M 2 (C), 1 2 Tr) are explicitly computed.

On density of transitive operator algebras

“…Fang, Hadwin and Ravichandran[4] obtained a general result containing our result.Proposition 3.5. (See [4].)…”

“…The author thanks the referee for his valuable comments on this paper and informing us the work by Fang, Hadwin and Ravichandran [4]. We would like to thank Professor L. Ge for his many fruitful suggestions and discussion on the transitive algebra question.…”

“…There have been many attempts to solve the invariant subspace problem relative to von Neumann algebras including Arveson [1], Fang, Hadwin and Ravichandran [4], Haagerup and Schultz [6], Pearcy and Salinas [9], Radjavi and Rosenthal [10]. In particular, Haagerup and Schultz [6] recently showed that if M is a II 1 -factor and if for any operator T ∈ M, its Brown's spectral distribution measure is not concentrated in one point, then T has a non-trivial closed invariant subspace affiliated with M. That is, there is a projection P in M such that P = 0, I and P T P = T P .…”

“…In particular, Haagerup and Schultz [6] recently showed that if M is a II 1 -factor and if for any operator T ∈ M, its Brown's spectral distribution measure is not concentrated in one point, then T has a non-trivial closed invariant subspace affiliated with M. That is, there is a projection P in M such that P = 0, I and P T P = T P . Recently, Fang, Hadwin and Ravichandran [4] gave a positive answer to the transitive algebra question in cases where a transitive algebra containing a finite W * -algebra is 2-fold transitive.…”

Transitive algebras and reductive algebras on reproducing analytic Hilbert spaces

“…This problem was firstly explicitly stated by Arveson [4], and it remains open up to now. The basic results in [4] led to research on the transitive algebra problem by a number of authors [1,3,7,16,20,23]. Recently, some interesting progress on the connection with other topics in operator theory has been made in [6].…”