Normalisers, nest algebras and tensor products

Abstract

We show that if A is the tensor product of finitely many continuous nest algebras, B is a CDCSL algebra and A and B have the same normaliser semi-group then either A = B or A * = B.

Normalisers of operator algebras and tensor product formulas

“…Within the classes of operator algebras which admit an affirmative answer to the above question, one is able to recover (up to adjoints) non-self-adjoint operator algebras by using self-adjoint objects, namely the normaliser semigroups. It was shown in [21] that question (1.1) has an affirmative answer if A is the tensor product of finitely many continuous nest algebras and B is a CDCSL algebra. In this paper, we extend this result by showing that the answer to question (1.1) is affirmative if B is any CSL algebra (while A is still the tensor product of finitely many continuous nest algebras).…”

“…We show that the assumption on the continuity of the nest algebras cannot be omitted, settling in this way question (1.1) in the negative. After dropping the condition that B is a CDCSL algebra, we cannot assume the presence of enough Hilbert-Schmidt operators in B, which was the main ingredient in the proofs in [21]. Instead, the main tool we use here is a tensor product formula for bimodules over maximal abelian self-adjoint algebras (masas) which we believe is interesting in its own right.…”

“…It follows from Lemma 3.1 in[21] that the span of the spaces of the form EB(H)F is weak* dense in A, and hence A ⊆ [(ST )A(ST ) * : S ∈ N e (A), T ∈ N P (A)] w *…”