Normalisers of operator algebras and tensor product formulas

Abstract

We establish a tensor product formula for bimodules over maximal abelian self-adjoint algebras and their supports. We use this formula to show that if $\mathcal{A}$ is the tensor product of finitely many continuous nest algebras, $\mathcal{B}$ is a CSL algebra and $\mathcal{A}$ and $\mathcal{B}$ have the same normaliser semigroup then either $\mathcal{A}=\mathcal{B}$ or ${\mathcal{A}}^{\ast }=\mathcal{B}$. We show that the result does not hold without the assumption that the nests be continuous, answering in the negative a question previously raised in the literature.