Rich lattices of multiplier topologies

Abstract

Each symmetrically-normed ideal \(\mathcal{I}\) of compact operators on a Hilbert space \(H\) induces a multiplier topology \(\mu^*_{\mathcal{I}}\) on the algebra \(\mathcal{B}(H)\) of bounded operators. We show that under fairly reasonable circumstances those topologies precisely reflect, strength-wise, the inclusion relations between the corresponding ideals, including the fact that the topologies are distinct when the ideals are.

Said circumstances apply, for instance, for the two-parameter chain of Lorentz ideals \(\mathcal{L}^{p,q}\) interpolating between the ideals of trace-class and compact operators. This gives a totally ordered chain of distinct topologies \(\mu^*_{p,q\mid 0}\) on \(\mathcal{B}(H)\), with \(\mu^*_{2,2\mid 0}\) being the \(\sigma \mbox{-}strong^*\) topology and \(\mu^*_{\infty,\infty\mid 0}\) the strict/Mackey topology. In particular, the latter are only two of a natural continuous family.