Rich lattices of multiplier topologies

Each symmetrically-normed ideal \(\mathcal{I}\) $\mathcal{I}$ of compact operators on a Hilbert space \(H\) $H$ induces a multiplier topology \(\mu^*_{\mathcal{I}}\) ${\mu }_{\mathcal{I}}^{\ast }$ on the algebra \(\mathcal{B}(H)\) $\mathcal{B}(H)$ 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 \(\mathcal{L}^{p,q}\) ${\mathcal{L}}^{p,q}$ interpolating between the ideals of trace-class and compact operators. This gives a totally ordered chain of distinct topologies \(\mu^*_{p,q\mid 0}\) ${\mu }_{p,q\mid 0}^{\ast }$ on \(\mathcal{B}(H)\) $\mathcal{B}(H)$ , with \(\mu^*_{2,2\mid 0}\) ${\mu }_{2,2\mid 0}^{\ast }$ being the \(\sigma \mbox{-}strong^*\) $\sigma \text{-}stron{g}^{\ast }$ topology and \(\mu^*_{\infty,\infty\mid 0}\) ${\mu }_{\infty ,\infty \mid 0}^{\ast }$ the strict/Mackey topology. In particular, the latter are only two of a natural continuous family.