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 $\sigma$-strong$^*$ topology and $\mu^*_{\infty,\infty\mid 0}$ the strict/Mackey topology. In particular, the latter are only two of a natural continuous family.