Rich lattices of multiplier topologies

Each symmetrically-normed ideal $\mathcal{I}$ of compact operators on a Hilbert space $H$ induces a multiplier topology ${\mu }_{\mathcal{I}}^{\ast }$ 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 ${\mu }_{p,q\mid 0}^{\ast }$ on $\mathcal{B}(H)$, with ${\mu }_{2,2\mid 0}^{\ast }$ being the $\sigma$-strong${}^{\ast }$ topology and ${\mu }_{\mathrm{\infty },\mathrm{\infty }\mid 0}^{\ast }$ the strict/Mackey topology. In particular, the latter are only two of a natural continuous family.