Rich lattices of multiplier topologies

Each symmetrically-normed ideal I of compact operators on a Hilbert space H induces a multiplier topology μ_I^∗ on the algebra 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 L^p,q interpolating between the ideals of trace-class and compact operators. This gives a totally ordered chain of distinct topologies μ_p,q∣0^∗ on B(H), with μ_2,2∣0^∗ being the σ-strong^∗ topology and μ_∞,∞∣0^∗ the strict/Mackey topology. In particular, the latter are only two of a natural continuous family.