Traces and symmetric linear forms

Abstract

The famous Calkin theorem says that there is a one-to-one correspondence between all ideals of operators on the separable infinite-dimensional Hilbert space and all symmetric ideals (invariant under permutations) of bounded scalar sequences: $${\mathfrak {A}} (H)\!\leftrightarrow \!{\mathfrak {a}} ({\mathbb {N}})$$ . Being unaware of my method from 1987/90 and working in the Banach space setting, F. Sukochev and his coauthors extended this relationship to a one-to-one correspondence between all continuous traces $$\tau $$ on $${\mathfrak {A}} (H)$$ and all continuous symmetric linear forms $$\psi $$ on $${\mathfrak {a}} ({\mathbb {N}})$$ . Combining this result with [4, Corollary 5.6] yields some kind of trace formulas: $$\begin{aligned} \tau (S) = \psi \big (\sigma _m (v_m|u_m) \big ), \quad \text{ where }\quad S = \sum _{m=1}^\infty \sigma _m^{\phantom {\star }}u_m^\star \otimes v_m^{\phantom {\star }}\end{aligned}$$ is a Schmidt representation of the operator $$S \!\in \!{\mathfrak {A}} (H)$$ . It will be shown that everything remains true without any topological assumption. However, more important than the generalization to the purely algebraic case is the elegance and simplicity of the new approach, which turns out to be very short and completely different from the original ones.