On optimality of constants in the Little Grothendieck Theorem

We explore the optimality of the constants making valid the recently established Little Grothendieck inequality for JB${}^{\ast }$-triples and JB${}^{\ast }$-algebras. In our main result we prove that for each bounded linear operator $T$ from a JB${}^{\ast }$-algebra $B$ into a complex Hilbert space $H$ and $\epsilon >0$, there is a norm-one functional $\phi \in B^{\ast }$ such that $$\mathrm{\parallel }Tx\mathrm{\parallel }\le (\sqrt{2}+\epsilon )\mathrm{\parallel }T\mathrm{\parallel }\mathrm{\parallel }x{\mathrm{\parallel }}_{\phi }forx\in B\mathrm{.}$$ The constant appearing in this theorem improves the best value known up to date (even for C${}^{\ast }$-algebras). We also present an easy example witnessing that the constant cannot be strictly smaller than $\sqrt{2}$, hence our main theorem is `asymptotically optimal’. For type I JBW${}^{\ast }$-algebras we establish a canonical decomposition of normal functionals which may be used to prove the main result in this special case and also seems to be of an independent interest. As a tool we prove a measurable version of the Schmidt representation of compact operators on a Hilbert space.