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$^*$-triples and JB$^*$-algebras. In our main result we prove that for each bounded linear operator $T$ from a JB$^*$-algebra $B$ into a complex Hilbert space $H$ and $\varepsilon>0$, there is a norm-one functional $\varphi\in B^*$ such that $$\|Tx\|\le(\sqrt{2}+\varepsilon)\|T\|\|x\|_\varphi\quad\mbox{ for }x\in B.$$ The constant appearing in this theorem improves the best value known up to date (even for C$^*$-algebras). We also present an easy example witnessing that the constant cannot be strictly smaller than $\sqrt2$, hence our main theorem is `asymptotically optimal'. For type I JBW$^*$-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.