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We explore the optimality of the constants making valid the recentlyestablished Little Grothendieck inequality for JB$^*$-triples andJB$^*$-algebras. In our main result we prove that for each bounded linearoperator $T$ 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 theconstant cannot be strictly smaller than $\sqrt2$, hence our main theorem is`asymptotically optimal'. For type I JBW$^*$-algebras we establish a canonicaldecomposition of normal functionals which may be used to prove the main resultin this special case and also seems to be of an independent interest. As a toolwe prove a measurable version of the Schmidt representation of compactoperators on a Hilbert space.

On optimality of constants in the Little Grothendieck Theorem