On optimality of constants in the Little Grothendieck Theorem
Volume 264 / 2022
Abstract
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 $\varepsilon \gt 0$, there is a norm-one functional $\varphi \in B^*$ such that $$ \|Tx\|\le (\sqrt {2}+\varepsilon )\|T\|\,\|x\|_\varphi \quad \ \text { for } x\in B. $$ 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 $\sqrt 2$, 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.
