On optimality of constants in the Little Grothendieck Theorem

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 epsilon > 0, there is a norm-one functional phi is an element of B* such that

parallel to Tx parallel to <= (root 2 + epsilon)parallel to T parallel to parallel to x parallel to(phi) for x is an element of 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 root 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.