@misc{10481/71999,
year = {2021},
month = {8},
url = {http://hdl.handle.net/10481/71999},
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.},
organization = {Spanish Ministry of Science, Innovation and Universities (MICINN)},
organization = {European Commission	PGC2018-093332-B-I00},
organization = {IMAG -Maria de Maeztu grant	CEX2020-001105-M/AEI/10.13039/501100011033},
organization = {Junta de Andalucia	FQM375	
A-FQM-242-UGR18},
publisher = {Instytut Matematyczny},
keywords = {Non-communicative little Grothendieck inequality},
keywords = {JB∗-algebra},
keywords = {JB∗- triple},
keywords = {Best constants},
title = {On optimality of constants in the Little Grothendieck Theorem},
author = {Kalenda, Ondrej F. K. and Peralta Pereira, Antonio Miguel and Pfitzner, Hermann},
}