On optimality of constants in the Little Grothendieck Theorem Kalenda, Ondrej F. K. Peralta Pereira, Antonio Miguel Pfitzner, Hermann Non-communicative little Grothendieck inequality JB∗-algebra JB∗- triple Best constants A. M. Peralta was partially supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European Regional Development Fund project no. PGC2018-093332-B-I00, the IMAG -Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033, and by Junta de Andalucia grants FQM375 and A-FQM-242-UGR18. 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. 2021-12-10T13:15:49Z 2021-12-10T13:15:49Z 2021-08-22 journal article Published version: Kalenda, O. F., Peralta, A. M., & Pfitzner, H. (2021). On optimality of constants in the Little Grothendieck theorem. Studia Mathematica (), 1-42. DOI: [10.4064/sm201125-23-8] http://hdl.handle.net/10481/71999 eng http://creativecommons.org/licenses/by-nc-nd/3.0/es/ open access Atribución-NoComercial-SinDerivadas 3.0 España Instytut Matematyczny