On optimality of constants in the Little Grothendieck Theorem

Ondřej F. K. Kalenda, Antonio M. Peralta, Hermann Pfitzner Studia Mathematica MSC: 46L70, 47A30, 17C65. DOI: 10.4064/sm201125-23-8 Published online: 22 November 2021

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.

Authors

  • Ondřej F. K. KalendaDepartment of Mathematical Analysis
    Faculty of Mathematics and Physics
    Charles University
    Sokolovská 86
    186 75 Praha 8, Czech Republic
    e-mail
  • Antonio M. PeraltaInstituto de Matemáticas de la
    Universidad de Granada (IMAG)
    Departamento de Análisis Matemático
    Facultad de Ciencias
    Universidad de Granada
    18071 Granada, Spain
    e-mail
  • Hermann PfitznerInstitut Denis Poisson
    Université d’Orléans
    Université de Tours
    Rue de Chartres, BP 6759
    F-45067 Orléans Cedex 2, France
    e-mail

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