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Jordan norms for multilinear maps on C*-algebras and Grothendieck's inequalities

Erik Christensen

Abstract

There exists a generalization of the concept, completely bounded norm for multilinear maps on C*-algebras. We will use the word, Jordan norm, for this norm. The Jordan norm of a multilinear map is obtained via factorizations of the map, such that bounded operators and Jordan homomorphisms form a long product, as in the case of a completely bounded multilinear map. We show that any bounded bilinear form on a pair of C*-algebras is Jordan bounded and its Jordan norm is at most twice the norm.

Jordan norms for multilinear maps on C*-algebras and Grothendieck's inequalities

Abstract

There exists a generalization of the concept, completely bounded norm for multilinear maps on C*-algebras. We will use the word, Jordan norm, for this norm. The Jordan norm of a multilinear map is obtained via factorizations of the map, such that bounded operators and Jordan homomorphisms form a long product, as in the case of a completely bounded multilinear map. We show that any bounded bilinear form on a pair of C*-algebras is Jordan bounded and its Jordan norm is at most twice the norm.
Paper Structure (5 sections, 8 theorems, 40 equations)

This paper contains 5 sections, 8 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. On completely bounded maps
  3. The Jordan norms
  4. The non commutative Grothendieck inequalities
  5. Positive bilinear forms

Key Result

Theorem 1.1

There exists a positive real $K_G^{\mathbb C}$ such that for any compact Hausdorff space $\Omega$ and any bounded bilinear form $B(f,g)$ on the continuous functions on $\Omega$ there exist probability measures $\mu$ and $\nu$ on $\Omega$ such that

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 11 more