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A new uniform structure for Hilbert $C^*$-modules

Denis Fufaev, Evgenij Troitsky

Abstract

We introduce and study some new uniform structures for Hilbert $C^*$-modules over an algebra $A$. In particular, we prove that in some cases they have the same totally bounded sets. To define one of them, we introduce a new class of $A$-functionals: locally adjointable functionals, which have interesting properties in this context and seem to be of independent interest. A relation between these uniform structures and the theory of $A$-compact operators is established.

A new uniform structure for Hilbert $C^*$-modules

Abstract

We introduce and study some new uniform structures for Hilbert -modules over an algebra . In particular, we prove that in some cases they have the same totally bounded sets. To define one of them, we introduce a new class of -functionals: locally adjointable functionals, which have interesting properties in this context and seem to be of independent interest. A relation between these uniform structures and the theory of -compact operators is established.
Paper Structure (6 sections, 19 theorems, 51 equations)

This paper contains 6 sections, 19 theorems, 51 equations.

Table of Contents

  1. Preliminaries and formulation of results
  2. The case of multipliers
  3. Proof of the main theorem
  4. Proof of the main theorem for ${\mathcal{N}}={\mathcal{A}}$
  5. Proof of the main theorem for ${\mathcal{N}}=\ell^2({\mathcal{A}})$
  6. Proof of the main theorem in the general case

Key Result

Lemma 1.1

Murphy For any state ${\varphi}$ on ${\mathcal{A}}$ and any $a\in{\mathcal{A}}$ one has $|{\varphi}(a)|^2\le {\varphi}(a^*a)$.

Theorems & Definitions (34)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6: Kasparov stabilization theorem
  • Definition 1.7
  • Example 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 24 more