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Locally adjointable operators on Hilbert $C^*$-modules

Denis Fufaev, Evgenij Troitsky

Abstract

In the theory of Hilbert $C^*$-modules over a $C^*$-algebra $A$ (in contrast with the theory of Hilbert spaces) not each bounded operator ($A$-homomorphism) admits an adjoint. The interplay between the sets of adjointable and non-adjointable operators plays a very important role in the theory. We study an intermediate notion of locally adjointable operator $F:M \to N$, i.e. such an operator that $F\circ g$ is adjointable for any adjointable $g: A \to M$. We have introduced this notion recently and it has demonstrated its usefulness in the context of theory of uniform structures on Hilbert $C^*$-modules. In the present paper we obtain an explicit description of locally adjointable operators in important cases.

Locally adjointable operators on Hilbert $C^*$-modules

Abstract

In the theory of Hilbert -modules over a -algebra (in contrast with the theory of Hilbert spaces) not each bounded operator (-homomorphism) admits an adjoint. The interplay between the sets of adjointable and non-adjointable operators plays a very important role in the theory. We study an intermediate notion of locally adjointable operator , i.e. such an operator that is adjointable for any adjointable . We have introduced this notion recently and it has demonstrated its usefulness in the context of theory of uniform structures on Hilbert -modules. In the present paper we obtain an explicit description of locally adjointable operators in important cases.
Paper Structure (8 theorems, 21 equations)

This paper contains 8 theorems, 21 equations.

Key Result

Lemma 6

A bounded ${\mathcal{A}}$-morphism $F:{\mathcal{K}} \to {\mathcal{N}}$ of Hilbert $C^*$-modules is adjointable if and only if, for any $y \in {\mathcal{N}}$, the morphism $F_y:{\mathcal{K}} \to {\mathcal{A}}$, $F_y(x)=\langle y, F(x)\rangle$ is adjointable.

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 6
  • proof
  • Corollary 7
  • Theorem 8
  • proof
  • ...and 9 more