Locally adjointable operators on Hilbert $C^{\ast }$-modules

Abstract

In the theory of Hilbert $C^{\ast }$-modules over a $C^{\ast }$-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 $C^{\ast }$-modules. In the present paper we obtain an explicit description of locally adjointable operators in important cases.

6 pages