Regular Operators on Hilbert C^*-modules

TL;DR

This paper investigates regular and semiregular operators on Hilbert C*-modules, establishing conditions under which semiregular operators are regular and exploring their extensions, with implications for abelian and liminal C*-algebras.

Contribution

It introduces the concept of semiregular operators on C*-algebras and proves their regularity in specific algebraic contexts, extending the understanding of operator regularity.

Findings

Semiregular operators are regular on abelian C*-algebras.

Semiregular operators are regular on subalgebras of compact operators.

Criteria for regular extensions of semiregular operators on liminal C*-algebras.

Abstract

A regular operator T on a Hilbert C^*-module is defined just like a closed operator on a Hilbert space, with the extra condition that the range of (I+T^*T) is dense. Semiregular operators are a slightly larger class of operators that may not have this property. It is shown that, like in the case of regular operators, one can, without any loss in generality, restrict oneself to semiregular operators on C^*-algebras. We then prove that for abelian C^*-algebras as well as for subalgebras of the algebra of compact operators, any closed semiregular operator is automatically regular. We also determine how a regular operator and its extensions (and restrictions) are related. Finally, using these results, we give a criterion for a semiregular operator on a liminal C^*-algebra to have a regular extension.