The Hoffman-Rossi theorem for operator algebras

TL;DR

This paper explores noncommutative analogs of the Hoffman-Rossi theorem within operator algebras, establishing conditions for extending homomorphisms to von Neumann algebras.

Contribution

It introduces a necessary and sufficient condition for positive weak* continuous extensions of homomorphisms in the noncommutative setting.

Findings

Established a condition on the range of homomorphisms for extension

Provided a noncommutative version of the Hoffman-Rossi theorem

Enhanced understanding of extensions in operator algebra theory

Abstract

We study possible noncommutative (operator algebra) variants of the classical Hoffman-Rossi theorem from the theory of function algebras. In particular we give a condition on the range of a contractive weak* continuous homomorphism defined on an operator algebra A, which is necessary and sufficient (in the setting we explain) for a positive weak* continuous extension to any von Neumann algebra containing A.