The Hoffman-Rossi theorem for operator algebras

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.