The Hoffman–Rossi Theorem for Operator Algebras
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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.
KeywordsOperator algebra Noncommutative function theory Extension of linear map Injective von Neumann algebra Conditional expectation
Mathematics Subject ClassificationPrimary 46A22 46L10 46L30 47L30 47L45 Secondary 46H10 46M10 46L51 46L52 47A20 47A57 47L50 47L55
Noncommutative Hoffman–Rossi theorems were a project suggested (and guided in its more technical parts, e.g. things involving von Neumann algebras) by the first author for the second author’s M.S. thesis  supervised by the third author. The present paper contains several advances made subsequent to that reference, including the main result. We also thank the referee for several comments. DB is supported by a Simons Foundation Collaboration Grant 527078.