Selfless C*-correspondences, operator valued C*-probability spaces and completely positive maps

Abstract

We develop a general theory of selflessness for C*-correspondences, with several applications. Specializing this theory to completely positive maps, in particular to conditional expectations, gives rise to a novel notion of relative selflessness. Among applications, the machinery developed here yields new examples of selfless C*-algebras, for instance among minimal tensor products and reduced crossed products, new examples of MF C*-algebras arising as reduced amalgamated free products, a conceptually new proof of Kirchberg's $\mathcal{O}_{\infty}$-absorption theorem, and the lack of ``phantom'' traces on ultrapowers.

Disclosure

“ng from crossed products. Acknowledgments. We are grateful to the Brin Mathematical Research Center for hosting the workshop “Recent developments in operator algebras” organized by the third author, wherein this work began. AI statement. ChatGPT was used for English language editing, proofreading, and grammatical corrections. All mathematical content was generated solely by the human authors. 2. Preliminaries Throughout, all C*-algebras are assu”

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2026-08-23T19:01:59.577474Z

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