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Progress in noncommutative function theory. (English) Zbl 1325.46001

Summary: In this expository paper, we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator valued) functions on their spaces of representations. We will show that these spaces of representations can be parameterized as unit balls of certain \(W^*\)-correspondences and the functions can be viewed as Schur class operator functions on these balls. We will provide evidence to show that the elements in these (non commutative) Hardy algebras behave very much like bounded analytic functions and the study of these algebras should be viewed as noncommutative function theory.

MSC:

46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46L08 \(C^*\)-modules
46L52 Noncommutative function spaces
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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