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Progress in noncommutative function theory

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Paul S. Muhly

  2. Department of Mathematics, Technion, Haifa, 32000, Israel

    Baruch Solel

Authors
  1. Paul S. Muhly
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  2. Baruch Solel
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Correspondence to Paul S. Muhly.

Additional information

To our esteemed friend and teacher, Richard V. Kadison, on the happy occasion of his 85th birthday

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Muhly, P.S., Solel, B. Progress in noncommutative function theory. Sci. China Math. 54, 2275–2294 (2011). https://doi.org/10.1007/s11425-011-4241-6

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  • DOI: https://doi.org/10.1007/s11425-011-4241-6

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