Involutive operator algebras

Abstract

Examples of operator algebras with involution include the operator \(*\)-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function algebras, triangular matrix algebras, (complexifications) of real operator algebras, and an operator algebraic version of the complex symmetric operators studied by Garcia, Putinar, Wogen, Zhu, and others. We investigate the general theory of involutive operator algebras, and give many applications, such as a characterization of the symmetric operator algebras introduced in the early days of operator space theory.

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Acknowledgements

This project grew out of [10], and we thank Jens Kaad and Bram Mesland for several ideas and perspectives learned there. We also thank Stephan Garcia–whose work on complex symmetric operators has influenced some results in our paper–for helpful conversations, and also Elias Katsoulis.

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Correspondence to David P. Blecher.

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Blecher, D.P., Wang, Z. Involutive operator algebras. Positivity 24, 13–53 (2020). https://doi.org/10.1007/s11117-019-00664-6

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Keywords

  • Operator algebras
  • Involution
  • Accretive operator
  • Ideal
  • Hereditary subalgebra
  • Interpolation
  • Complex symmetric operator

Mathematics Subject Classification

  • Primary 46K50
  • 46L52
  • 47L07
  • 47L30
  • 47L75
  • Secondary 32T40
  • 46J15
  • 46L07
  • 46L85
  • 47B44
  • 47L25
  • 47L45