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On classifying monotone complete algebras of operators

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Abstract

We give a classification of “small” monotone complete C *-algebras by order properties. We construct a corresponding semigroup. This classification filters out von Neumann algebras; they are mapped to the zero of the classifying semigroup. We show that there are 2c distinct equivalence classes (where c is the cardinality of the continuum). This remains true when the classification is restricted to special classes of monotone complete C *-algebras e.g. factors, injective factors, injective operator systems and commutative algebras which are subalgebras of ℓ∞. Some examples and applications are given.

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Authors and Affiliations

  1. 2-7-5 Yoshinari, Aoba-ku, Sendai, 989-3205, Japan

    Kazuyuki Saitô

  2. Mathematical Sciences, University of Aberdeen, Aberdeen, AB24 3UE, Scotland, UK

    J. D. Maitland Wright

  3. Christ Church, University of Oxford, Oxford, OX1 1DP, England, UK

    J. D. Maitland Wright

Authors
  1. Kazuyuki Saitô
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  2. J. D. Maitland Wright
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Correspondence to J. D. Maitland Wright.

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Communicated by P. de Lucia.

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Saitô, K., Maitland Wright, J.D. On classifying monotone complete algebras of operators. Ricerche mat. 56, 321–355 (2007). https://doi.org/10.1007/s11587-007-0021-6

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  • DOI: https://doi.org/10.1007/s11587-007-0021-6

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