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Commutativity of operator algebras

David P. Blecher

TL;DR

This work analyzes reversibility and symmetry for operator algebras beyond the unital setting, showing that while reversibility does not always imply commutativity, several robust sufficient conditions guarantee commutativity for nonunital algebras. A central theme is the use of the injective envelope and ternary ring of operators (TRO) techniques to place A in a standard position and extract a canonical commutative structure, from which 3‑commutativity follows. The paper also furnishes compelling counterexamples based on canonical anticommutation relations (CAR) to delineate limits of existing conjectures, and develops essential-extension and Wedderburn-type results clarifying the finite-dimensional landscape. Additional sections connect these ideas to Jordan operator algebras and derive a suite of further sufficient conditions, including a structural reduction to nilpotent components. Overall, the work advances understanding of when reversible or symmetric operator algebras must be commutative and highlights rich structural phenomena in low dimensions and in essential extensions.

Abstract

We call an operator algebra A {\em reversible} if A with reversed multiplication is also an abstract operator algebra (in the modern operator space sense). This class of operator algebras is intimately related to the {\em symmetric operator algebras}: the subalgebras of B(H) on which the transpose map is a complete isometry. In previous work we studied the unital case, where reversibility is equivalent to commutativity. We give many sufficient conditions under which a nonunital reversible or symmetric operator algebra is commutative. We also give many complementary results of independent interest, and solve a few open questions from previous papers. Not every reversible or symmetric operator algebra is commutative, however we show that they all are 3-commutative. That is, order does not matter in the product of three or more elements from A. The proof of this relies on a technical analysis involving the injective envelope. Indeed nonunital algebras are often enormously more complicated than unital ones in regard to the topics we consider. On the positive side, our considerations raise very many questions even for low dimensional matrix algebras, some of which are of a computational nature and might be suitable for undergraduate research. The canonical anticommutation relations from mathematical physics play a significant role.

Commutativity of operator algebras

TL;DR

This work analyzes reversibility and symmetry for operator algebras beyond the unital setting, showing that while reversibility does not always imply commutativity, several robust sufficient conditions guarantee commutativity for nonunital algebras. A central theme is the use of the injective envelope and ternary ring of operators (TRO) techniques to place A in a standard position and extract a canonical commutative structure, from which 3‑commutativity follows. The paper also furnishes compelling counterexamples based on canonical anticommutation relations (CAR) to delineate limits of existing conjectures, and develops essential-extension and Wedderburn-type results clarifying the finite-dimensional landscape. Additional sections connect these ideas to Jordan operator algebras and derive a suite of further sufficient conditions, including a structural reduction to nilpotent components. Overall, the work advances understanding of when reversible or symmetric operator algebras must be commutative and highlights rich structural phenomena in low dimensions and in essential extensions.

Abstract

We call an operator algebra A {\em reversible} if A with reversed multiplication is also an abstract operator algebra (in the modern operator space sense). This class of operator algebras is intimately related to the {\em symmetric operator algebras}: the subalgebras of B(H) on which the transpose map is a complete isometry. In previous work we studied the unital case, where reversibility is equivalent to commutativity. We give many sufficient conditions under which a nonunital reversible or symmetric operator algebra is commutative. We also give many complementary results of independent interest, and solve a few open questions from previous papers. Not every reversible or symmetric operator algebra is commutative, however we show that they all are 3-commutative. That is, order does not matter in the product of three or more elements from A. The proof of this relies on a technical analysis involving the injective envelope. Indeed nonunital algebras are often enormously more complicated than unital ones in regard to the topics we consider. On the positive side, our considerations raise very many questions even for low dimensional matrix algebras, some of which are of a computational nature and might be suitable for undergraduate research. The canonical anticommutation relations from mathematical physics play a significant role.

Paper Structure

This paper contains 11 sections, 25 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reversible and symmetric operator algebras
  4. A 'standard position', and three-commuting algebras
  5. Counterexamples
  6. Essential extensions of operator algebras
  7. Jordan operator algebras
  8. Other sufficient conditions
  9. Wedderburn theory of finite dimensional algebras
  10. The commutator ideal and Lie ideals
  11. Ternary matrix algebras

Key Result

Theorem 2.1

Operator algebra products on an operator space $X$ are in bijective correspondence with elements $z \in {\rm Ball}(I(X))$ for which $X z^* X \subseteq X$. For such $z$ the associated operator algebra product on $X$ is $x z^* y$.

Theorems & Definitions (50)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Theorem 2.5
  • proof
  • Proposition 2.6
  • ...and 40 more