bibkey: blecher2025commutativity
authors: David P. Blecher
year: 2025
title: "Commutativity of operator algebras"
doi: 10.1016/j.exmath.2025.125742
url: "https://arxiv.org/abs/2509.17673v2"
claim: "The paper studies reversible operator algebras, those whose reversed multiplication with the same matrix norms is again an abstract operator algebra, shows that they are 3-commutative, constructs an anticommuting noncommutative reversible subalgebra of M_4 from the canonical anticommutation relations, and conjectures that up to unitary equivalence it is the only noncommutative reversible 4×4 matrix algebra."
strata_touched:
  - D5/S3/HilbertOperators/ReversibleFourByFourNonuniqueness
license: citation-only
triage: anchor

Blecher, Commutativity of operator algebras

D. P. Blecher, Expositiones Mathematicae 43(6) (2025), DOI 10.1016/j.exmath.2025.125742; arXiv:2509.17673v2. Quotations are from the arXiv v2 TeX source.

An abstract operator algebra is reversible if it “with reversed multiplication (but with same matrix norms) is also an abstract operator algebra”. Example 5.2 takes $U=e_{13}+e_{24}$ and $V=e_{12}-e_{34}$ in $M_4$, which satisfy the canonical anticommutation relations, and $A=\mathrm{Span}\{U,V,UV\}$; $A$ is anticommuting, hence reversible, and noncommutative. The example ends:

We conjecture that up to unitary equivalence $A$ is the only noncommutative reversible $4\times 4$ matrix algebra.

Verified locator

Source: https://arxiv.org/abs/2509.17673v2

• DOI: https://doi.org/10.1016/j.exmath.2025.125742 (journal version; title, author, journal, volume and issue from Crossref).
• URL: https://arxiv.org/abs/2509.17673v2. The e-print member 3commfarx.tex has 1118 lines and md5 af6ec2d293b318abc844f9be27f97a3e. Locations:
  • the definition of reversibility, l. 114 and 145;
  • Example 5.2 (the CAR algebra $A$ in $M_4$), l. 646–682;
  • the conjecture, l. 684.