Commutators in selfless C$^\ast$-algebras

This note concerns bounded derivations on maximal triangular operator algebras on a Hilbert space. Given any bounded derivation $δ$ on a maximal triangular algebra whose invariant lattice is continuous at 1, an operator which is shown to implement $δ$ is constructed explicitly. For a general reducible maximal triangular algebra the same construction yields an operator which is shown to implement any $δ$, if and only if $δ$ obeys an additional triple product rule. This work is based on unpublished parts of the author's dissertation [19] and describes a variant of the proof of a more general result in [21] and is thus in effect a footnote to that work. To the best of the author's knowledge the constructive proof and triple product rule have not appeared elsewhere. The work here is not set in the context of the large body of subsequent research, and no claims are made regarding its relationship to later developments.

math.OA↗

arXiv2026

An Approximate-Diagonal Characterization of Amenable Actions on C*Algebras

We introduce relative $h$-diagonals for crossed products $A\rtimes_αΓ$ by discrete groups, which are positive-form approximate diagonals in the Haagerup tensor product taken relative to $A$. We show that $α$ is amenable if and only if $A\rtimes_αΓ$ admits a relative $h$-diagonal, an analogue of Johnson's theorem for $L^{1}(G)$.

math.OA↗

arXiv2026

Failure of noncommutative Rota's theorem with $L\log L$ endpoint

We construct a factorizable Markov map $T$ on a hyperfinite finite von Neumann algebra and a positive element $x\in L\log L$ for which the sequence $T^n(T^*)^n x$ does not converge bilaterally almost uniformly. This answers a question of Anantharaman-Delaroche \cite{AD2006}.

math.OA↗