Isomorphisms, Moduli, and Cohomological Dimension for Twisted Triangular Banach Algebras
Sara Behnamian, Fatemeh Fogh
TL;DR
This work introduces twisted triangular Banach algebras $\mathcal{T}_\sigma(A,B;X)$ built from diago nal blocks $A,B$, a Banach $A$-$B$ bimodule $X$, and automorphisms $\sigma=(\sigma_A,\sigma_B)$, and analyzes both their isomorphism classes and homological properties. The authors establish a complete isomorphism classification: $\mathcal{T}_\sigma(A,B;X)\cong\mathcal{T}_\tau(A,B;X)$ iff there exist $\alpha\in\mathrm{Aut}(A)$, $\beta\in\mathrm{Aut}(B)$, an $(\alpha,\beta)$-bimodule isomorphism $u$, and a $\tau$-cocycle $\theta$ satisfying diagonal conjugacy, bimodule equivariance, and a cocycle identity; transport-of-structure shows how $\tau$-cocycles relate to $\sigma$-cocycles. Cohomologically, the square-zero extension yields sharp bounds for the Hochschild cohomological dimension: $$\max\{\mathrm{cd}(A),\mathrm{cd}(B), \mathbf{1}_{\{X\neq0\}}\} \le \mathrm{cd}(\mathcal{T}_\sigma) \le \max\{\mathrm{cd}(A),\mathrm{cd}(B), 1\},$$ with equality in key acyclic cases, and a precise formula $\mathrm{cd}(\mathcal{T}_\sigma)=\max\{\mathrm{cd}(A),\mathrm{cd}(B), \mathbf{1}_{\{X\neq 0\}}\}$ in general. Amenability is constrained: $\mathcal{T}_\sigma$ is amenable only in the degenerate case $X=0$ with amenable diagonals; however, twisting introduces obstructions to weak amenability even when the diagonals are amenable, and a twisted version of the Forrest–Marcoux criterion provides a positive path to weak amenability under vanishing twisted first cohomology. In group-algebra and commutative settings the results reveal new dynamical invariants and explicit shear-cohomology behaviour, clarifying how automorphism dynamics and cocycle data shape the extension theory. Overall, the paper develops a coherent extension-theoretic framework for twisted triangular Banach algebras and identifies both obstructions and conditions under which homological properties survive twisting, with clear directions for further study in noncommutative cases and explicit computation of shear cohomology.
Abstract
We introduce and study twisted triangular Banach algebras T_sigma(A,B;X), built from Banach algebras A,B, a Banach A-B bimodule X, and a pair of automorphisms sigma=(sigma_A,sigma_B). This construction extends the classical triangular framework by incorporating twisted module actions on the off-diagonal block. We obtain a complete isomorphism classification: T_sigma is isomorphic to T_tau precisely when the diagonal twists are conjugate, the bimodule admits an (alpha,beta)-equivariant isomorphism, and a shear cocycle satisfies a natural identity. In the case of group algebras, the classification detects conjugacy classes inside Aut(G), yielding new dynamical invariants absent from the untwisted setting. On the homological side, we establish sharp bounds for the Hochschild cohomological dimension and deduce that T_sigma is amenable only when X=0 and both A,B are amenable. Thus twisting enriches the classical theory while preserving extension-theoretic control of cohomology.