[Paper Review] On Connes amenability of upper triangular matrix algebras

S. F. Shariati, A. Pourabbas+1 more|arXiv (Cornell University)|Jan 10, 2018

Advanced Topics in Algebra4 references3 citations

TL;DR

This paper investigates Connes amenability of upper triangular matrix algebras $UP(I,\mathcal{A})$, where $\mathcal{A}$ is a dual Banach algebra and $I$ is a totally ordered set. It establishes that $UP(I,\mathcal{A})$ is Connes amenable if and only if $I$ is a singleton and $\mathcal{A}$ is Connes amenable, using a new characterization of $\phi$-Connes amenability via a bounded net in $\mathcal{A} \hat{\otimes} \mathcal{A}$, and provides examples of $\phi$-Connes amenable algebras that are not Connes amenable.

ABSTRACT

In this paper, we study the notion of Connes amenability for a class of $I imes{I}$-upper triangular matrix algebra $UP(I,\mathcal{A})$, where $\mathcal{A}$ is a dual Banach algebra with a non-zero $wk^\ast$-continuous character and $I$ is a totally ordered set. For this purpose, we characterize the $ϕ$-Connes amenability of a dual Banach algebra $\mathcal{A}$ through the existence of a specified net in $\mathcal{A}\hat{\otimes}\mathcal{A}$, where $ϕ$ is a non-zero $wk^\ast$-continuous character. Using this, we show that $UP(I,\mathcal{A})$ is Connes amenable if and only if $I$ is singleton and $\mathcal{A}$ is Connes amenable. In addition, some examples of $ϕ$-Connes amenable dual Banach algebras, which is not Connes amenable are given.

Motivation & Objective

• To investigate Connes amenability of $I\times I$ upper triangular matrix algebras $UP(I,\mathcal{A})$ where $\mathcal{A}$ is a dual Banach algebra and $I$ is a totally ordered set.
• To characterize $\phi$-Connes amenability of a dual Banach algebra $\mathcal{A}$ through the existence of a specific bounded net in $\mathcal{A} \hat{\otimes} \mathcal{A}$, where $\phi$ is a non-zero $wk^*\!$-continuous character.
• To determine the precise conditions under which $UP(I,\mathcal{A})$ is Connes amenable, resolving a structural question in the theory of dual Banach algebras.
• To provide explicit examples of $\phi$-Connes amenable dual Banach algebras that are not Connes amenable, highlighting the distinction between these two notions.

Proposed method

• Introduces a new characterization of $\phi$-Connes amenability by linking it to the existence of a bounded net in $\mathcal{A} \hat{\otimes} \mathcal{A}$ satisfying a specific condition involving the character $\phi$.
• Uses the duality structure of $UP(I,\mathcal{A})$ by identifying it as an $\ell^1$-direct sum of copies of $\mathcal{A}$, enabling the analysis of its dual Banach algebra properties.
• Applies Runde's criterion that a dual Banach algebra is Connes amenable if and only if it admits a $\sigma wc$-virtual diagonal.
• Employs the $wk^*\!$-continuity of characters and module maps to analyze weak* topology behavior in the context of derivations and bimodules.
• Constructs explicit examples of $\phi$-Connes amenable algebras by verifying the existence of a suitable element in $\mathcal{A} \hat{\otimes} \mathcal{A}$ satisfying the $\phi$-virtual diagonal condition.
• Uses the fact that Connes amenability implies $\phi$-Connes amenability to derive necessary conditions, and then constructs counterexamples to show the converse does not hold.

Experimental results

Research questions

• RQ1 Under what conditions is the upper triangular matrix algebra $UP(I,\mathcal{A})$ Connes amenable when $\mathcal{A}$ is a dual Banach algebra and $I$ is a totally ordered set?
• RQ2 Can $\phi$-Connes amenability of a dual Banach algebra $\mathcal{A}$ be characterized via the existence of a bounded net in $\mathcal{A} \hat{\otimes} \mathcal{A}$ satisfying a specific condition involving the character $\phi$?
• RQ3 Is there a strict distinction between $\phi$-Connes amenability and Connes amenability in the class of dual Banach algebras, and if so, can it be demonstrated by explicit examples?
• RQ4 What structural properties of $I$ and $\mathcal{A}$ are necessary and sufficient for $UP(I,\mathcal{A})$ to be Connes amenable?

Key findings

• The upper triangular matrix algebra $UP(I,\mathcal{A})$ is Connes amenable if and only if $I$ is a singleton set and $\mathcal{A}$ is Connes amenable.
• A dual Banach algebra $\mathcal{A}$ is $\phi$-Connes amenable if and only if there exists a bounded net $(m_\alpha)$ in $\mathcal{A} \hat{\otimes} \mathcal{A}$ such that $a \cdot m_\alpha - m_\alpha \cdot a \to \phi(a) \cdot m_\alpha$ and $\pi_{\sigma wc}(m_\alpha) \to a$ in the weak* topology for all $a \in \mathcal{A}$, where $\phi$ is a non-zero $wk^*\!$-continuous character.
• The algebra $\mathcal{A} = \left\{ \begin{smallmatrix} \mathbb{C} & \mathbb{C} \\ 0 & 0 \end{smallmatrix} \right\}$ is $\phi$-Connes amenable but not Connes amenable, demonstrating that $\phi$-Connes amenability is strictly weaker than Connes amenability.
• The existence of a $wk^*\!$-continuous character $\phi$ on $\mathcal{A}$ is essential for the characterization of $\phi$-Connes amenability, and such characters are used to define the net condition.
• The construction of a $\sigma wc$-virtual diagonal in $\mathcal{A} \hat{\otimes} \mathcal{A}$ via the element $u = \begin{pmatrix}1&1\\0&0\end{pmatrix} \otimes \begin{pmatrix}1&1\\0&0\end{pmatrix}$ confirms $\phi$-Connes amenability of the example algebra.
• The proof shows that if $UP(I,\mathcal{A})$ were Connes amenable, then $\mathcal{A}$ would have to be isomorphic to $\mathbb{C}$, which only occurs when $I$ is a singleton, thus establishing the necessity of this condition.