[Paper Review] Connes-amenability and normal, virtual diagonals for measure algebras, II

Volker Runde|arXiv (Cornell University)|Nov 20, 2001

Advanced Operator Algebra Research18 references4 citations

TL;DR

This paper establishes the equivalence of three properties for the measure algebra $M(G)$ of a locally compact group $G$: amenability of $G$, Connes-amenability of $M(G)$, and the existence of a normal, virtual diagonal in $M(G)$. Using a net of positive functions in $L^1(G)$ satisfying a uniform approximation condition, the authors construct a limit point in the dual of a space of continuous functions vanishing at infinity, which serves as a normal, virtual diagonal, thereby proving the equivalence via ultralimits and weak*-continuity arguments.

ABSTRACT

We prove that the following are equivalent for a locally compact group $G$: (i) $G$ is amenable; (ii) $M(G)$ is Connes-amenable; (iii) $M(G)$ has a normal, virtual diagonal.

Motivation & Objective

• To resolve the open question of whether Connes-amenability and the existence of a normal, virtual diagonal are equivalent for general dual Banach algebras, focusing on measure algebras $M(G)$.
• To extend previous partial results by proving that for $M(G)$, Connes-amenability and the existence of a normal, virtual diagonal are equivalent precisely when $G$ is amenable.
• To construct a normal, virtual diagonal for $M(G)$ using a net of $L^1$-functions satisfying a uniform approximation condition related to amenability.
• To establish a complete characterization of $M(G)$'s structural properties in terms of group amenability, using weak*-topological and duality techniques.

Proposed method

• Construct a net $(m_α)$ in $M(G \times G)$ using positive $L^1$-functions $(f_\alpha)$ satisfying the amenability condition $\|\delta_x * f_\alpha - f_\alpha\| \to 0$ uniformly on compact sets.
• Define a limit element $\rm M$ in the dual of $\operatorname{\cal SC}_0(G \times G)$ via an ultralimit over the net $(m_\alpha)$, ensuring weak*-continuity.
• Verify that $\Delta \rm M = \delta_e$ by showing the evaluation of $\rm M$ on the multiplication map yields the Dirac measure at identity.
• Prove that $\rm M$ is bimodule-invariant by showing $\|\mu \cdot \rm M - \rm M \cdot \mu\| \to 0$ for all $\mu \in M(G)$, using Fubini's theorem and norm estimates.
• Use the inner regularity of $|\mu|$ and the uniform convergence condition $\| (\delta_y \otimes \delta_e) * m_\alpha - m_\alpha * (\delta_e \otimes \delta_y) \| \to 0$ to conclude the bimodule invariance.
• Leverage the identification of $\mathcal{L}^{2}_{w^*}(M(G), \mathbb{C})^*$ with $\operatorname{\cal SC}_0(G \times G)$ to embed the construction into the correct dual space.

Experimental results

Research questions

• RQ1 Is Connes-amenability of $M(G)$ equivalent to the existence of a normal, virtual diagonal for all locally compact groups $G$?
• RQ2 Can a normal, virtual diagonal be explicitly constructed for $M(G)$ when $G$ is amenable?
• RQ3 Does the existence of a net of $L^1$-functions satisfying the uniform approximation condition $\|\delta_x * f_\alpha - f_\alpha\| \to 0$ on compact sets imply the existence of a normal, virtual diagonal?
• RQ4 Is the converse true: does the existence of a normal, virtual diagonal in $M(G)$ imply amenability of $G$?
• RQ5 What is the role of weak*-continuity and duality in characterizing derivations and bimodule structures for measure algebras?

Key findings

• The existence of a normal, virtual diagonal in $M(G)$ is equivalent to the amenability of $G$.
• Connes-amenability of $M(G)$ is equivalent to the amenability of $G$, extending prior results for discrete and compact groups.
• A normal, virtual diagonal for $M(G)$ is constructed as an ultralimit of measures $m_\alpha$ derived from $L^1$-nets satisfying the amenability condition.
• The construction ensures that $\Delta \rm M = \delta_e$, verifying the required property of a virtual diagonal.
• The bimodule invariance $\mu \cdot \rm M = \rm M \cdot \mu$ holds in the limit, as the norm difference tends to zero due to uniform convergence on compact sets.
• The proof establishes that $M(G)$ is Connes-amenable if and only if $G$ is amenable, completing the equivalence chain for measure algebras.