[Paper Review] Subgroups and strictly closed invariant C*-subalgebras

Pekka Salmi|arXiv (Cornell University)|Oct 25, 2011

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper establishes a one-to-one correspondence between closed subgroups of a locally compact group $G$ and strictly closed, left invariant C*-subalgebras of $\mathrm{C}_b(G)$, and for amenable $G$, between closed subgroups and strictly closed invariant C*-subalgebras of the multiplier algebra of the reduced group C*-algebra $\mathrm{C}_r^*(G)$. The key contribution is showing that the strict topology, rather than the norm topology, is the appropriate framework for such characterizations in C*-algebraic harmonic analysis.

ABSTRACT

We characterise the strictly closed left invariant C*-subalgebras of the C*-algebra C_b(G) of bounded continuous functions on a locally compact group G. On the dual side, we characterise the strictly closed invariant C*-subalgebras of the multiplier algebra of the reduced group C*-algebra C*_r(G) when G is amenable. In both cases, these C*-subalgebras correspond to closed subgroups of G.

Motivation & Objective

• To characterize strictly closed, left invariant C*-subalgebras of $\mathrm{C}_b(G)$ for a locally compact group $G$.
• To extend this characterization to the multiplier algebra of the reduced group C*-algebra $\mathrm{C}_r^*(G)$ when $G$ is amenable.
• To resolve the limitation of norm-closed subalgebras, which only correspond to compact or open subgroups, by introducing the strict topology.
• To establish a duality between closed subgroups of $G$ and strictly closed invariant C*-subalgebras in the multiplier algebra setting.

Proposed method

• Use of the strict topology on multiplier algebras, generated by seminorms $x \mapsto \|xa\|$ and $x \mapsto \|ax\|$ for $a$ in the original C*-algebra.
• Application of the strict Stone–Weierstrass theorem to show that a strictly closed subalgebra separating points must be the full algebra.
• Construction of a strictly continuous $*$-isomorphism $\pi: \mathrm{C}_b(G/H) \to \{f \in \mathrm{C}_b(G) : R_s f = f \text{ for all } s \in H\}$.
• Use of strict continuity of the left regular representation $\lambda: G \to \mathrm{M}(\mathrm{C}_r^*(G))$ and its extension to multiplier algebras.
• Leveraging the existence of bounded approximate identities in $\mathrm{A}(G)$ to extend results from compactly supported elements to general elements in the multiplier algebra.
• Proof that the support condition $\operatorname{supp}x \subseteq H$ characterizes the invariant subalgebra $X$ when $H$ is closed and $G$ is amenable.

Experimental results

Research questions

• RQ1 Which C*-subalgebras of $\mathrm{C}_b(G)$ correspond to closed subgroups of $G$ under the strict topology?
• RQ2 How do strictly closed, invariant C*-subalgebras of $\mathrm{M}(\mathrm{C}_r^*(G))$ relate to closed subgroups when $G$ is amenable?
• RQ3 Why does the norm topology fail to capture all subgroups in the C*-algebraic setting, and how does the strict topology resolve this?
• RQ4 Is the $*$-isomorphism between $\mathrm{M}(\mathrm{C}_r^*(H))$ and the subalgebra $\{x \in \mathrm{M}(\mathrm{C}_r^*(G)) : \operatorname{supp}x \subseteq H\}$ strictly closed in $\mathrm{M}(\mathrm{C}_r^*(G))$?
• RQ5 Can every functional on $\mathrm{M}(\mathrm{C}_r^*(H))$ be extended to a strictly continuous functional on $\mathrm{M}(\mathrm{C}_r^*(G))$?

Key findings

• There is a one-to-one correspondence between closed subgroups $H$ of a locally compact group $G$ and strictly closed, left invariant C*-subalgebras $X \subseteq \mathrm{C}_b(G)$, where $X = \{f \in \mathrm{C}_b(G) : R_s f = f \text{ for all } s \in H\}$.
• For amenable $G$, there is a one-to-one correspondence between closed subgroups $H$ and strictly closed, invariant C*-subalgebras $X \subseteq \mathrm{M}(\mathrm{C}_r^*(G))$, where $X = \{x \in \mathrm{M}(\mathrm{C}_r^*(G)) : \operatorname{supp}x \subseteq H\}$.
• The strict topology is essential for this duality, as norm-closed subalgebras only capture compact or open subgroups.
• The map $\pi: \mathrm{C}_b(G/H) \to \{f \in \mathrm{C}_b(G) : R_s f = f \text{ for all } s \in H\}$ is a strictly continuous $*$-isomorphism.
• The range of the embedding $\pi: \mathrm{M}(\mathrm{C}_r^*(H)) \to \mathrm{M}(\mathrm{C}_r^*(G))$ is strictly dense in the corresponding subalgebra $X$, but it is not known whether it is strictly closed.
• The failure of strict homeomorphism for $\pi$ implies that not all functionals on $\mathrm{M}(\mathrm{C}_r^*(H))$ extend strictly to $\mathrm{M}(\mathrm{C}_r^*(G))$, as shown by counterexamples in groups like the affine group on $\mathbb{R}$.