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The canonical map from the full to the reduced group C star algebra

Statement

Assume the Axiom of Choice. Let $G$ be an LCH group and let $λ_{G}$ be its left regular representation on $L^{2} (G)$ (Left and right regular unitary representations of an LCH group, The regular representations are unitary, strongly continuous, and the left one is faithful). The integrated form of $λ_{G}$ extends to a surjective star-homomorphism $λ_{G} : C^{*} (G) ↠ C_{r}^{*} (G),$ sending the canonical image of $f \in L^{1} (G)$ to $λ_{G} (f)$ (The full (maximal) group C star algebra, The reduced group C star algebra, Integrated forms are contractive nondegenerate star representations of L one). Conversely, for every unitary representation $π$ whose kernel contains the kernel of $λ_{G}$ on $C^{*} (G)$ , the assignment $λ_{G} (f) \mapsto π (f)$ defines a star-homomorphism $C_{r}^{*} (G) \to C_{π}^{*} (G)$ , where $C_{π}^{*} (G)$ is the norm closure of ${π (f) : f \in L^{1} (G)}$ in $B (H_{π})$ . In particular $C_{r}^{*} (G)$ is a quotient of $C^{*} (G)$ .

Facts & Assumptions

Given: AC; an LCH group $G$ ; the full and reduced group C*-algebras; the integrated form $λ_{G} (f)$ of the left regular representation; the unitary representations of $G$ and their extended representations of $C^{*} (G)$ .

[F1]

The left regular representation is a strongly continuous unitary representation, so $f \mapsto λ_{G} (f)$ is a nondegenerate star-representation of $L^{1} (G)$ (The regular representations are unitary, strongly continuous, and the left one is faithful, Integrated forms are contractive nondegenerate star representations of L one).

[F2]

Every nondegenerate star-representation of $C^{*} (G)$ restricts to a nondegenerate star-representation of $L^{1} (G)$ and conversely every unitary representation extends uniquely to $C^{*} (G)$ (Nondegenerate representations of the full group C star algebra are unitary representations).

[F3]

Star-homomorphisms between C*-algebras have closed image, and a star-homomorphism factors uniquely through any C*-quotient by an ideal contained in its kernel; injective star-homomorphisms are isometric (Quotients of C star algebras by closed two-sided ideals). Star-homomorphisms are contractive (Positive calculus and order estimates in a C star algebra).

[F4]

$C_{r}^{*} (G)$ is by definition the norm closure of ${λ_{G} (f) : f \in L^{1} (G)}$ in $B (L^{2} (G))$ (The reduced group C star algebra).

Proof

technique · direct

Given: AC, an LCH group $G$ , the integrated forms $λ_{G} (f)$ and $π (f)$ , and the C*-algebras $C^{*} (G)$ , $C_{r}^{*} (G)$ , $C_{π}^{*} (G)$ .

1.1F1F2F3F4

The integrated form of the regular representation extends to a star-homomorphism $Λ : C^{*} (G) \to B (L^{2} (G))$ with $Λ (f) = λ_{G} (f)$ on $L^{1} (G)$ , and $Λ$ is surjective onto $C_{r}^{*} (G)$ : the extension exists by the universal property [F2] applied to the unitary representation $λ_{G}$ , its image contains $λ_{G} (L^{1} (G))$ and is closed by [F3], hence contains the norm closure $C_{r}^{*} (G)$ by [F4], while by construction the image is contained in $C_{r}^{*} (G)$ .

2.1F2F3F4

Let $π$ be a unitary representation of $G$ with $\ker λ_{G} \subseteq \ker π$ (kernels of the extended representations on $C^{*} (G)$ ). The assignment $λ_{G} (f) \mapsto π (f)$ for $f \in L^{1} (G)$ is well defined and linear, multiplicative and star-preserving where defined, because $λ_{G} (f) = λ_{G} (h)$ means $f - h \in \ker λ_{G} \subseteq \ker π$ ; To justify its relative norm bound, factor the bounded extension $π$ through $Q = C^{*} (G) / \ker λ_{G}$ . The induced map $\dot{π}$ is bounded because $∥ \dot{π} (a + \ker λ_{G}) ∥ \leq ∥ π ∥ \inf_{k \in \ker λ_{G}} ∥ a + k ∥$ . The map $\dot{Λ} : Q \to C_{r}^{*} (G)$ induced by step 1.1 is injective, surjective and isometric by [F3]. Hence $T = \dot{π} \circ {\dot{Λ}}^{- 1}$ is a bounded star-homomorphism, and is contractive by the C*-quotient supplier [F3]. Its image lies in $C_{π}^{*} (G)$ by density of the integrated forms. Its restriction is exactly $λ_{G} (f) \mapsto π (f)$ , proving the assertion without inferring relative boundedness from two separate $L^{1}$ bounds.

3.1F2F3step 1.1step 2.1∎

In particular $C_{r}^{*} (G)$ is a quotient of $C^{*} (G)$ , namely the quotient by the closed two-sided ideal $\ker λ_{G}$ , and the second assertion of the statement is the factorization of any representation whose kernel contains that ideal through this quotient. The Axiom of Choice is inherited from the full-norm completion and the representation correspondence; the quotient and density arguments use no further choice (The Axiom of Choice).

Depends on

Used by

Full and reduced group C star algebras of a finite group Example

Dependency tree · two levels

61 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bachir Bekka, Pierre de la Harpe and Alain Valette, Kazhdan's Property (T) (Cambridge University Press 2008; author-hosted complete text) (standard reference, not scraped)
Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters (arXiv:1912.07262v1, 16 December 2019) (standard reference, not scraped)