Topological groups, Lie groups Definition

Group C*-algebra

Builds on Locally compact group Group algebra of a locally compact group Unitary representation

Definition

Let $G$ be a locally compact group with left Haar measure $\mu$ , and let $L^1(G)$ be its group algebra. For a unitary representation $\pi$ of $G$ on a Hilbert space $H$ and $f \in L^1(G)$ , let $\pi(f)$ be the bounded operator on $H$ given by $\ip{\pi(f)u}{v} = \int_G f(x) \ip{\pi(x)u}{v} \, d\mu(x) \quad (u, v \in H).$ For $f \in L^1(G)$ put $\norm{f}_* = \sup_\pi \norm{\pi(f)},$ where $\pi$ runs over all unitary representations of $G$ . The group C*-algebra $C^*(G)$ is the completion of $L^1(G)$ for the norm $\norm{\cdot}_*$ . The reduced group C*-algebra $C^*_r(G)$ is the closure of $\set{\lambda(f) : f \in L^1(G)}$ in the bounded operators on $L^2(G)$ , where $\lambda$ is the left regular representation.

Remarks

The supremum is finite, since $\norm{\pi(f)} \le \norm{f}_1$ , and $\norm{\cdot}_*$ is a norm on $L^1(G)$ , since the left regular representation is injective on $L^1(G)$ . Both $C^*(G)$ and $C^*_r(G)$ are C*-algebras.

Example: for $G = \Z$ , the Fourier transform $f \mapsto \sum_n f(n) z^n$ identifies $C^*(\Z)$ and $C^*_r(\Z)$ with the algebra $C(\mathbb{T})$ of continuous functions on the circle with the supremum norm.

Non-example: $L^1(G)$ with its own norm $\norm{\cdot}_1$ is in general not a C*-algebra. In $\ell^1(\Z)$ , let $f = \delta_0 + \delta_1 - \delta_2$ . Then $f^* * f = 3\delta_0 - \delta_2 - \delta_{-2}$ , so $\norm{f^* * f}_1 = 5$ , while $\norm{f}_1^2 = 9$ , and the C*-identity $\norm{f^* f} = \norm{f}^2$ fails.

There is always a surjective homomorphism $C^*(G) \to C^*_r(G)$ ; it is an isomorphism exactly when $G$ is amenable.

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Group algebra of a locally compact group concept
Locally compact group concept
Unitary representation concept
Group C*-algebra definition

Tag	`7PY20WMZDZ`
URI	`m2://7PY20WMZDZ`
Kind	definition
Writing	W0 stub
MSC	22D25, 46L05, 22D15
Defines	Group C*-algebra
Uses	Locally compact group, Group algebra of a locally compact group, Unitary representation
Changed	2 November 2026

Group C*-algebra

Definition

Remarks

Reading path

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