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The complex vector space \(\mathbb C[\Gamma ]\) consists of finitely supported functions \(a:\Gamma \to \mathbb C\), with pointwise vector operations. Write \(\operatorname {supp}(a)=\{ g:a(g)\ne 0\} \) and let \(\delta _g\) be \(1\) at \(g\) and \(0\) elsewhere. Define

These sums are finite. The proposed multiplicative identity is \(\delta _e\).

Define the complex-linear map

This is a finite sum of bounded operators, and \(\rho (\delta _g)=\lambda _g\).

For a nonnegative family, define its sum as the supremum of its finite partial sums. Put

with pointwise vector operations and \(\langle \xi ,\eta \rangle =\sum _h\overline{\xi (h)}\eta (h)\). The notation \(\delta _g\) also denotes the corresponding vector in \(\ell ^2(\Gamma )\); the ambient space determines its meaning.

The left regular representation is the group homomorphism \(\lambda :\Gamma \to U(\ell ^2(\Gamma ))\), \(g\mapsto \lambda _g\), where \(U(H)\) denotes the unitary operators on \(H\). It is distinct from the linear extension defined below.

For \(g\in \Gamma \), prescribe

The next lemma verifies that this defines an operator on \(\ell ^2(\Gamma )\).

Define

with inherited operations and norm. The closure is in the operator norm, not the weak or strong operator topology.

For \(a\in \mathbb C[\Gamma ]\), put \(\| a\| _r=\| \rho (a)\| \).

Convolution and involution preserve finite support, with

Moreover,

For \(a\in \mathbb C[\Gamma ]\), \(\xi \in \ell ^2(\Gamma )\) and \(h\in \Gamma \),

Each \(\lambda _g\) is a complex-linear isometry, and

where the final norm is the operator norm.

The operations make \(\mathbb C[\Gamma ]\) a unital complex \(*\)-algebra: convolution is associative and bilinear, its identity is \(\delta _e\), and

Regard \(a\in \mathbb C[\Gamma ]\) also as a finitely supported \(\ell ^2\)-vector. Then

In particular, \(\rho \) is injective.

The map \(\rho \) is a unital complex \(*\)-algebra homomorphism:

The inner product is well defined, and \(\ell ^2(\Gamma )\) is a complex Hilbert space. The delta vectors are orthonormal and have dense finite linear span. For every \(h\), \(|\xi (h)|\leq \| \xi \| _2\) and \(\langle \delta _h,\xi \rangle =\xi (h)\).

Each \(\lambda _g\) is unitary, with \(\lambda _g^{-1}=\lambda _{g^{-1}}=\lambda _g^*\).

With codomain restricted to \(C_r^*(\Gamma )\), \(\rho \) is an isometric unital \(*\)-homomorphism for \(\| \cdot \| _r\) and has dense range. It extends uniquely to an isometric \(*\)-isomorphism from the norm completion of \((\mathbb C[\Gamma ],\| \cdot \| _r)\) onto \(C_r^*(\Gamma )\).

The algebra \(C_r^*(\Gamma )\) is the norm closure of \(\operatorname {span}_{\mathbb C}\{ \lambda _g:g\in \Gamma \} \). It is the smallest norm-closed \(*\)-subalgebra of \(\mathcal B(\ell ^2(\Gamma ))\) containing every \(\lambda _g\).

If \(\Gamma \) is countable, then \(C_r^*(\Gamma )\) is separable.

The inherited operations make \(C_r^*(\Gamma )\) a unital complex \(C^*\)-algebra with identity \(I=\lambda _e\).

For the one-element group, \(z\mapsto zI\) is an isometric unital \(*\)-isomorphism \(\mathbb C\to C_r^*(\{ e\} )\).

The reduced norm is a norm, with

Completeness of \(\mathbb C[\Gamma ]\) in this norm is not asserted.