If \((a*b)(h)\ne 0\), some summand is nonzero, so \(g\) belongs to \(\operatorname {supp}(a)\) and \(g^{-1}h\) belongs to \(\operatorname {supp}(b)\). Thus \(h\) lies in the indicated finite product set. The other assertions follow by evaluating at each group element.

Bilinearity and conjugate-linearity follow from the defining formulas. Using the finite delta expansion, associativity reduces to \(\delta _{(gk)l}=\delta _{g(kl)}\), the identity laws reduce to \(eg=g=ge\), and reversal under involution reduces to \((gk)^{-1}=k^{-1}g^{-1}\). Inversion twice and complex conjugation twice give \((a^*)^*=a\).

Finite-sum Cauchy–Schwarz, followed by a supremum over finite sets, gives \(\sum _h|\overline{\xi (h)}\eta (h)|\leq \| \xi \| _2\| \eta \| _2\). Thus the inner-product sum is absolutely convergent. The inequality \(|z+w|^2\leq 2|z|^2+2|w|^2\) ensures closure under addition; the inner-product identities follow coordinatewise.

For completeness, a Cauchy sequence \((\xi _n)\) is Cauchy in each coordinate. Define \(\xi (h)=\lim _n\xi _n(h)\). If \(\| \xi _n-\xi _m\| _2\leq \varepsilon \) for \(n,m\geq N\), taking \(m\to \infty \) on any finite set \(F\) gives \(\sum _{h\in F}|\xi _n(h)-\xi (h)|^2\leq \varepsilon ^2\). Taking the supremum proves \(\xi -\xi _n\in \ell ^2(\Gamma )\) and \(\| \xi -\xi _n\| _2\leq \varepsilon \), hence \(\xi \in \ell ^2(\Gamma )\) and norm convergence.

The delta formulas are immediate from the inner product. For \(\varepsilon {\gt}0\), choose a finite \(F\) whose complement has \(\sum _{h\notin F}|\xi (h)|^2{\lt}\varepsilon ^2\). Then \(\sum _{h\in F}\xi (h)\delta _h\) approximates \(\xi \) within \(\varepsilon \). This argument does not require an enumeration of \(\Gamma \).

The bijection \(h\mapsto g^{-1}h\) preserves finite subsets and their supremum, so \(\sum _h|\xi (g^{-1}h)|^2=\sum _h|\xi (h)|^2\). This proves well-definedness and norm preservation; linearity is coordinatewise. Also \((\lambda _g\lambda _k\xi )(h)=\xi (k^{-1}g^{-1}h)=\xi ((gk)^{-1}h)\). The identity and delta formulas follow by evaluation. Norm preservation gives \(\| \lambda _g\| \leq 1\), and the unit vector \(\delta _e\) gives the reverse inequality.

The multiplication identities give the two-sided inverse \(\lambda _{g^{-1}}\). Reindexing the absolutely convergent inner product gives \(\langle \lambda _g\xi ,\eta \rangle =\langle \xi ,\lambda _{g^{-1}}\eta \rangle \), which identifies the adjoint.

Apply the finite operator sum to \(\xi \) and evaluate at \(h\). The operator-norm triangle inequality gives \(\| \rho (a)\| \leq \sum _g|a(g)|\| \lambda _g\| =\| a\| _1\).

Distribute finite sums and group terms with \(gk=h\):

Conjugate-linearity of adjoints and \(\lambda _g^*=\lambda _{g^{-1}}\) give

Linearity is built into the definition; \(\lambda _e=I\) gives unitality.

In \((\rho (a)\delta _e)(h)=\sum _g a(g)\delta _e(g^{-1}h)\), only \(g=h\) contributes, giving \(a(h)\). Since \(\| \delta _e\| _2=1\), \(\| a\| _2=\| \rho (a)\delta _e\| _2\leq \| \rho (a)\| \). The coordinate bound gives the last inequality. If \(\rho (a)=0\), all coefficients of \(a\) vanish.

Homogeneity and the triangle inequality come from the operator norm; faithfulness gives positive definiteness. The homomorphism identities transfer submultiplicativity, isometry of the involution, and the \(C^*\)-identity from bounded operators. The remaining bounds were proved above.

The image of \(\rho \) is a unital \(*\)-subalgebra. Its closure is closed under linear combinations and adjoints by continuity. If \(A_n\to A\) and \(B_n\to B\) in the image, \((\| A_n\| )\) is bounded and

This proves closure under multiplication; \(I\) already lies in the image. The closure is complete as a closed subspace of the Banach space of bounded operators. The \(C^*\)-identity is inherited from that algebra.

Isometry follows from the reduced norm and density from the definition of the closure. Completeness gives a unique isometric linear extension to the completion. Its range is closed by completeness and dense because it contains \(\rho (\mathbb C[\Gamma ])\), hence is the whole target. Submultiplicativity and isometry of involution extend the algebra operations to the completion. Passing to limits shows that the extension preserves multiplication, involution, and the unit.

The image of \(\rho \) is exactly the finite linear span of the generators. Any closed \(*\)-subalgebra containing them contains this span and its closure. Conversely, \(C_r^*(\Gamma )\) is such a subalgebra.

The finite sums \(\sum _g q_g\lambda _g\) with \(q_g\in \mathbb Q+i\mathbb Q\) form a countable set. Approximate coefficients on any fixed finite support by such rationals. The bound \(\| \rho (a)-\rho (b)\| \leq \| a-b\| _1\) proves density in \(\rho (\mathbb C[\Gamma ])\), hence in its closure.

Evaluation at \(e\) identifies \(\ell ^2(\{ e\} )\) with \(\mathbb C\). Then \(\rho (z\delta _e)=zI\) and \(\| zI\| =|z|\). The image is complete and hence closed, so taking its closure adds nothing.