The reduced group C star algebra

Assume the Axiom of Choice. Let $G$ be an LCH group with a fixed left Haar measure, let ${\lambda }_G$ 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, Hilbert space), and for $f\in L^1(G)$ let ${\lambda }_G(f)\in \mathcal{B}(L^2(G))$ be the integrated form of ${\lambda }_G$ at $f$ (The integrated form of a unitary representation, A bounded linear operator between normed spaces). The reduced group C*-algebra $C_r^{\ast }(G)$ is the norm closure in $\mathcal{B}(L^2(G))$ of $\{{\lambda }_G(f):f\in L^1(G)\};$ it is the C*-subalgebra of $\mathcal{B}(L^2(G))$ generated by the integrated left regular representation.

• Well-definedness. By Integrated forms are contractive nondegenerate star representations of L one the map $f\mapsto {\lambda }_G(f)$ is multiplicative and star-preserving, so its image is a complex $\ast$-subalgebra of the C*-algebra $\mathcal{B}(L^2(G))$. The norm closure of a $\ast$-subalgebra of a C*-algebra is again a $\ast$-subalgebra — sums, products and adjoints of limits are the limits of the corresponding sums, products and adjoints by continuity of the algebra operations and of the adjoint — and it is complete as a closed subset of the complete space $\mathcal{B}(L^2(G))$ (C star algebra, C star algebra generated by a normal operator). Hence $C_r^{\ast }(G)$ with the inherited operations is a C*-algebra, a C*-subalgebra of $\mathcal{B}(L^2(G))$.
• Quotient description. With $N:=\{f\in L^1(G):{\lambda }_G(f)=0\}$ the null ideal of the norm $\mathrm{\parallel }f{\mathrm{\parallel }}_r:=\mathrm{\parallel }{\lambda }_G(f)\mathrm{\parallel }$, the integrated form identifies $C_r^{\ast }(G)$ with the completion of the quotient normed algebra $L^1(G)\mathrm{/}N$; the identity map on the dense subspace shows that this completion is the same C*-algebra as the norm closure above ([BeHV–08, F.4.6]). In particular $\mathrm{\parallel }f{\mathrm{\parallel }}_r\le \mathrm{\parallel }f{\mathrm{\parallel }}_1$ for every $f\in L^1(G)$, since the integrated form is contractive.
• Choice. The Axiom of Choice is inherited from the Haar measure, the regular representation and the integrated form; the closure argument adds no further choice (The Axiom of Choice).