The full (maximal) group C star algebra

Assume the Axiom of Choice. Let $G$ be an LCH group with a fixed left Haar measure and, for $f\in L^1(G)$, put $\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}:={\sup }_{\pi }\mathrm{\parallel }\pi (f)\mathrm{\parallel },$ the supremum running over the unitary equivalence classes of strongly continuous unitary representations of $G$, with $\pi (f)$ the integrated form (Integrated forms are contractive nondegenerate star representations of L one). Let $N:=\{f\in L^1(G):\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}=0\}$. The full (maximal) group C*-algebra $C^{\ast }(G)$ is the completion of the quotient $L^1(G)\mathrm{/}N$ in the norm induced by $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{C^{\ast }}$; the quotient and completion maps compose to a canonical map $L^1(G)\to C^{\ast }(G)$ with dense image which is a $\ast$-homomorphism (Banach star-algebra without a required unit, C star algebra).

• The supremum is over a set. The individual numbers $\mathrm{\parallel }\pi (f)\mathrm{\parallel }$ depend only on the unitary equivalence class of $\pi$. For any representation $\pi$ and unit vector $\xi$, the closed span of $\{\pi (g)\xi \}$ is an invariant closed subspace whose representation is the GNS representation of the normalized coefficient $g\mapsto ⟨\pi (g)\xi ,\xi ⟩$, and operator norms are tested on unit vectors; by the pointed-cyclic correspondence every such class is the GNS class of an element of $P_1(G)\subseteq {\mathbb{C}}^G$ (Normalized positive type and pointed cyclic unitary representations, GNS construction for a continuous positive-type function). Hence the supremum may be taken over the set $P_1(G)$ of continuous normalized positive-type functions, and it is a supremum of a set of nonnegative real numbers.
• Well-definedness is proved, not assumed. The finiteness $\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_1$, the submultiplicativity and star properties of $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{C^{\ast }}$, the fact that $N$ is a closed two-sided $\ast$-ideal, and the C*-identity on the completion are established in Well-definedness of the full group C star norm and its zero ideal, which defines its seminorm locally and is a prerequisite of this definition. The definition itself is the standard maximal (enveloping) norm of [BeHV–08, F.4.3] and [BeH–19, 8.B.1].
• Choice. The Axiom of Choice is inherited from the GNS construction and the completion chain; the definition adds no further choice (The Axiom of Choice).