Well-definedness of the full group C star norm and its zero ideal

The universal supremum is set-sized. Given a representation $\pi$ and a unit vector $\xi$, its cyclic subspace $M=\stackrel{}{\mathrm{span}}\{\pi (g)\xi :g\in G\}$ is invariant. Its orthogonal complement is invariant as well, because $\pi$ is unitary, so its projection commutes with every $\pi (g)$. The weak integral identity then shows that $\pi (f)M\subseteq M$. By [F5] the restricted pointed representation is equivalent to the GNS triple of its normalized coefficient in $P_1(G)$. Thus $\mathrm{\parallel }\pi (f)\xi \mathrm{\parallel }$ is bounded by the supremum of the integrated norms of these GNS representations. Taking the supremum over unit vectors, and observing that every GNS representation is itself eligible, proves equality with the universal supremum. The zero representation contributes only zero, and $P_1(G)$ is nonempty because it contains the constant function $1$.

$\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{C^{\ast }}$ is a finite submultiplicative $\ast$-seminorm on $L^1(G)$: for each $f$, F1 gives $\mathrm{\parallel }\pi (f)\mathrm{\parallel }\le \mathrm{\parallel }f{\mathrm{\parallel }}_1$ for every $\pi$, so $\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_1<\mathrm{\infty }$; $\mathrm{\parallel }\alpha f{\mathrm{\parallel }}_{C^{\ast }}=\mathrm{\mid }\alpha \mathrm{\mid }\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}$ by linearity; $\mathrm{\parallel }f+h{\mathrm{\parallel }}_{C^{\ast }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}+\mathrm{\parallel }h{\mathrm{\parallel }}_{C^{\ast }}$ by the operator triangle inequality before taking the supremum; $N$ contains $0$; for $f,h\in L^1(G)$ and each $\pi$, $\mathrm{\parallel }\pi (f\ast h)\mathrm{\parallel }=\mathrm{\parallel }\pi (f)\pi (h)\mathrm{\parallel }\le \mathrm{\parallel }\pi (f)\mathrm{\parallel }\mathrm{\parallel }\pi (h)\mathrm{\parallel }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}\mathrm{\parallel }h{\mathrm{\parallel }}_{C^{\ast }}$, so $\mathrm{\parallel }f\ast h{\mathrm{\parallel }}_{C^{\ast }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}\mathrm{\parallel }h{\mathrm{\parallel }}_{C^{\ast }}$; and $\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_{C^{\ast }}={\sup }_{\pi }\mathrm{\parallel }\pi (f{)}^{\ast }\mathrm{\parallel }={\sup }_{\pi }\mathrm{\parallel }\pi (f)\mathrm{\parallel }=\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}$ by F1 and F2.

$N=\{f:\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}=0\}$ is a closed two-sided $\ast$-ideal of $L^1(G)$: it is a linear subspace by the seminorm identities, and it is closed in the $L^1$ norm because $\mathrm{\mid }\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}-\mathrm{\parallel }h{\mathrm{\parallel }}_{C^{\ast }}\mathrm{\mid }\le \mathrm{\parallel }f-h{\mathrm{\parallel }}_{C^{\ast }}\le \mathrm{\parallel }f-h{\mathrm{\parallel }}_1$; if $f\in N$ and $h\in L^1(G)$ then step 1.2 gives $\mathrm{\parallel }f\ast h{\mathrm{\parallel }}_{C^{\ast }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}\mathrm{\parallel }h{\mathrm{\parallel }}_{C^{\ast }}=0$ and $\mathrm{\parallel }h\ast f{\mathrm{\parallel }}_{C^{\ast }}\le \mathrm{\parallel }h{\mathrm{\parallel }}_{C^{\ast }}\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}=0$, so $N$ is a two-sided ideal; and $f\in N$ implies $\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_{C^{\ast }}=\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}=0$, so $N$ is a $\ast$-ideal. Hence $L^1(G)\mathrm{/}N$ is a normed $\ast$-algebra with the induced norm and involution.

The C*-identity holds on $L^1(G)$ and descends to the quotient: for every $f$, $\mathrm{\parallel }{f}^{\ast }\ast f{\mathrm{\parallel }}_{C^{\ast }}={\sup }_{\pi }\mathrm{\parallel }\pi (f{)}^{\ast }\pi (f)\mathrm{\parallel }={\sup }_{\pi }\mathrm{\parallel }\pi (f){\mathrm{\parallel }}^2=\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{\ast }}^2$, using multiplicativity, $\pi (f^{\ast })=\pi (f{)}^{\ast }$ F1 and the C*-identity in $\mathcal{B}(H)$ F2; in particular $\mathrm{\parallel }[f{]}^{\ast }[f]\mathrm{\parallel }=\mathrm{\parallel }[f]{\mathrm{\parallel }}^2$ for the coset $[f]$ in $L^1(G)\mathrm{/}N$.

Put $B=L^1(G)\mathrm{/}N$ with its induced norm. Apply F4 to its norm metric, and define addition, scalar multiplication, multiplication and involution on Cauchy-sequence classes termwise. These operations are well defined: Cauchy sequences are bounded, and $\mathrm{\parallel }{x}_n{y}_n-x_m{y}_m\mathrm{\parallel }\le \mathrm{\parallel }{x}_n\mathrm{\parallel }\mathrm{\parallel }{y}_n-y_m\mathrm{\parallel }+\mathrm{\parallel }{x}_n-x_m\mathrm{\parallel }\mathrm{\parallel }{y}_m\mathrm{\parallel }$ shows that products are Cauchy; the same estimate for equivalent representatives shows independence of representatives. The isometry of the involution from step 1.2 gives both its preservation of Cauchy sequences and independence of representatives; addition and scalar multiplication follow from their norm inequalities. The norm is $\mathrm{\parallel }[x_n]\mathrm{\parallel }={\lim }_n\mathrm{\parallel }{x}_n\mathrm{\parallel }$, so submultiplicativity passes to the limit, as do the vector-space and star-algebra identities. Thus the complete metric space is a Banach $\ast$-algebra with dense isometric copy of $B$. Finally step 3.1 gives $\mathrm{\parallel }[x_n{]}^{\ast }[x_n]\mathrm{\parallel }={\lim }_n\mathrm{\parallel }{x}_n^{\ast }{x}_n\mathrm{\parallel }={\lim }_n\mathrm{\parallel }{x}_n{\mathrm{\parallel }}^2=\mathrm{\parallel }[x_n]{\mathrm{\parallel }}^2$. This is a C*-algebra by F4.

The Axiom of Choice is inherited from the integrated-form and completion suppliers of F1–F4, as declared in the definition of the universal seminorm (The Axiom of Choice).