GNS representation of a continuous unitary character

The homomorphism law gives $\chi (e)=\chi (e{)}^2$. Since $\mathrm{\mid }\chi (e)\mathrm{\mid }=1$, the value is nonzero, so cancellation gives $\chi (e)=1$. Applying the homomorphism law to $g^{-1}g=e$ gives $\chi (g^{-1})=\chi (g{)}^{-1}=\overline{\chi (g)}$ by [A2]. [A1, A2] 2.1 For any $n\ge 1$, any $g_1,\dots ,g_n\in G$ (including repetitions), and any $c_1,\dots ,c_n\in \mathbb{C}$, the matrix quadratic form is ${\sum }_{i,j=1}^n\overline{c_i}\chi (g_i^{-1}{g}_j)c_j={\mid {\sum }_{j=1}^n{c}_j\chi (g_j)\mid }^2\ge 0.$ Thus the matrix is positive semidefinite. The given continuity of $\chi$ and $\chi (e)=1$ show $\phi \in P_1(G)$; zero coefficients are covered by the same identity. [A2, A6, step 1.1, algebra] 2.2 The pairing $⟨z,w⟩=z\bar{w}$ is the stated complex inner product, its induced norm is $\mathrm{\mid }z\mathrm{\mid }$, and the complex plane is complete; hence $\mathbb{C}$ is a complex Hilbert space by [A4]. For each $g$, multiplication by $\chi (g)$ is complex-linear by [A5] and is an isometry because $\mathrm{\mid }\chi (g)z\mathrm{\mid }=\mathrm{\mid }z\mathrm{\mid }$; multiplication by $\chi (g^{-1})$ is its inverse. The homomorphism law makes ${\pi }_{\chi }$ a representation. For fixed $z$ and $g_0$, $\mathrm{\parallel }{\pi }_{\chi }(g)z-{\pi }_{\chi }(g_0)z\mathrm{\parallel }=\mathrm{\mid }\chi (g)-\chi (g_0)\mathrm{\mid }\mathrm{\mid }z\mathrm{\mid }⟶0$ as $g\to g_0$, by continuity in [A3]. Thus it is strongly continuous. Since ${\pi }_{\chi }(e)1=1$, the orbit span contains $1$ and is all of $\mathbb{C}$; also $⟨{\pi }_{\chi }(g)1,1⟩=\chi (g)=\phi (g)$. [A3, A4, A5, step 1.1] 2.3 Define $L_{\chi }(f)={\sum }_{x}f(x)\chi (x)$ on finitely supported $f$. Using [A2] and step 1.1, $B_{\chi }(f,h)={\sum }_{x,y}f(x)\overline{h(y)}\chi (y{)}^{-1}\chi (x)=L_{\chi }(f)\overline{L_{\chi }(h)}\mathrm{.}$ Therefore $B_{\chi }(f,f)=\mathrm{\mid }{L}_{\chi }(f){\mathrm{\mid }}^2$ and $N_{\chi }=\ker L_{\chi }$. The map $[f]\mapsto L_{\chi }(f)$ is a well-defined linear isometry from the quotient onto $\mathbb{C}$: it is onto because $L_{\chi }(z{\delta }_e)=z$. For each $g$, $L_{\chi }({\delta }_g)=\chi (g)$ and $L_{\chi }(\chi (g){\delta }_e)=\chi (g)$, so injectivity on the quotient gives $[{\delta }_g]=\chi (g)[{\delta }_e]$. Left translation satisfies $L_{\chi }(L_{g}f)=\chi (g)L_{\chi }(f)$, in agreement with the scalar action ${\pi }_{\chi }(g)$ from [A7]. [A2, A6, A7, step 1.1, algebra] 3.1 By step 2.1, $\phi$ meets the input hypotheses of the GNS construction in [A7]. Its canonical triple is cyclic, strongly continuous, and has diagonal coefficient $\phi$. Step 2.2 gives the same properties and coefficient for $({\pi }_{\chi },\mathbb{C},1)$. The pointed uniqueness theorem in [A7] therefore gives the unique unitary intertwiner carrying the vector $1$ to the canonical GNS vector. [A5, A7, step 2.1, step 2.2] 4.1 The only choice used is AC, declared in the example, through the GNS completion/action and pointed uniqueness inputs in [A7]; [A8] identifies the precise reduction AC $\Rightarrow$ DC $\Rightarrow$ Countable Choice used for completion and bounded extensions. The finite matrix, scalar representation, and quotient calculations in steps 1.1–3.1 use no choice.