Positive type on a discrete group: the identity mass, characters, and the regular GNS model

For $n\ge 1$, $g_i^{-1}{g}_j=e$ exactly when $g_i=g_j$; partitioning the finite list by its distinct values gives ${\sum }_{i,j=1}^n\overline{c_i}{c}_j{\delta }_e(g_i^{-1}{g}_j)={\sum }_{h\in \{g_1,\dots ,g_n\}}{\mid {\sum }_{j:g_j=h}{c}_j\mid }^2\ge 0$. This includes repeated group elements and zero coefficients.

Every orbit map $g\mapsto \sigma (g)v$ is continuous because its domain is discrete, so $\sigma$ is strongly continuous. Choose an orthonormal basis $(e_k{)}_{k<d}$ of $V$. Since $V\mathrm{\ne }\{0\}$, its dimension $d$ is nonzero; write $d=\mathrm{succ}(m)$. From $1=\mathrm{succ}(0)$ and the recursive addition law, induction gives $1+m=\mathrm{succ}(m)$, so $1\le d$ by the natural-order definition. Its canonical real scalar $d_{\mathbb{R}}$ is positive, and its image $d_{\mathbb{C}}$ in $\mathbb{C}$ is nonzero.

The function ${\delta }_e$ is continuous because $\mathrm{\Gamma }$ is discrete, and ${\delta }_e(e)=1$. Step 1.1 proves its finite-matrix test, so ${\delta }_e$ is of positive type.

For $k<d$ set ${\varphi }_k(g)=⟨\sigma (g)e_k,e_k⟩$. By [F5], each ${\varphi }_k$ is continuous and of positive type. If $\sigma (g)e_k={\sum }_{\mathrm{\ell }<d}{a}_{\mathrm{\ell }k}(g)e_{\mathrm{\ell }}$, orthonormality gives ${\varphi }_k(g)=a_{kk}(g)$; the trace definitions in [F7] consequently give $\mathrm{tr}(\sigma (g))={\sum }_{k<d}{\varphi }_k(g)$.

By [F8], $\sigma (e)=I_V$; its matrix in this basis is the $d$ by $d$ identity, so [F6, F7] give $\mathrm{tr}(\sigma (e))=d$. Hence ${\chi }_{\sigma }(e)=d_{\mathbb{C}}\mathrm{/}{d}_{\mathbb{C}}=1$ by the field embedding [F10]. Also ${\chi }_{\sigma }$ is continuous, since it is a function from a discrete domain.

For any finite test $(g_i,c_i{)}_{i=1}^n$ with $n\ge 1$, step 2.2 gives ${\sum }_{i,j=1}^n\overline{c_i}{c}_j{\chi }_{\sigma }(g_i^{-1}{g}_j)=d_{\mathbb{R}}^{-1}{\sum }_{k<d}({\sum }_{i,j=1}^n\overline{c_i}{c}_j{\varphi }_k(g_i^{-1}{g}_j))$. Each parenthesized value is a nonnegative real by [F5]; their finite sum is nonnegative, and $d_{\mathbb{R}}^{-1}>0$ by [F9]. The embedding in [F10] identifies this real nonnegative value with the displayed complex quadratic form, so [F1] shows that ${\chi }_{\sigma }$ is of positive type.

For finitely supported $f,h:\mathrm{\Gamma }\to \mathbb{C}$, [F2] gives $B_{{\delta }_e}(f,h)={\sum }_{x,y}f(x)\overline{h(y)}{\delta }_e(y^{-1}x)={\sum }_{x}f(x)\overline{h(x)}$. This is positive definite: if $f\mathrm{\ne }0$, a nonzero coordinate contributes a strictly positive squared modulus while every other term is nonnegative. Hence the GNS null space is $\{0\}$, and the quotient is this finite-support inner-product space.

Assume AC. By [F11], Countable Choice holds; [F12] gives a Hilbert completion $\widehat{E}$ of the inner-product space in step 3.2, with the finite-support functions embedded densely. On that dense subspace define $L_{g}f(x)=f(g^{-1}x)$. This is linear, and reindexing $x=gy$ gives $⟨L_{g}f,L_{g}h⟩={\sum }_{x}f(g^{-1}x)\overline{h(g^{-1}x)}={\sum }_{y}f(y)\overline{h(y)}=⟨f,h⟩$. Thus $L_g$ is bounded with norm bound one; by [F13] it extends uniquely to a linear contraction ${\lambda }_{\mathrm{\Gamma }}(g)$ on $\widehat{E}$. This is the completion stage in the GNS construction for ${\delta }_e$.

On the dense finite-support subspace, $L_g{L}_h=L_{gh}$ and $L_g{L}_{g^{-1}}=L_{g^{-1}}{L}_g=I$. The continuous extensions obey the same identities on $\widehat{E}$; since each extension and its inverse are contractions, each is an isometry and hence unitary. Therefore $g\mapsto {\lambda }_{\mathrm{\Gamma }}(g)$ is strongly continuous because $\mathrm{\Gamma }$ is discrete. The point masses $({\delta }_x{)}_{x\in \mathrm{\Gamma }}$ are orthonormal and their span is dense in $\widehat{E}$, so they form a complete orthonormal family. By [F14] the Fourier coefficient map $U:\widehat{E}\to {\mathrm{\ell }}^2(\mathrm{\Gamma },\mathbb{C})$ is unitary and sends ${\delta }_x$ to the standard coordinate vector $e_x$. For each $g,x$, $U{\lambda }_{\mathrm{\Gamma }}(g){\delta }_x=e_{gx}$, which is the standard left translation of $e_x$; density and continuity imply that $U$ intertwines ${\lambda }_{\mathrm{\Gamma }}$ with the left regular representation on ${\mathrm{\ell }}^2(\mathrm{\Gamma })$. Finally ${\lambda }_{\mathrm{\Gamma }}(g){\delta }_e={\delta }_g$ and $⟨{\lambda }_{\mathrm{\Gamma }}(g){\delta }_e,{\delta }_e⟩=⟨{\delta }_g,{\delta }_e⟩={\delta }_e(g)$; the orbit spans a dense subspace, so ${\delta }_e$ is cyclic and this is its GNS representation.

The positivity and normalization proofs in steps 1.1–3.1 use no choice. AC is used only through Countable Choice in [F12]–[F14] for the Hilbert completion, bounded extensions, and the standard ${\mathrm{\ell }}^2$ coordinate identification. The extensions and Fourier coefficient map are unique, so the action and unitary equivalence require no further choice. All three claims are established.