Definition 32.4.2 (Positive-Definite Function).label Let $G$ be a locally compact group and $\phi \in BC(G; \complex)$, then the following are equivalent:
- (1)
The mapping $f \mapsto \dpn{f, \phi}{L^1(G; \complex)}$ is a positive linear functional on $L^{1}(G; \complex)$.
- (2)
For each $n \in \natp$, $\seqf{\lambda_j}\subset \complex$ and $\seqf{x_j}\subset G$,
\[\sum_{i = 1}^{n} \sum_{j = 1}^{n} \lambda_{i} \ol{\lambda_j}\phi(x_{j}^{-1}x_{i}) \ge 0\]
If the above holds, then $\phi$ is positive-definite.
The set $P(G)$ is the space of positive-definite functions on $G$. The set $P_{0}(G)$ is the space of positive-definite functions of norm at most 1. Finally, the set $S(G)$ of positive-definite functions of norm 1 is the state space of $G$.
Proof, [Proposition 3.35, Fol16]. (1) $\Rightarrow$ (2): Using Proposition 31.4.10, let $\angles{\psi_\alpha}_{\alpha \in A}\subset L^{1}(G; \complex)$ be an approximate identity such that for all $U \in \cn_{G}(1)$, there exists $\alpha_{0} \in A$ with $\supp{\psi_\alpha}\subset U$ for all $\alpha \ge \alpha_{0}$.
For each $\alpha \in A$, $\seqf{\lambda_j}\subset \complex$, and $\seqf{x_j}\subset G$, let $f_{\alpha} = \sum_{j = 1}^{n} \lambda_{j} L_{x_j}\psi_{\alpha}$, then by continuity of $\phi$,
(2) $\Rightarrow$ (1): Let $f \in C_{c}(G; \complex)$ and $K = \supp{f}$, then $F(x, y) = f(x)\ol{f(y)}\phi(y^{-1}x)$ is uniformly continuous by Proposition 31.1.2. As such, there exists a partition $K = \bigsqcup_{j = 1}^{n} U_{j}$ and $\seqf{x_j}\subset G$ such that $|F(x_{i}, x_{j}) - F(x, y)| \le \eps$ for all $(x, y) \in U_{i} \times U_{j}$ and $1 \le i, j \le n$. Thus
As
and the above holds for all $\eps > 0$, $\int_{G} (f^{*} * f) \phi \ge 0$. By density of $C_{c}(G; \complex)$ in $L^{1}(G; \complex)$ and continuity of the convolution product, $\int_{G} (g^{*} * g) \phi \ge 0$ for all $g \in L^{1}(G; \complex)$.$\square$
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