Positive-definite function on a group - Imported Knowlpedia Content

Definition

Let $G$ be a topological group. A continuous function $\varphi:G\to\mathbb C$ is positive-definite if, for every finite choice $g_1,\ldots,g_n\in G$ and $c_1,\ldots,c_n\in\mathbb C$ ,

\sum_{i,j=1}^n c_i\overline{c_j}\, \varphi(g_j^{-1}g_i)\geq 0.

Equivalently, the matrix $\bigl[\varphi(g_j^{-1}g_i)\bigr]_{i,j}$ is positive semidefinite for every finite tuple. The inequality is real despite being written with complex entries. Continuity is part of the topological-group definition used here; on an abstract group, the same positivity condition can be imposed without it.

Representation-theoretic characterization

Every function of the form $\varphi(g)=\langle\pi(g)\xi,\xi\rangle$ , where $\pi$ is a strongly continuous unitary representation, is positive-definite. Conversely, the Gelfand–Naimark–Segal construction realizes every continuous positive-definite function in this way with a cyclic vector, uniquely up to unitary equivalence when the cyclic realization is minimal Folland, §3.3.

Basic properties

Positivity gives $\varphi(e)\geq0$ , $\varphi(g^{-1})=\overline{\varphi(g)}$ , and $\lvert\varphi(g)\rvert\leq\varphi(e)$ . If $\varphi(e)=1$ , the associated cyclic vector can be chosen to have norm one. Products and nonnegative linear combinations of positive-definite functions are again positive-definite.

Conventions and scope

The displayed formula uses an inner product linear in its first variable. Positive-definiteness does not imply conjugation invariance, so a positive-definite function need not be an class function. For abelian groups, normalized continuous examples include characteristic functions arising as Fourier transforms of probability measures.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §3.3 on positive-definite functions and cyclic representations.
Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan's Property (T), Cambridge University Press, 2008. DOI record. Relevant: Appendix C on positive-definite functions.