GNS construction for a positive-definite function

Core idea

Let $G$ be a topological group and let $\varphi:G\to\mathbb C$ be a continuous positive-definite function. The GNS construction for $\varphi$ produces a Hilbert space $\mathcal H_\varphi$ , a strongly continuous unitary representation $\pi_\varphi$ , and a cyclic vector $\xi_\varphi$ such that

\varphi(g)=\langle\pi_\varphi(g)\xi_\varphi,\xi_\varphi\rangle .

This pointed cyclic representation is unique up to a unique unitary intertwiner carrying $\xi_\varphi$ to the other distinguished vector. Moreover, $\lVert\xi_\varphi\rVert^2=\varphi(e)$ , so the vector is a unit vector exactly when $\varphi$ is normalized.

Construction

Let $V_0$ be the vector space spanned by formal symbols $\{\delta_x:x\in G\}$ . With the inner product linear in the first variable, set

\left\langle\sum_i c_i\delta_{x_i},\sum_jd_j\delta_{y_j}\right\rangle_\varphi =\sum_{i,j}c_i\overline{d_j}\,\varphi(y_j^{-1}x_i).

Positive definiteness makes this form positive semidefinite. Quotient $V_0$ by its null space and complete. Left translation $\pi_\varphi(g)\delta_x=\delta_{gx}$ preserves the form, and $\xi_\varphi=[\delta_e]$ . Continuity of $\varphi$ , followed by density of the symbol span, gives strong continuity.

Cyclicity and uniqueness

The orbit of $\xi_\varphi$ contains every symbol class, so its linear span is dense. If $(\pi,\mathcal H,\xi)$ is another cyclic realization of $\varphi$ , the rule

\sum_i c_i\pi_\varphi(x_i)\xi_\varphi\longmapsto \sum_i c_i\pi(x_i)\xi

preserves inner products. It therefore extends to the unique unitary intertwiner required above. This is the group form of the Gelfand–Naimark–Segal construction Folland, Theorem 3.20.

Degenerate and normalized cases

If $\varphi(e)=0$ , positive definiteness forces $\varphi=0$ , and the construction yields the zero Hilbert space. Otherwise one may normalize by $\varphi(e)$ . The GNS representation is irreducible precisely when the normalized function is an extreme point of the convex set of normalized continuous positive-definite functions; cyclicity alone does not imply irreducibility.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Theorem 3.20 and the discussion of cyclic representations.
Jacques Dixmier, $C^*$ -Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: Chapter 2 on positive forms and representations.