Strongly continuous unitary representation

Let $G$ be a topological group and $\mathcal H$ a complex Hilbert space. A strongly continuous unitary representation of $G$ on $\mathcal H$ is a group homomorphism

\pi:G\longrightarrow \mathcal U(\mathcal H)

such that, for every $\xi\in\mathcal H$ , the orbit map $g\mapsto\pi(g)\xi$ is continuous in the norm of $\mathcal H$ . Thus “strongly” refers to pointwise norm continuity of the operators, not norm continuity of $g\mapsto\pi(g)$ in the operator norm. The representation is unitary because every $\pi(g)$ preserves the Hilbert-space inner product.

Equivalent continuity tests

Strong continuity is equivalent to continuity of the action map $G\times\mathcal H\to\mathcal H$ , $(g,\xi)\mapsto\pi(g)\xi$ . The weak–strong continuity theorem says it is also equivalent to continuity of every matrix coefficient $g\mapsto\langle\pi(g)\xi,\eta\rangle$ : weak continuity implies strong continuity by applying the coefficient identity to $\|\pi(g)\xi-\pi(g_0)\xi\|^2$ . These equivalences require unitarity, which supplies uniform norm control.

Examples and consequences

Every norm-continuous homomorphism $G\to\mathcal U(\mathcal H)$ is strongly continuous, but the converse can fail in infinite dimension. For a locally compact group, the left regular representation on $L^2(G)$ , given by $(\lambda(g)f)(x)=f(g^{-1}x)$ , is strongly continuous. Strong continuity is precisely the regularity needed to obtain infinitesimal generators from one-parameter subgroups.

Conventions and scope

The modulation example satisfies $\|M_t-I\|_{\mathrm{op}}=2$ for every $t\ne0$ , despite strong continuity. A smooth map into the operator-norm unitary group imposes a stronger condition and has bounded infinitesimal operators.

Some authors say simply “unitary representation” and include strong continuity in the term. For a discrete group the condition is automatic. A finite-dimensional representation of a Lie group is often formulated using smoothness; the present notion is designed to include infinite-dimensional Hilbert spaces and does not assert operator-norm continuity.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §3.1 on unitary representations and continuity.
Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan's Property (T), Cambridge University Press, 2008. Author record. Relevant: Appendix A on unitary representations.