Weak and strong continuity for unitary representations

Statement

Let $G$ be a topological group and let $\pi:G\to U(H)$ be a group homomorphism on a Hilbert space. The following conditions are equivalent:

$\pi$ is strongly continuous;
every matrix coefficient $g\mapsto\langle\pi(g)\xi,\eta\rangle$ is continuous; and
every diagonal coefficient $g\mapsto\langle\pi(g)\xi,\xi\rangle$ is continuous.

In conditions 2 and 3, it is enough to require continuity at the identity. Thus weak operator continuity and strong operator continuity agree for unitary representations, even though the weak operator topology and strong operator topology differ on general operator families.

Proof mechanism

Strong continuity immediately implies coefficient continuity. Polarization recovers all coefficients from diagonal ones. Conversely, unitarity gives

\lVert\pi(g)\xi-\xi\rVert^2 =2\lVert\xi\rVert^2- 2\operatorname{Re}\langle\pi(g)\xi,\xi\rangle,

so continuity of the diagonal coefficient at the identity forces continuity of the orbit map there. Translation then gives continuity at every group element. This is the argument in Folland, opening of §3.1.

Why unitarity matters

The fixed norm $\lVert\pi(g)\xi\rVert=\lVert\xi\rVert$ turns weak convergence of an orbit to $\xi$ into norm convergence through the displayed identity. Without a uniform norm-preserving hypothesis, weak continuity of an operator-valued homomorphism need not imply strong continuity by this argument.

Conventions and scope

Here "weakly continuous" refers to continuity of every scalar matrix coefficient, not to weak containment of representations. No local compactness or finite-dimensionality assumption on $G$ or $H$ is needed.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: the opening discussion of §3.1 on equivalent continuity conditions.