Weakly continuous unitary representation

Let $G$ be a topological group and $H$ a complex Hilbert space. A group homomorphism

\pi:G\longrightarrow U(H)

is a weakly continuous unitary representation if every matrix coefficient

g\longmapsto\langle\pi(g)\xi,\eta\rangle

is continuous for all $\xi,\eta\in H$ . Equivalently, $\pi$ is continuous when $U(H)$ carries the weak operator topology. By the weak–strong continuity equivalence, this is equivalent to strong continuity; no local compactness or separability assumption is needed. Its coefficient functions are therefore continuous complex-valued functions on $G$ .

Why weak continuity implies strong continuity

Fix $g_0\in G$ and $\xi\in H$ . Unitarity gives

\|\pi(g)\xi-\pi(g_0)\xi\|^2 =2\|\xi\|^2-2\operatorname{Re}\langle\pi(g)\xi,\pi(g_0)\xi\rangle.

Weak continuity makes the right-hand side tend to zero as $g\to g_0$ . Hence every orbit map $g\mapsto\pi(g)\xi$ is norm-continuous. Strong continuity plainly implies weak continuity by continuity of the inner product.

Scope of the equivalence

The equivalence relies on the fixed norm of unitary operators. For general bounded-operator-valued representations, weak-operator continuity need not imply strong-operator continuity without additional assumptions. “Weakly continuous” here refers to weak operator topology, not to the Banach-space weak topology on $B(H)$ or the ultraweak topology.

Matrix coefficients

Weak continuity is often convenient because it can be checked through scalar-valued functions. Positive-definite functions arise as diagonal coefficients $g\mapsto\langle\pi(g)\xi,\xi\rangle$ , and polarization recovers all matrix coefficients from diagonal ones. Thus continuity of all diagonal coefficients is another equivalent test.

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 of matrix coefficients.