Weakly continuous unitary representation - Imported Knowlpedia Content

Definition

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 Folland, §3.1.

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.