Continuity criteria for unitary representations

If $\pi$ is strongly continuous, [A1] makes every mixed matrix coefficient continuous on $G$. In particular, every diagonal coefficient is continuous at $e$ on every total subset.

Now suppose the diagonal coefficients are continuous at $e$ for a total subset $S$. Fix $v\in S$, put $q(g)=⟨\pi (g)v,v⟩$, $r=\mathrm{\parallel }v{\mathrm{\parallel }}^2=q(e)$, and $z(g)=q(g)-r$. The homomorphism law gives $\pi (e)=I$. Using [A2] and [A4], $\mathrm{\parallel }\pi (g)v-v{\mathrm{\parallel }}^2=\mathrm{\parallel }\pi (g)v{\mathrm{\parallel }}^2-⟨\pi (g)v,v⟩-⟨v,\pi (g)v⟩+\mathrm{\parallel }v{\mathrm{\parallel }}^2=2r-q(g)-\overline{q(g)}=-(z(g)+\overline{z(g)})\mathrm{.}$ This is a nonnegative real number. By [A6] and [A7], $0\le \mathrm{\parallel }\pi (g)v-v{\mathrm{\parallel }}^2=\mid z(g)+\overline{z(g)}\mid \le \mathrm{\mid }z(g)\mathrm{\mid }+\mathrm{\mid }\overline{z(g)}\mathrm{\mid }=2\mathrm{\mid }z(g)\mathrm{\mid }\mathrm{.}$ Continuity of $q$ at $e$ in the metric of [A10] gives, for each $\epsilon >0$, a neighborhood of $e$ on which $2\mathrm{\mid }z(g)\mathrm{\mid }<{\epsilon }^2$. Then $\mathrm{\parallel }\pi (g)v-v{\mathrm{\parallel }}^2<{\epsilon }^2$, so [A5] and [A9] give $\mathrm{\parallel }\pi (g)v-v\mathrm{\parallel }<\epsilon$. No nonzero-vector hypothesis was used, so this also covers $v=0$.

If $w={\sum }_{j=1}^n{\lambda }_j{v}_j$ is a finite complex linear combination of elements of $S$, then linearity and [A5] give $\mathrm{\parallel }\pi (g)w-w\mathrm{\parallel }\le {\sum }_{j=1}^n\mathrm{\mid }{\lambda }_j\mathrm{\mid }\mathrm{\parallel }\pi (g)v_j-v_j\mathrm{\parallel }⟶0(g\to e)\mathrm{.}$ by step 1.2. Only finitely many orbit maps occur, so intersecting their neighborhoods proves convergence of the sum. If $n=0$, then $w=0$ and the orbit difference is identically zero.

For any $x\in H$ and $\epsilon >0$, totality supplies a finite-span vector $w$ with $\mathrm{\parallel }x-w\mathrm{\parallel }<\epsilon \mathrm{/}3$. Step 2.1 gives a neighborhood of $e$ on which $\mathrm{\parallel }\pi (g)w-w\mathrm{\parallel }<\epsilon \mathrm{/}3$. For such $g$, the triangle inequality and the isometry property in [A2] give $\mathrm{\parallel }\pi (g)x-x\mathrm{\parallel }\le \mathrm{\parallel }\pi (g)(x-w)\mathrm{\parallel }+\mathrm{\parallel }\pi (g)w-w\mathrm{\parallel }+\mathrm{\parallel }w-x\mathrm{\parallel }=2\mathrm{\parallel }x-w\mathrm{\parallel }+\mathrm{\parallel }\pi (g)w-w\mathrm{\parallel }<\epsilon \mathrm{.}$ This proves continuity at $e$ for every orbit map. If $S=\mathrm{\varnothing }$, totality means the only available finite sum is $0$ and still supplies the required approximation; step 2.1 and the same estimate apply (and force $H=\{0\}$).

Fix $x\in H$ and $g_0\in G$. As $g\to g_0$, continuity of multiplication in [A3] gives $h=g_0^{-1}g\to e$. The homomorphism law and the isometry $\pi (g_0)$ yield $\mathrm{\parallel }\pi (g)x-\pi (g_0)x\mathrm{\parallel }=\mathrm{\parallel }\pi (g_0)(\pi (h)x-x)\mathrm{\parallel }=\mathrm{\parallel }\pi (h)x-x\mathrm{\parallel }⟶0.$ by step 3.1. Thus every orbit map is norm-continuous on $G$, which is strong continuity.

If every matrix coefficient is continuous, its diagonal coefficients are continuous at $e$ on any total subset, so steps 1.2–4.1 prove strong continuity. Conversely step 1.1 proves that strong continuity implies both coefficient conditions. Hence all three conditions are equivalent.