Theorem 1. Let $G$ be a topological group, and let $\pi$ be a (group-theoretic) representation of $G$ in a dual Fréchet space $E$ by conjugate linear operators.

The representation $\pi$ is continuous in the weak${}^*$ operator topology if and only if, for some $q$, $0\leqslant q<1$, for any neighbourhood $U$ of the zero element in $E$ and its polar $\mathring{U}$ in the predual space $E_*$, and for any vector $\xi$ in $U$ and any element $f\in\mathring{U}$, there exists a neighbourhood $V$ of the identity element of $G$ such that the inequality $|f(\pi(g)\xi-\xi)|\leqslant q$ holds for all $g\in V$.

Definition 1. Let $G$ be a topological group, and let $\pi$ be a (not necessarily weakly$^*$ continuous) representation of $G$ in a locally convex space $E$ dual to a locally convex space $E_*$. By a weak variation $\omega(\pi;\xi;f;V)\geqslant0$ of the representation $\pi$ in a neighbourhood $V$ of the identity element $e$ of $G$ on the vector $\xi\in E$ and the functional $f\in E_*$ we mean the upper bound $\sup_{g\in V}|f(\pi(g)\xi-\xi)|$; by a weak variation $\omega(\pi;\xi;f)\geqslant0$ of the representation $\pi$ at the identity element $e$ of $G$ on the vector $\xi\in E$ and the functional $f\in E^*$ we mean the greatest lower bound of the quantities $\sup_{g\in V}|f(\pi(g)\xi-\xi)|$ over all neighbourhoods $V\subset G$ of $e\in G$, that is, $\omega(\pi;\xi;f)=\inf_{V\ni e}\omega(\pi;\xi;f;V)=\inf_{V\ni e}\,\sup_{g\in V}|f(\pi(g)\xi-\xi)|$; by the weak variation $\omega(\pi)\geqslant0$ of $\pi$ at the identity element $e$ of $G$ (or, briefly, the weak variation of $\pi$) we mean the least upper bound over all neighbourhoods $U$ of the zero element of $E$, all vectors $\xi\in U$, and all functionals $f\in E_*$ in the polar $\mathring{U}$ of $U$ of the lower bounds for the quantities $\sup_{g\in V}|f(\pi(g)\xi-\xi)|$ over all neighbourhoods $V\subset G$ of the identity element $e\in G$:

Definition 2. Let $G$ be a topological group and let $\pi$ be a (group-theoretic) representation of $G$ in a Fréchet space $E$. We say that $\pi$ is locally uniformly bounded if there exists a neighbourhood $U_0$ of the identity element $e$ of $G$ such that the restriction of $\pi$ to $U_0$ is an equicontinuous family of continuous linear operators in $E$.

Lemma 1. Let $G$ be a topological group, and let $\pi$ be a locally uniformly bounded representation of $G$ in a locally convex space $E$. Let $E$ be a dual space, let the operators of the representation $\pi$ be dual, and let $\omega(\pi)$ be the weak variation of the representation $\pi$ introduced in Definition 1.

If $a\in\mathbb{R}$, $a>0$, then the inequality $\omega(\pi)\leqslant a$ holds if and only if, for any $b>a$ and any neighbourhood of zero $U$ in the space $E$ of the representation $\pi$, there exists a neighbourhood $V\subset G$ of the identity element $e\in G$ such that $|f(\pi(g)\xi-\xi)|< b$ for all $f$ in the polar $\mathring{U}$ of $V$, all vectors $\xi\in V$, and all $g\in V$. If the value $\omega(\pi)$ is finite, then the representation $\pi$ is locally uniformly bounded. The representation $\pi$ is continuous in the weak$^*$ operator topology if and only if $\omega(S)=0$.

Lemma 2. Let $G$ be a locally compact group, and let $G_i$, $i=1,\dots,n$, be a finite collection of metrizable locally compact groups equipped with continuous homomorphisms $\varphi_i G_i\to G$ such that the restriction $\varphi|^{}_V$ of the map $\varphi$ of the product $G_1\times \dots \times G_n$ to $G$ defined by $\varphi(g_1,\dots,g_n)=\varphi_1(g_1)\cdots\varphi_n(g_n)$, where $g_i\in G_i$ for $i=1,\dots,n$, to some neighbourhood $V$ of the identity element $e$ of $G_1\times \dots \times G_n$ is a homeomorphism. Let $\pi$ be a (group-theoretic) representation of $G$ in a Fréchet space.

If $\pi$ is continuous in the weak$^*$ operator topology on the image in $G$ of every subgroup of $G_i$, then the representation $\pi$ is weakly$^*$ continuous.

In particular, if $G$ is a connected Lie group and the representation $\pi$ is continuous in the weak$^*$ operator topology on every one-parameter subgroup of $G$, then it is continuous in the weak$^*$ operator topology.

Corollary 1. Let $G$ be a Lie group, and let $\pi$ be its (group-theoretic) representation in a Fréchet space dual to a locally convex space.

If $\pi$ is continuous in the weak${}^*$ operator topology on every one-parameter subgroup of $G$, then it is continuous in the weak${}^*$ operator topology.

Definition 3. Let $E$ be a Fréchet space dual to some locally convex space $E_*$, and let ${\mathcal L}(E)$ be the space of continuous linear operators on $E$. By the weak${}^*$ operator topology in ${\mathcal L}(E)$ we mean the topology defined by the family of seminorms of the form $T\mapsto|f(Tx)|$, where $T\in{\mathcal L}(E)$, $x\in E$, and $f\in E_*$.

Lemma 3. Let $E$ be a Fréchet space dual to a locally convex space, let $G$ be a topological group, and let $\pi$ be a locally uniformly bounded (group-theoretic) representation of $G$ in $E$ by weakly${}^*$ continuous linear operators.

If $\omega^*(\pi)<1$, then $\omega^*(\pi)=0$, and the representation $\pi$ is continuous in the weak${}^*$ operator topology.