Strong operator topology - Imported Knowlpedia Content

Definition

Let $H$ be a Hilbert space. The strong operator topology on $\mathcal B(H)$ is the locally convex topology generated by the seminorms

p_\xi(T)=\|T\xi\|,\qquad \xi\in H.

Thus a net $(T_i)$ converges strongly to $T$ exactly when

\|T_i\xi-T\xi\|\longrightarrow 0

for every $\xi\in H$ . Equivalently, it is the topology of pointwise convergence when $\mathcal B(H)$ is viewed as a space of functions $H\to H$ and the target $H$ carries its norm topology. The use of nets is essential when the topology is not first countable.

Neighborhoods and comparison

A basic neighborhood of $T$ is specified by finitely many vectors $\xi_1,\ldots,\xi_n$ and $\varepsilon>0$ :

\{S:\|(S-T)\xi_j\|<\varepsilon\text{ for }1\leq j\leq n\}.

Operator-norm convergence implies strong convergence, which in turn implies weak-operator convergence. In finite-dimensional $H$ , these operator topologies agree with the norm topology; in infinite dimension they are distinct.

Algebraic operations

Addition and scalar multiplication are strongly continuous, and multiplication is separately strongly continuous. Multiplication is jointly strongly continuous on norm-bounded sets. The adjoint operation is not strongly continuous in general: for the unilateral shift $S$ on $\ell^2(\mathbb N)$ , $S^{*n}\to0$ strongly while $(S^{*n})^*=S^n$ does not converge strongly to $0$ . Requiring both $T_i\to T$ and $T_i^*\to T^*$ gives the strong-star topology.

Operator-algebraic role

For a unital $*$ -subalgebra $A\subseteq\mathcal B(H)$ , the von Neumann bicommutant theorem identifies the strong-operator closure of $A$ with its weak-operator closure and its bicommutant $A''$ . Consequently, von Neumann algebras may be characterized as strongly closed unital $*$ -subalgebras of $\mathcal B(H)$ .

Continuity of representations

A unitary representation $g\mapsto\pi(g)$ is strongly continuous precisely when $g\mapsto\pi(g)\xi$ is norm-continuous for each vector $\xi$ . On the unitary group, strong convergence automatically entails strong convergence of adjoints, so multiplication and inversion behave continuously. This special bounded setting should not be confused with the behavior of the adjoint map on all of $\mathcal B(H)$ .

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I, American Mathematical Society, 1997. Publisher record. Relevant: §5.1 on operator topologies and closures.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter I, §3 and Chapter III, §2 on operator topologies and the bicommutant theorem.