Strong-* operator topology - Imported Knowlpedia Content

Definition

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

q_\xi(T)=\bigl(\|T\xi\|^2+\|T^*\xi\|^2\bigr)^{1/2}, \qquad \xi\in H.

Thus a net $T_i$ converges strong- $*$ to $T$ exactly when

T_i\xi\to T\xi \quad\text{and}\quad T_i^*\xi\to T^*\xi

in norm for every $\xi\in H$ . A concrete von Neumann algebra inherits this topology from $B(H)$ . It refines the strong operator topology by requiring pointwise control of adjoints as well as operators.

Characterizing convergence

Strong- $*$ convergence is equivalent to simultaneous strong convergence of $T_i$ and $T_i^*$ . Consequently the involution $T\mapsto T^*$ is continuous. Multiplication is jointly strong- $*$ continuous on norm-bounded subsets: if $S_i\to S$ and $T_i\to T$ strong- $*$ , with both nets bounded, then $S_iT_i\to ST$ strong- $*$ Takesaki, Chapter I, §3.

Comparison with other topologies

Strong- $*$ convergence implies strong convergence, but the converse can fail because adjoints need not preserve strong limits. For example, on $\ell^2(\mathbb N)$ , the powers $S^{*n}$ of the backward shift converge strongly to zero, while their adjoints $S^n$ do not. Hence $S^{*n}$ does not converge strong- $*$ to zero.

On norm-bounded sets, strong- $*$ and ultrastrong-star convergence agree. Globally, the ultrastrong-star topology is strictly finer in infinite dimension because it can test square-summable families of vectors at once.

Conventions and scope

The symbols “strong- $*$ ,” “strong star,” and “SOT-star” all refer here to the topology generated by individual vectors. They should not be used for the $\sigma$ -strong-star topology on an unbounded set. For abstract von Neumann algebras, the ultrastrong-star formulation is intrinsic, whereas a strong- $*$ topology is attached to a concrete representation.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. Springer DOI record. Relevant: Chapter I, §3 on strong, strong-star, weak, and related operator topologies.