Strict topology on a multiplier algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra and $M(A)$ its multiplier algebra. The strict topology on $M(A)$ is the locally convex topology generated, for each $a\in A$ , by the seminorms

p_a(m)=\lVert ma\rVert,\qquad q_a(m)=\lVert am\rVert.

Thus a net $(m_i)$ converges strictly to $m$ exactly when

\lVert(m_i-m)a\rVert\to0 \quad\text{and}\quad \lVert a(m_i-m)\rVert\to0

for every $a\in A$ . The topology uses the distinguished essential ideal $A\subseteq M(A)$ ; it is generally weaker than the multiplier norm topology.

Approximate identities and strict density

If $(e_\lambda)$ is an approximate identity of $A$ , then

e_\lambda\longrightarrow1_{M(A)}

strictly. More generally, $me_\lambda\to m$ and $e_\lambda m\to m$ strictly for every $m\in M(A)$ . Consequently the canonical copy of $A$ is strictly dense in $M(A)$ , even though it is norm closed there. This is the basic mechanism by which the multiplier unit is approximated from the nonunital algebra Pedersen, §3.12.

If $A$ is unital, choosing $a=1_A$ among the defining seminorms shows that the strict topology equals the norm topology on $M(A)=A$ . The distinction therefore matters primarily for nonunital algebras.

Extension of nondegenerate maps

A nondegenerate $*$ -homomorphism $\phi:A\to M(B)$ has a unique unital extension

\overline{\phi}:M(A)\longrightarrow M(B)

that is strictly continuous and agrees with $\phi$ on $A$ . It can be recovered from any approximate identity by strict limits. This extension theorem is one reason strict, rather than norm, continuity is built into the morphism theory of multiplier algebras Lance, Chapter 2.

Concrete model and comparison

For a Hilbert space $H$ ,

M(\mathcal K(H))=\mathcal B(H).

On norm-bounded subsets of $\mathcal B(H)$ , the strict topology determined by $\mathcal K(H)$ agrees with the strong-star operator topology: both $T_i\xi\to T\xi$ and $T_i^*\xi\to T^*\xi$ for every $\xi\in H$ . This gives a useful concrete model, but strict topology is defined through two-sided norm multiplication by elements of $A$ , not by selecting a Hilbert-space representation.

References

Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §3.12 on multiplier algebras, strict topology, and approximate identities.
E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. DOI record. Relevant: Chapter 2 on multipliers, strict convergence, and extensions of nondegenerate homomorphisms.