When is a $*$-homomorphism between multiplier algebras strictly continuous?

Question

(This question was posted on MSE here but didn't get any answers.)

The strict topology on the multiplier algebra M(A) of a C*-algebra A is that generated by the seminorms

$$ x\mapsto \|ax\|\quad x\mapsto \|xa\| \qquad (a\in A, x\in M(A)) $$

Whereas a ∗-homomorphism $\phi : M(A)\to M(B)$ between two multiplier algebras is necessarily norm-continuous, if I understand things correctly it will not always be continuous with respect to the strict topologies on either side. Where is there a good reference for this?

On the other hand an easily-proven theorem states that $\phi$ is strictly continuous if the image of $\phi$ contains B. This is not necessary, however; take $\phi : \mathcal{B}(\ell^2)\to\mathcal{B}(\ell^2)$ to be the map $x\mapsto sxs^*$ where $s$ is the unilateral shift. This is strictly continuous even though its image doesn't contain $\mathcal{K}(\ell^2)$. Are there other conditions which guarantee $\phi$ to be strictly continuous?

I'm particularly interested in the case where $\phi$ maps A into B, and both are nonunital. Is this enough to show that $\phi$ is strictly continuous?

Answer 1

Non-degenerate *-homorphism from $A$ (or $M(A)$) to $M(B)$ are strict (where non-degenerate means that $\phi(A)B$ is total in $B$). An important property of strict maps $\phi:A\to M(B)$ is that they possess a unique strict extension $\tilde{\phi}:M(A)\to M(B)$.

A good reference is

E.C. Lance, Hilbert C∗-modules, London Mathematical Society Lecture Notes Series 210, Cambridge University Press, Cambridge, 1995.