(This question was posted on MSE [here](https://math.stackexchange.com/questions/300276/when-is-a-homomorphism-between-multiplier-algebras-strictly-continuous) 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?**



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


