# ct.category theory – Pushout of generalised morphisms \$C^*\$-algebras

There is a known construction of pushout of $$C^*$$-algebras, or rather, the amalgamated free product, which is universal for commutative squares of $$*$$-homomorphisms. Jensen and Thomsen in their book Elements of KK-theory give, in an appendix, a detailed treatment for the free product (i.e the coproduct), for instance. However, $$*$$-homomorphisms are not the only morphisms between $$C^*$$-algebras that are reasonable to consider. For instance, a non-proper map $$Xto Y$$ between locally compact Hausdorff spaces induces a $$*$$-homomorphism $$C_0(Y) to M(C_0(X)) simeq C_b(X)$$, which is continuous for the strict topology on $$C_0(Y)$$. This motivates somewhat the more general type of morphism, where we can take a map $$Arightsquigarrow B$$ between $$C^*$$-algebras to be a strictly continuous $$*$$-homomorphism $$Ato M(B)$$, where $$M(B)$$ is the multiplier algebra of $$B$$. Any $$*$$-homomorphism $$Ato B$$ gives one of these more general morphisms, namely the composite $$Ato B hookrightarrow M(B)$$. It is a fun fact that as a set $$M(B)$$ is the completion in the strict topology of $$B$$, though we regard it as being equipped with both the strict topology and its (Banach) $$C^*$$-topology. These generalised morphisms compose by using the universal property of the strict completion, and so we have a category.

I’m interested in the pushout, in this category, of an arbitrary generalised morphism and a generalised morphism arising from an injective $$*$$-homomorphism, in particular the inclusion of a full corner.

Conjecture: A full corner inclusion pushes out to also give (the generalised morphism arising from) a full corner inclusion.