Do *-homomorphisms have support projections?

Question

Consider a surjective $*$-homomorphism of unital $\mathrm{C}^*$-algebras $\pi:\mathcal{A}\to\mathcal{B}$. Is it possible to define a (central) support for such a map?

I imagine the way this works is that $\ker \pi$ is a $\mathrm{C}^*$ideal in $\mathcal{A}$ and then $(\ker \pi)^{**}$ is an ideal in $\mathcal{A}^{**}$, a von Neumann algebra, so: $$(\ker \pi)^{**}=\mathcal{A}^{**}p,$$ for some central projection in $\mathcal{A^{**}}$.

Then I imagine that when we extend $\pi:\mathcal{A}\to\mathcal{B}$ to $\pi^{**}:\mathcal{A}^{**}\to \mathcal{B}^{**}$, we have stuff like, for $q:=1_{\mathcal{A}^{**}}-p$, for all $a\in \mathcal{A}$ (embedded in the bidual): $$\pi^{**}(a)=\pi^{**}(qa)=\pi^{**}(aq)=\pi^{**}(qaq),$$ and maybe $p$ is the largest projection in $(\ker \pi)^{**}$.

Question:

Does this all check out? Is there an identification of $\mathcal{A}p\subset \mathcal{A}^{**}$ with $\mathcal{B}\subset \mathcal{B}^{**}$?

Thanks for any help.

Answer 1

The support projection of a unitary representation of a C$^*$-algebra is defined in [1, Definition III.2.11]. You can use it for $\pi$ composed with a universal representation of $\mathcal B$ to relate to your picture. You also have $\mathcal A q \simeq \mathcal B$: $\mathcal A q$ is a quotient of $\mathcal A$, and the kernel of quotient map $\pi'$ satisfies $V(\pi') = V(\pi)$ in the notation of Takesaki. Then you can for example use his Proposition 2.12.

1. Takesaki, M., Theory of operator algebras I., Encyclopaedia of Mathematical Sciences 124. Operator Algebras and Non-Commutative Geometry 5. Berlin: Springer (ISBN 3-540-42248-X/hbk). xix, 415 p. (2002). ZBL0990.46034.