What is a support projection in a $C^*$-algebra?

Question

Let $A$ a $C^*$-algebra and consider $a\in A$ self adjoint and $ax=xa$ for all $x\in A$. I want to know:

-what is the support projection of $a$?
-what is the definition of the support projection of an element in a $C^*$-algebra in genereal?

I used google and I searched in books, but I'm not sure about the correct definition.

Answer 1

There is no such thing; a C $^*$-algebra can be projectionless. What usually happens is that $A $ is represented in $B (H) $ and then the support projection of $a $ is the orthogonal projection onto $ aH $, which belongs to $A''$.