Existence of central cover for a representation of a C*-algebra

Question

I've been trying to learn the basics about the representation theory of C*-algebras and came across the following in Pedersen's C*-algebras and their Automorphism Groups:

With each (non-degenerate) representation $(\pi,H)$ of a C*-algebra $A$ we associate the projection $c(\pi)$ in the centre of $A''$ for which $A''c(\pi)$ is isomorphic to $\pi(A)''$. We say that $c(\pi)$ is the central cover $(\pi,H)$.

This is where $A''$ is the enveloping von Neumann algebra of $A$ and $\pi(A)''$ the strong closure of $\pi(A)$ in $\mathcal{B}(H)$.

My question is, why does $c(\pi)$ necessarily exist? This is probably covered earlier in the book, but I've had trouble finding it.

Edit: A comment about uniqueness would also be appreciated.

Answer 1

Consider the given representation as sitting inside the universal representation. Then there exists a projection $E$ in the commutant of the enveloping von Neumann algebra which projects onto the representation subspace. Now consider the family $F$ of central projections that contain $E$. This is clearly non-empty as the identity is always one such projection. Consider all the ranges of $G\in F$, which are closed subspaces of the universal representation Hilbert space and take their intersection. The associated projection lives in the centre of the enveloping von Neumann algebra and is, by definition, the central cover of $\pi$, which also happens to be the smallest central projection that contains $E$. Uniqueness should then follow from this minimality requirement.

Answer 2

Just to make a link with other things:

• If A were a von Neumann algebra and $\pi$ a normal representation, then its kernel would be an weakly closed two sided ideal. Then there exists a unique central projection $c$ such that $ \mathrm{Ker}(\pi) = c A = A c$ and thus $\pi(A) \cong (1-c) A$. (cf. for example "C*- algebras and W*-algebras" S. Sakai, 1.10.1 p.24, left and right ideal version, a little check to do)

Then given a $C^*$-algebra, one can always get a von Neumann algebra by considering the universal envelopping alg.