TITLE: A variant of the Stone-Weierstrass theorem?
QUESTION [6 upvotes]: I would like to ask specialists in C*-algebras if the following variant of the Stone-Weierstrass theorem is true.
Suppose $A$ is a C*-algebra and $C$ is its center. Since $C$ is a commutative C*-algebra, there exists a compact space $T$ such that $C$ is isomorphic to the algebra $C(T)$ of continuous functions on $T$. Does this mean that there exists a C*-algebra $B$ such that 
1) $A$ is isomorphic to a closed subalgebra in the algebra $C(T,B)$ of continuous mappings $f:T\to B$ with the pointwise algebraic operations (and the topology of uniform convergence on $T$), and
2) this isomorphism turns $C$ into the algebra of scalar mappings, i.e. the mappings of the form $f(x)=\lambda(x)\cdot 1_B$, where $1_B$ is the identity in $B$, and $\lambda(x)$ $\in$ $\mathbb{C}$ for all $x\in T$.
EDIT 21-03-12: All the C*-algebras here are supposed to be unital, excuse me for not mentioning this from the very beginning!

REPLY [10 votes]: The statement of the Dauns-Hofmann theorem is actually too weak to get bundles of $C^*$-algebras. For completeness, let me state it:
Let $A$ be a $C^*$-algebra. For each $P \in Prim(A)$, let $\pi_P \colon A \to A/P$ be the quotient map. Then there is an isomorphism $\phi$ of $C_b(Prim(A))$ onto the center $ZM(A)$ of the multiplier algebra $M(A)$ such that for all $f \in C_b(Prim(A))$ and $a \in A$
$$
\pi_P(\phi(f)a) = f(P)\pi_P(a)
$$
for every $P \in Prim(A)$. Usually one writes $f \cdot a = \phi(f)a$. 
So, the best you could hope for is some kind of sheaf of $C^*$-algebras over the primitive ideal space. Getting local triviality in general is kind of hopeless, I think. A reading recommendation for these matters would be the book "Morita Equivalence and Continuous-Trace $C^*$-algebras" by Raeburn and Williams. 
For continuous trace $C^*$-algebras things are quite different. These are all Morita equivalent (or stably isomorphic) to sections in a bundle of compact operators!