Definition 1. Let $B\subset A$ be C*-algebra. A projection from A onto B is a linear map $E: A \rightarrow B$ such that $E(b)=b$ for every $b\in B$. A conditional expectation from A onto B is a contractive completely positive projection $E$ from $A$ onto $B$ such that $E(bxb')=bE(x)b'$ for every $x\in A$ and $b, b'\in B$.
Theorem 2 (Tomiyama). Let $B\subset A$ be C*-algebra and $E$ be a projection from $A$ onto $B$. Then, the following are equivalent:
(1) $E$ is a conditional expectation;
(2) $E$ is contractive completely positive;
(3) $E$ is contractive.
Proof. We only have to prove that the last condition implies the first, so assume $E$ is contractive. Passing to double duals, we may assume that $A$ and $B$ are von Neumann algebras. We first prove that $E$ satisfies $E(bxb')=bE(x)b'$ for every $x\in A$ and $b, b'\in B$..........
My question: I can not understand why we can regard a C*-algebra as a von Neumann algebra? Could someone explain to me in detail? Many thanks.