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.
You have canonical embeddings $A\hookrightarrow A^{**}$, $B\hookrightarrow B^{**}$. The embedding is given by $a(f)=f(a)$ for all $a\in A$, $f\in A^*$. Restriction allows us to embed $B^{**}$ into $A^{**}$.
Now, a map $E:A\to B$ like the conditional expectation admits a dual map $E^*:B^*\to A^*$ given by $(E^*f)(a)=f(E(a))$. And similarly we can obtain a double dual map $E^{**}:A^{**}\to B^{**}$. When restricted to $A$, the following happens; for $a\in A$, $g\in B^*$, $$ E^{**}(a)(g)=a(E^*g)=(E^*g)(a)=g(E(a))=(E(a))g. $$ This shows that $E^{**}|_A=E$. Since $\|E^{**}\|=\|E\|$, we get that $E^{**}$ is contractive, and now one performs that proof for $E^{**}$, $A^{**}$, $B^{**}$ to show that $E^{**}$ is completely positive. As $E$ is a restriction of $E^{**}$, it will also be completely positive.