Suppose, that $A$ is a $C^{\ast}$-algebra. It follows, according to Sakai's book, that double dual of $A$ is also $C^{\ast}$-algebra. I'm not quite sure if I understand the proof correctly. Author proves that $A^{\ast \ast}=\pi(A)''$ where $\pi$ is universal representation showing first that the corresponding preduals coincide. But preduals ale only Banach spaces, so the question rises: how do we define multiplication and involution in the second dual of $A$? I've heard that can be done in natural way (not using the universal representation), using so called Goldstein theorem but I don't know what precisely this theorem states. If we follow this 'natural' procedure via Goldstein theorem and simultaneously use universal representation we get that double dual of $A$ and $\pi(A)''$ are isomorphic as BANACH SPACES-but is it true that they are isomorhic as $C^{\ast}$-algebras (in general it seems that this need not to be true as $\ell^{\infty}$ and $L^{\infty}[0,1]$ are isomorhic as Banach spaces but not as $C^{\ast}$-algebras).
Thank You for Your answers-still there is one detail which I don't understand, namely-we have isomorphism of Banach spaces $T:A^{\ast \ast} \to \pi(A)''$, this isomorphism preserves multiplication when restricted to $A$ and $A$ is dense in $A^{\ast \ast}$ as well as $T(A)$ in $\pi(A)''$. Morever $T$ is $\ast$ weak continuous. But it seems to me that multiplication IS NOT continuos with respect to weak $\ast$ topology, so I'm not sure if we have enough information to conclude that $T$ preserves multiplication in general ( for example preserving $\ast$ structure can be proved via Goldstine-we define $x^{\ast}$ for $x \in A^{\ast \ast}$ as $\lim_{i}a_i^{\ast}$ where $a_i \to x$ is suitable net. As involution $\ast$ on $A$ is continuos with respect to $\ast$ weak topology, therefore it can be extended to whole $A^{\ast \ast}$ and $T$ preserves this operation-how to repeat this argument for the multiplication if we have lack of continuity?)
