TITLE: Double dual of C*-algebra
QUESTION [5 upvotes]: 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?)

REPLY [5 votes]: As Yemon says, it is the Arens product on the double dual of a Banach algebra that you're thinking of. Yes, $A^{**}$ and $\pi(A)''$ are isometrically isomorphic as Banach spaces and $*$-isomorphic as C*-algebras. This is surely in Sakai's book that you're already looking at.
A related question that you didn't quite ask is whether two C*-algebras which are isometrically isomorphic as Banach spaces must be $*$-isomorphic as C*-algebras. The answer is no --- there are C*-algebras which are not $*$-isomorphic to their opposite algebras (change the order of the product and leave everything else, including norm, alone). However, isometric C*-algebras will have the same state space and this implies that they are Jordan isomorphic, and roughly speaking, switching the order of the product is the worst thing that can happen. See the book State Spaces of Operator Algebras by Alfsen and Schultz.
The "right" version of this result is that two C*-algebras that are completely isometric must be $*$-isomorphic.