Consider the monomorphism A→A** of Banach spaces.
Here A** denotes the second dual of A as a Banach space.
The Banach space A** is a von Neumann algebra with the predual being A*.
See Section 1.17 in Sakai's [C*-algebras and W*-algebras](http://gen.lib.rus.ec/get?nametype=orig&md5=EEDE446B49C1D34D58D4942BD30F8A38).

We have a commutative square of Banach spaces consisting of morphisms
A→A*\*→B*\* and A→B→B*\*.
Thus we can pull back the ultraweak topology on A** to A
and obtain a functorial topology on C*-algebras due to the commutativity of the square above.

Henceforth denote by A* the dual of A in the new topology and
by A** the dual of A* in the norm topology

If A is a von Neumann algebra, then the functorial topology coincides with the ultraweak topology
and the canonical morphism A→A** is an isomorphism.

On the other hand, if the canonical morphism A→A** is an isomorphism, then A has a predual,
therefore it is a von Neumann algebra.