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.
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.