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.

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

Unfortunately, if A is a von Neumann algebra, then the functorial topology does not coincide with the ultraweak topology and the canonical morphism A→A** is not an isomorphism.

We can fix this problem by composing the monomorphism A→A** with the multiplication by a certain central projection. However, the definition of this central projection relies on the fact that A is a von Neumann algebra and I don't see any way to extend it to arbitrary C*‑algebras.