2 added 296 characters in body

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 C*‑algebras and W*-algebrasW*‑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 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 coincides does not coincide with the ultraweak topology and the canonical morphism A→A** is not an isomorphism.

On the other hand, if

We can fix this problem by composing the canonical morphism monomorphism A→A** is an isomorphism, then A has with the multiplication by a predualcertain central projection. However, therefore it 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.

1

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.