Is there a way to equip every C*-algebra A with a functorial topology such that
the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra?
Here A** denotes the dual of A* in some other functorial topology.
Of course if we restrict our attention to von Neumann algebras from the beginning,
then the σ-weak topology on A and the norm topology on A* answer the question.
Best Answer
This isn't quite an answer, but it might lead to one. Takesaki, III, Thm 3.16 shows that a C*-algebra A is a W*-algebra (i.e. non-spacial version of a von Neumann algebra) if and only if A is monotone closed and admits sufficiently many normal positive linear functionals.
Monotone closed == bounded increasing net of self-adjoints has a supremum.
Normal == Positive functional which respects the supremum. I guess we can then define normal to mean: positive and negative parts etc. are normal.
So on the class of Monotone Closed C*-algebras, we could let A* be the space of normal functionals, and give A* to norm topology. Then I think A=A** only when A is a von Neumann algebra (or otherwise A won't even inject into A**).
What I don't see is how to extend this to all C*-algebras.