Concrete and abstract von Neumann algebras

We used the word ‘concrete’ in the opening paragraphs of the first section of this book, to indicate that we were looking at a concrete realisation or representation (as operators on Hilbert space) of a more abstract object. The abstract notion is as follows: suppose M is a C*-algebra – i.e., a Banach *-algebra, where the involution satisfies ∥x*x∥ = ∥x∥2 for all x in M; suppose further that M is a dual space as a Banach space – i.e., there exists a Banach space M* (called the pre-dual of M) such that M is isometrically isomorphic, as a Banach space, to the dual Banach space (M*)*; let us temporarily call such an M an ‘abstract von Neumann algebra’.

It turns out – cf. [Tak1], Corollary III.3.9 – that the pre-dual of an abstract von Neumann algebra is uniquely determined up to isometric isomorphism; hence it makes sense to define the σ-weak topology on M as σ(M, M*), the weak* topology on M defined by M*.

The natural morphisms in the category of von Neumann algebras are *-homomorphisms which are continuous relative to the σ-weak topology (on range as well as domain); such maps are called normal homomorphisms.