Appendix
Published online by Cambridge University Press: 21 October 2009
Summary
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.
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- Introduction to Subfactors , pp. 133 - 150Publisher: Cambridge University PressPrint publication year: 1997