Categorical Aspects of von Neumann Algebras and AW ∗ - algebras Sander

We take a look at categorical aspects of von Neumann algebras, constructing products, coproducts, and more general limits, and colimits. We shall see that exponentials and coexponentials do not exist, but there is an adjoint to the spatial tensor product, which takes the role of coexponent. We then introduce the class of AW*-algebras and try to see to what extend these categorical constructions are still valid. Introduction The Gelfand duality between commutative unital C∗-algebras and compact Hausdorff spaces (see Proposition 2.16) has led to the idea that we can interpret general C∗algebras as generalized (non commutative) topological spaces. A simialar theorem is valid for von Neumann algebras; every commutative von Neumann algebra is isomorphic to the continuous functions on some hyperstonean space. This leads one to the idea that one can interpret von Neumann algebras as generalized (non commutative) measure spaces. In [5], A. Kornell studies the category of von Neumann algebras and interprets the dual category as a set-like category whose objects he calls quantum collections, in which quantum-mechanical computations can be made. This is inspired by the embedding of sets (seen as topological spaces with discrete topology) in the opposite of the category of von Neumann algebras via X 7→ ◦`∞(X). First, the category W∗ of von Neumann algebras and unital normal ∗-homomorphims is studied and this category has nice properties. It has products, coproducts, equalizers, coequalizers, and general limits and colimits. It does, however, not have exponents and coexponents. The non existence of coexponents in W∗ is the same as the non existence of exponentials in the opposite category, so this cuts ties with Set, the category of sets and functions. To remedy this, it is shown that instead of coexponents (which are left adjoints to the coproduct) there does exist a construction mimicing that of a coexponent, and this is a left adjoint to the spatial tensor product, making W∗ a closed monoidal category. A special case of this adjunction is the following formula: Hom(M∗N ,C) ∼= Hom(M,N ), which shows that any normal unital ∗-homomorphism between von Neumann algebras M and N comes from some homomorphic state on the free exponentials M∗N . Kornell then procedes to the category of von Neumann algebras and unital completely positive maps and shows that in this category there is a surjective natural transformation Hom(M∗N ,C)→ Hom(M,N ). This shows that any quantum operation is induced by a state on the free exonential. It becomes a natural question to ask if these constructions are special to von Neumann algebras, or if there is some larger class of operator algebras in which we can perform the same categorical constructions. In this paper, we try to do this for the catgory of AW ∗-algebras and AW ∗ morphisms. In the first chapter, we explain the basics of category theory and introduce the constructions we wish to study. We follow, in the second chapter, with the basics of operator