Commutative W*-algebras as a Markov Category

We show that the opposite of the category of commutative W*-algebras, with positive unital maps as morphisms, (CW*AlgPU) is a Markov category. We do this by showing that the comonad of which CW*AlgPU is the coKleisli category is commutative, where the chosen tensor product is the coproduct (of CW*Alg, since CW*AlgPU doesn’t have one). It follows, by the duality between commutative W*-algebras and measure spaces, that the corresponding monad on the category of compact complete strictly localizable measure spaces is commutative.
On the way, we give a universal property in CW*AlgPU for the colimits of CW*Alg in terms of “true continuity” of positive-operator-valued measures. This is essential to describe the positive unital maps out of a CW∗Alg coproduct. We can then explicitly calculate the coproduct L^∞([0, 1]) + L^∞([0, 1]) as L^∞([0, 1])^{2^ℵ_0).