A three-functor formalism for commutative von Neumann algebras
Andre G. Henriques, Thomas A. Wasserman
TL;DR
The paper constructs a unitary, bi-involutive three-functor formalism for commutative von Neumann algebras and their modules, positioning the opposite of the CvNa category as the space of spaces and using $A$-Mod with fusion over $A$ as the target categories. It identifies the duality functor $\mathbb D_X$ with a contravariant involution on $A$-modules and shows base-change and projection are expressed via the fiber-product isomorphism $L^2(A)\boxtimes_C L^2(B) \cong L^2(A \ast_C B)$, integrating with the monoidal fusion structure. The framework yields an involutive (and dagger) three-functor formalism in the $\mathrm{W}^*$-categorical setting, where $f^{\bigstar}$ and $f_{\bigstar}$ play roles reminiscent of $f^*,f^!$ and $f_*,f_!$, respectively, and satisfy an exotic adjoint relationship appropriate for $\mathrm{W}^*$-categories. As an application, the authors prove Fell absorption for representations of measure groupoids, showing that the regular representation tensorially absorbs any faithful unitary representation, demonstrated via the fiber-product formalism and $L^2$-density constructions. The work extends the six-functor philosophy into a non-separable, operator-algebraic setting and provides a robust toolkit for analyzing representations and morphisms in measure-theoretic contexts.
Abstract
A three-functor formalism is the half of a six-functor formalism that supports the projection and base change formulas. In this paper, we provide a three-functor formalism for commutative von Neumann algebras and their modules. Using the Gelfand-Naimark theorem, this gives rise to a three-functor formalism for measure spaces and measurable bundles of Hilbert spaces. We use this to prove Fell absorption for unitary representations of measure groupoids. The three-functor formalism for commutative von Neumann algebras takes values in W*-categories, and we discuss in what sense it is a unitary three-functor formalism.