This is a question that I have been working on recently together with Robert Furber and Bas Westerbaan. Let me sketch what we know so far, starting with the case of (infinite) direct sums, treated in this paper. These are a special case of direct integrals which already display some of the essential features.

In the following, it may help to keep the example of the category of Hilbert spaces in mind, or more generally the category of normal representations of any W*-algebra. These categories contain Hilbert spaces of all dimensions, regardless of separability.

## Infinite direct sums

Let $I$ be an indexing set for a family of objects $(A_i)_{i
\in I}$ of which we want to take the direct sum $\oplus_{i\in I} A_i$. For another object $B$, consider the set
$$\oplus_i \mathsf{C}(A_i,B) := \left\{(f_i : A_i \to B)_{i\in I} \mid \sum_i f_i f_i^* < \infty \right\}$$
as a normed space under the $\ell^2$-norm
$$\|(f_i)\| := \|\sum_i f_i f_i^* \|^{1/2}.$$
It is not hard to show that this is actually a Banach space. Moreover, this construction is clearly functorial in $B$.

In line with the reindexing which has already been discussed, we use:

**Definition**: The *direct sum* $\oplus_i A_i$ is any object which represents the $\mathsf{Ban}$-enriched functor $\oplus_i \mathsf{C}(A_i,-) : \mathsf{C} \to \mathsf{Ban}$.

While the representability of a $\mathsf{Ban}$-enriched functor a priori characterizes the representing object up to unique isomorphism, we have shown that this unique isomorphism must actually be *unitary*. So although our universal property does not assume any compatibility with the involution/dagger, this turns out to come out naturally. This will be obvious to anyone who knows that if $u$ is an invertible element in a C*-algebra with $\|u\| = \|u^{-1}\| = 1$, then $u$ is unitary.

Although our definition may look like the definition of a (weighted) limit at first, I do not think that this is actually the case, even for finite $I$.

We have then shown the equivalence with the earlier definition of direct sums in W*-categories given by Ghez, Lima and Roberts:

**Proposition:** An object $A$ is a direct sum $\oplus_i A_i$ if and only if there is a family of morphisms $(\kappa_i : A_i \to A)_{i\in I}$ such that $\kappa_i^* \kappa_j = \delta_{ij}$ and
$$\sum_i \kappa_i \kappa_i^* = 1,$$
with convergence in the ultraweak topology on $\mathsf{C}(A,A)$.

As one would expect, this is an equivalence of structures on $A$ rather than properties of $A$. The "if" part is the more difficult direction.

In the category of Hilbert spaces, this clearly recovers the usual direct sums, and more generally so in any category of normal representations of a W*-algebra.

As was already known to Ghez, Lima and Roberts, we thus conclude that the existence of direct sums is still a property of a W*-category rather than extra structure.

## Direct integrals

Having treated direct sums as a warm-up, we can now try to tackle direct integrals. This seems to be more of the same, but with many tricky pitfalls that we need to steer clear of, and I'm not sure whether we've managed to do that yet. The following items describe where we're currently at. **I will update this answer as soon as we have an actually working definition. Further contributions by MOers will be welcome, just drop me an email.**

The role of the indexing set $I$ is now played by a complete strictly localizable compact measure space $(X,\Sigma,\mu)$. This class is large enough for every commutative W*-algebra to be of the form $L^\infty(X,\Sigma,\mu)$, regardless of separability.

For a given W*-category $\mathsf{C}$, we then want to construct a W*-category $L^\infty(X,\mathsf{C})$, whose objects are suitably defined measurable families of objects $(A_x)_{x\in X}$ indexed by our measure space. To this end, we can try to work with a generalization of measurable fields of Hilbert spaces, as follows. For $(A_x)_{x\in X}$ an arbitrary family of objects and $B$ any other object, let us say that a family of morphisms $(f_x : B \to A_x)_{x\in X}$ is measurable if the map $x\mapsto f_x^* f_x$ is measurable. The measurable families of morphisms form a subfunctor of the functor $\prod_{x\in X} \mathsf{C}(-,A_x) : \mathsf{C}^{\mathrm{op}} \to \mathsf{Set}$. A *measurable field of objects* could now be a further subfunctor of this measurable families functor, satisfying properties for every $B$ analogous to those of measurable fields of Hilbert spaces (see e.g.~Dixmier A69).

One could then hope that the definition of direct integral will be similar to the direct sum case: any object which represents the $\mathsf{Ban}$-enriched functor
$$B \longmapsto \int_X^\oplus \mathsf{C}(A_x,B)\, d\mu(x) := \left\{ (f_x : A_x \to B)_{x\in X} \textrm{ in the subfunctor} \: \bigg| \: \int_X f_x f_x^* \, d\mu(x) < \infty \right\},$$
again carrying the $L^2$-norm. **As per the discussion below, starting with Simon Henry's comment, it doesn't quite work like this yet.**

Since working with this definition of measurable fields of objects will be cumbersome in practice, one can try to add an additional layer of abstraction by assuming that $\mathsf{C}$ itself is a category internal to (large and suitably nice) measurable spaces, and then simply require the map $x\mapsto A_x$ to be measurable. Then a family $(f_x : B \to A_x)$ is part of the further subfunctor if and only if it is measurable as a map $X \to \mathrm{Mor}(\mathsf{C})$.

The resulting category of measurable families of objects, $L^\infty(X,\mathsf{C})$, should again be a W*-category. For $\mathsf{C} = \mathsf{Hilb}$, we should recover Yetter's categories of measurable fields of Hilbert spaces at least insofar as we should have good arguments for any potential differences. Then forming direct integrals should correspond to the analogous universal property as in the case of direct sums.

An important consistency check will be the following: if $\mathsf{C}$ has direct integrals $\int_X^\oplus A_x \, d\mu(x)$, then such a direct integral should come equipped with a normal representation of $L^\infty(X)$. Moreover, we then expect to get an equivalence of W*-categories between $L^\infty(X,\mathsf{C})$ and normal representations of $L^\infty(X)$ in $\mathsf{C}$, generalizing the spectral theorem.

**Summary:** We don't have a complete treatment of direct integrals yet. Concerning the necessary extra structure on the W*-category, it seems that we either need no extra structure at all, resulting in a cumbersome definition of direct integrals; but probably we want to require the collections of objects and morphisms to come equipped with suitable $\sigma$-algebras, in a way that makes the structure maps measurable, in order to get a potentially more usable definition.

locally trivialdirect integrals reduce to $\oplus$. And at least according to Wikipedia, direct integrals reduce "in some sense" to locally trivial ones. $\endgroup$ – Tim Campion♦ Apr 9 '20 at 17:59