What is the Categorified GelfandNaimark Theorem?
Posted by Urs Schreiber
Bruce Bartlett has made a very fruitful observation, dealing with which requires a good understanding of a couple of details of
John Baez
HigherDimensional Algebra II: 2Hilbert Spaces
qalg/9609018.
Since we seem to need those details  and since I keep forgetting them  I’ll give a very condensed review of the main points.
The key definitions (stripped of the pedagogical background information) are this:
An $H^*$algebra is an algebra with a nicely behaved Hilbert space structure on it, modeled on the example of the algebra $\mathrm{End}(H)$ for $H$ some Hilbert space.
This is categorified by first defining a 2Hilbert space as a well behaved $\mathrm{Hilb}$enriched category, and then putting an algebra structure on that in a more or less obvious way.
More precisely:
 Definition 5. An $H^*$algebra is a Hilbert space $A$ equipped with an associative unital algebra structure and an antilinear involution $*:A \to A$ compatible with taking the adjoint of the operators of left and right multiplication of $A$ with itself.

Definition 2. An $H^*$category $C$ is
 a $\mathrm{Hilb}_\mathbb{C}$enriched category
 with a $*$structure $* : C \to C$
 that induces an antinatural transformation $* : \mathrm{Hom}_C \to \bar \mathrm{Hom}_C \,,$ where $\bar{(\cdot)} : \mathrm{Hilb} \to \mathrm{Hilb}$ is switching the complex structure.
 Definition 9. A 2Hilbert space is an abelian $H^*$category.
 Definition 38. A (braided/symmetric) 2$H^*$algebra is 2Hilbert space with a (coherently weak) (braided/symmetrically braided) associative unital algebra structure on it.
The real interest is in super Hilbert spaces and their categorification. The role of the grading involution is played in the categorified setup by the balancing:
 Definition 44. In a braided monoidal category, the balancing $x \stackrel{b_x}{\to} x$ on any object is the morphism whose tangle diagram is a single looping.
Recalling
 Proposition 57 (DoplicherRoberts reconstruction)
the goal is to slightly generalize this, such as to obtain a decent categorification of the GelfandNaimark theorem, which says that any $C^*$ algebra is isomorphic to functions on (= representations of) its spectrum.
Spectrum and representation have a rather obvious categorification:
 Definition 62. A (compact) supergroupoid $G$ is a (compact) groupoid with an involution $\beta : \mathrm{Id}_G \to \mathrm{Id}_G \,.$
 Definition 60. The spectrum $\mathrm{Spec}(H)$ of a 2$H^*$algebra $H$ is the category of functors $H \to \mathrm{SuperHilb}_\mathbb{C} \,.$
 Definition 63. The category $\mathrm{Rep}(G)$ of representations of a supergroupoid $G$ is the category of functors $\rho : G \to \mathrm{SuperHilb}_\mathbb{C}$ which send the involution on $G$ to the grading involution on $\mathrm{SuperHilb}_\mathbb{C}$.
The desired categorification of GelfandNaimark now says
 Theorem 64. (generalized DoplicherRoberts theorem) Every symmetric 2$H^*$ algebra is equivalent to the representations of its spectrum $H \simeq \mathrm{Rep}(\mathrm{Spec}(H)) \,.$
It is natural to make the
 Conjecture (p. 53) The 2categories $\mathrm{CptSupGrpd}$ of compact supergroupoids and $2H^*\mathrm{Alg}$ of 2$H^*$algebras are equivalent, with $\mathrm{CptSupGrpd} \stackrel{\mathrm{Rep}}{\to} 2H^*\mathrm{Alg}$ and $\mathrm{CptSupGrpd} \stackrel{\mathrm{Spec}}{\leftarrow} 2H^*\mathrm{Alg}$ being weak inverses.
Michael Müger did some similarsounding constructions for DoplicherRoberts #  but I am too lazy to try to compare the details.
more general gradings
There are a couple of possible generalizations that suggest themselves.
For instance, I would be interested in seeing analogous constructions for gradings more general than the $\mathbb{Z}_2$grading used so far.
Here are two motivations:
What is called the balancing above can be taken to be the twist in ribbon categories, I think.
(Compare for instance equations (2.8) and (2.12) of hepth/0204148).
That twist has in general eigenvalues not just in $\{+1,1\}$ but in $U(1)$. For instance for ribbon categories used in rational 2D CFT #, the twist acts on a simple object $U$ by multiplication with
where $\Delta_U$ is the “conformal weight” of $U$.
The Kcohomology of $X$ can be regarded as a kind of decategorification of the derived category of certain sheaves on $X$ #. This (very) roughly amounts to passing from $\mathbb{Z}$graded vector spaces to $\mathbb{Z}_2$graded vector spaces.
So if Kcohomology is related to $[P_1(X,b_{\mathbb{Z}_2}), \mathrm{Hilb}_\mathbb{C}^{\mathbb{Z}_2}]$, maybe we eventually want to pass to $[P_1(X,b_{\mathbb{Z}}), \mathrm{Hilb}_\mathbb{C}^{\mathbb{Z}}]$.
(Here I am using the notation from these comments.)
Is anything known about such generalizations?