What is the Categorified Gelfand-Naimark Theorem?

October 11, 2006

What is the Categorified Gelfand-Naimark Theorem?

Posted by Urs Schreiber

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Bruce Bartlett has made a very fruitful observation, dealing with which requires a good understanding of a couple of details of

John Baez
Higher-Dimensional Algebra II: 2-Hilbert Spaces
q-alg/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.

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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 2-Hilbert 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 CC 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 2-Hilbert space is an abelian $H^*$ -category.
Definition 38. A (braided/symmetric) 2- $H^*$ -algebra is 2-Hilbert 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 (Doplicher-Roberts reconstruction)

the goal is to slightly generalize this, such as to obtain a decent categorification of the Gelfand-Naimark 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 Gelfand-Naimark now says

Theorem 64. (generalized Doplicher-Roberts 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 2-categories $\mathrm{CptSupGrpd}$ 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 similar-sounding constructions for Doplicher-Roberts # - but I am too lazy to try to compare the details.

Posted at October 11, 2006 5:01 PM UTC

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more general gradings

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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 hep-th/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

(1) $\exp(-2\pi i \Delta_U)\mathrm{Id}_U \,,$

where $\Delta_U$ is the “conformal weight” of $U$ .
The K-cohomology 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 K-cohomology 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?

Posted by: urs on October 11, 2006 8:36 PM | Permalink | Reply to this

Re: more general gradings

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Urs wrote:

What is called the balancing above can be taken to be the twist in ribbon categories, I think.

Yes: what some people called ribbon tensor categories, Joyal and Street called “balanced” braided monoidal categories. You probably can guess what I mean, but let me zip through the precise definitions. Different people use different terminology, but right now I’ll talk like a category theorist:

If you have a braided monoidal category we say an object $x$ has a dual $x^*$ if the functor $a \mapsto x \otimes a$ has a left adjoint given by $a \mapsto x^* \otimes a$ If such $x^*$ exists it’s unique up to isomorphism. There are two opportunities to get mixed up between left and right in the previous paragraph, and in a mere monoidal category we have to be careful: we have left and right duals for objects. But, in a braided monoidal category the functor $a \mapsto x \otimes a$ is naturally isomorphic to the functor $a \mapsto a \otimes x$ so we don’t get that distinction. That’s one reason I’m assuming our monoidal category is braided.

A braided monoidal category is compact if every object has a dual. One can then define a contravariant functor $x \mapsto x^*$

A compact braided monoidal category is balanced if it is equipped with a balancing: a natural isomorphism $\beta_x : x \to x^{**}$ to the identity functor. In string diagrams, we can draw $\beta_x$ as a full twist in a ribbon labelled by the object $x$ . The naturality condition I just mentioned is then equivalent to a more complicated but visually appealing condition involving the braiding; see page 3 here.

So, yes: if you wanted to generalize from the “super-Hilbert spaces” described above to more general setups that allow not just bosons and fermions but also “anyons”, we should replace our symmetric 2- $H^*$ -algebra by a braided 2- $H^*$ -algebra. In Theorem 46 and Corollary 47 of HDA2, we see that any such gadget can be made in to a balanced braided monoidal $*$ -category where the balancing is unitary: $\beta_x \beta_x^* = 1$ $\beta_x^* \beta_x = 1$ And, it can be done in a unique way.

For a simple object $x$ (one that’s not a direct sum of others), $\beta_x$ is just a phase, called the balancing phase of $x$ . When our braided 2- $H^*$ -algebra is actually symmetric, this phase is $\pm 1$ , which determines whether $x$ is bosonic or fermionic. But, in general, it can be any phase. Hence the term “anyons”.

All this is a lot of fun, and very important in 3d topological quantum field theory.

So, I indeed wondered how to generalize the theorems you cite to this braided case, but I could never figure out how. The basic idea seems obvious enough: we want every braided 2- $H^*$ -algebra to be the category of representations of a “compact quantum groupoid”. But, I don’t know how to define the quoted phrase in such a way that I can prove a result like this.

The main problem is generalizing the Doplicher-Roberts theorem from groups to quantum groups. Tannaka-Krein reconstruction works just as well for quantum groups as for groups, but Doplicher-Roberts reconstruction is (seemingly) harder.

However, I haven’t thought about this since Deligne found a slick new proof of Doplicher-Roberts reconstruction for groups starting from Tannaka-Krein reconstruction. Maybe someone has already figured out how to generalize his idea to quantum groups. That would overcome a big obstacle.

Posted by: John Baez on October 19, 2006 1:58 AM | Permalink | Reply to this

Re: more general gradings

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John said:

So, I indeed wondered how to generalize the theorems you cite to this braided case, but I could never figure out how. The basic idea seems obvious enough: we want every braided 2-H *-algebra to be the category of representations of a “compact quantum groupoid”. But, I don’t know how to define the quoted phrase in such a way that I can prove a result like this.

The main problem is generalizing the Doplicher-Roberts theorem from groups to quantum groups. Tannaka-Krein reconstruction works just as well for quantum groups as for groups, but Doplicher-Roberts reconstruction is (seemingly) harder.

Is the possibility of a Doplicher-Roberts-style theorem for compact quantum group(oids?) conjectured in a paper anywhere? I want to cite it. (And read it, too, of course.)

Posted by: Jamie Vicary on March 27, 2008 5:47 PM | Permalink | Reply to this

Re: What is the Categorified Gelfand-Naimark Theorem?

Based only on the title of this posting, I mention

G.V. Kondratiev, Strict ∞-Categories. Concrete Duality.

who indeed treats Gelfand-Naimark categorically as one instance of a more general duality

jim

Posted by: jim stasheff on November 13, 2006 9:22 PM | Permalink | Reply to this

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Excerpt: Paolo Bertozzini, Roberto Conti and Wicharn Lewkeeratiyutkul on categorified spaces and the many-object version of C-star algebras.
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Re: What is the Categorified Gelfand-Naimark Theorem?

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I’ve noticed that, on this blog, people use “Gelfand-Naimark theorem” as the name for the result about commutative C*-algebras being the algebras of functions on their spectra, or its categorification. But I always call this the “Gelfand representation theorem”, and reserve “Gelfand-Naimark theorem” for the result that an arbitrary C*-algebra isometrically *-embeds in the algebra of bounded operators on a Hilbert space, which is quite different. Wikipedia agrees with me, but that doesn’t mean anything. Can somebody who’s familiar with the history of this set me straight?

Posted by: Jamie Vicary on February 25, 2008 4:29 PM | Permalink | Reply to this

Re: What is the Categorified Gelfand-Naimark Theorem?

Had a quick look in Bonsall and Duncan and, contrary to what I remembered, their index uses G-N theorem to refer to the representation theorem for arbitrary C*-algebras (confusingly this theorem uses what is usually called the GNS construction). So you and Wikipedia are vindicated on that score.

I’m not keen on the wikipedia entry for “Gelfand representation theorem”, because it puts all the emphasis on the commutative C*-case, and really the Gelfand representation is a much more general notion. The extra constraints obscure how it is a spiritual descendant of, say, the Nulstellensatz f’rinstance. It also elides the fact that the Gelfand topology isn’t always the same as the hull-kernel topology (they do agree for commutative C*-algebras, but that’s somehow a boon rather than falling out of naturality).

I notice I haven’t answered your question, may look this up later…

Posted by: Yemon Choi on February 25, 2008 10:23 PM | Permalink | Reply to this

Re: What is the Categorified Gelfand-Naimark Theorem?

Yemon, thanks for those comments! Seeing as you and other Café patrons are undoubtedly losing sleep over this, I thought I’d let you know the results of my sleuthing.

The original paper is available to read in full at Google Book Search, from a proceedings volume celebrating its 50th birthday:

C*-algebras: 1943-1993, A Fifty Year Celebration

Anyway, it seems that the “Gelfand representation theorem” is due solely to Gelfand, but is just an embedding theorem, not an isomorphism theorem. It takes Gelfand and Naimark quite a bit of work to prove it’s an isomorphism for the case of commutative C*-algebras. In the same paper, they also prove the embedding theorem for arbitrary C*-algebras into the ring of bounded operators on a Hilbert space.

So historically, both theorems have good reason to be called the “Gelfand-Naimark theorem”. Perhaps the best way to cope with this is to refer to the “Gelfand representation”, and use the fact that it’s an isomorphism for C*-algebras, but remember that this isomorphism wasn’t proven by Gelfand alone.

Posted by: Jamie Vicary on March 29, 2008 2:09 PM | Permalink | Reply to this

Re: What is the Categorified Gelfand-Naimark Theorem?

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Over on the Noncommutative geometry blog Masoud Khalkali, apparently prompted by the discussion above, has posted a detailed comment with a reply:

On Gelfand-Naimark theorems

He writes:

There seems to be some inconsistency, mainly in the Internet, in naming one of the main theorems proved by Gelfand and Naimark in their foundational 1943 paper:

On the imbedding of normed rings into the ring of operators in Hilbert space. Rec. Math. [Mat. Sbornik] N.S. 12(54), (1943). 197–213

I have seen the result in question referred to as Gelfand-Naimark’ or Gelfand’s theorem. Also, talking to younger people I get a sense of confusion as to how it should be called. Before getting to the theorem in question, let me indicate why this paper is so important. This paper is fundamental for the following 4 reasons: […]

Posted by: Urs Schreiber on April 9, 2008 8:58 PM | Permalink | Reply to this

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October 11, 2006