Wednesday, April 9, 2008

On Gelfand-Naimark Theorems

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

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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:

1) C*-algebras were abstractly defined in this paper for the first time ever (in their main axioms they had two extra conditions, which as authors themselves indicated, but were unable to prove, were redundant. It took some 17 years to reach to the current concise formulation of the main axiom, called the C*-identity-see below. This needs another post to explain and I hope we can get to that in another time. This book gives a detailed account of this circle of ideas). Together with Murray-von Neumann's series of papers on Rings of Operators, or what later came to be called von Neumann algebras (1936-1943), the Gelfand-Naimark paper formed the foundaton stone of operator algebras and, eventually, noncommutative geometry.

2) Commutative C*-algebras were fully characterized in this paper as algebras of continuous functions on compact spaces. This is the theorem that concerns us in this post and we shall get to that later.

3) General (i.e. not necessarily commutative) C*-algebras were shown to admit a faithful embedding in the algebra of bounded operators on a Hilbert space.

4) The notion of state on a C*-algebra was introduced (but not under its current name) and used in the proof of 3) . This was later streamlined by I. Segal in 1947 and today we talk of the GNS (Gelfand-Naimark-Segal) construction.

Now the theorem in item 2) above (let us call it CGNT for commutatve Gelfand-Naimark theorem') only appears as a Lemma in the paper, in page 3, and was not even mentioned in the introduction! For sure it was needed for the proof of the noncommutative theorem, but they also mention that it is of independent interst as well. Obviously the authors felt, as reflected in their title, that the noncommutative result in the main theorem of the paper. I have seen CGNT referred to as the Gelfand-Naimark theorem or Gelfand's isomorphism theorem. Operator algebra books correctly call it the commutative Gelfand-Naimark theorem. Wickepdia calls it Gelfand's reprsenation theorem and reseves Gelfand-Naimark for the noncommutative theorem in item 3) above.

The proof of the CGNT is based on Gelfand's theory of commutative Banach algebras and in fact is one of its landmark applications. Another, earlier, major success of the theory was Gelfand's surprizingly short and elegant proof of Wiener's 1/f theorem: if a function f has an absolutely convergent Fourier series and is nowhere zero then its inverse 1/f has an absolutely convergent Fourier series as well.

A (complex, unital) Banach algebra is a complex algebra, equipped with a complete normed vector space structure. Futhermore, the norm and the multipcative structure are related by the identity $||xy|| \leq ||x|| \, ||y||$. A basic commutative example to keep in mind is the algebra C(X) of complex valued continuous functions on a compact Hausdorff space under pointwise addition and multiplication and the sup norm. But there are many other commutative examples of different natures. Important notions introduced in Gelfand's theory were that of the spectrum of an element and the fact that it is always non-empty; the spectrum of the algebra; and the Gelfand transform. The spectrum Spec (A) consists of multiplicative linear maps A....> C (charcaters or C-points of A). Being a subset of the dual space of A, it inherits a natural compact Hausdorff topology. Spec (A) can also be described as the set of maximal ideals of A: to a character associate its kernel...... The Gelfand transform is the more or less tautological map:
$\Gamma: A\longrightarrow C(spec(A))$ $\Gamma(a)(\chi)= \chi(a)$
It is clearly an algebra map and is contractive (norm decreasing), but it need not be faithful. In the special case when A is the group algebra of an abelian group it reduces to Fourier transform. Finding the right class of commutative Banach algebras for which $\Gamma$ is an isometric isomorphism is what is acheived by CGNT. A C*-algebra is an involutive Banach algebra which satisfies the C*-identity $||x*x||=||x||^2$

It is hard to exagerate the importance of the C*-identity. It has many implications, e.g. the uniqueness of the C*- norm, continuity of involutive algebra maps,....... Typically, for an involutive Banach algebra we have just an inequality x*x\leq x^2. The C*-identity puts
C*-algebras in a very special place among all Banach algebras, rather similar to the privileged position of Hilbert spaces among all Banach spaces. The world of Banach spaces is wild, but the Hilbertian universe is tame!

A few comments are in order here:

i) These days everything must be categorical' (I am afraid this is utterly out of date now and I should say categorified'!-but let us be pedantic). In fact the CGNT goes a long way towards establishing an equivalence between the categories of commutative unital C*-algebras and compact Hausdorff spaces. Let us call these categories (with appropriate notion of morphism in each case) A and S. We have two functors
Spec: A^o .......>S and C: S ^o..............>A

(o means the dual or opposite category), assigning the spectrum and the algebra of complex valued continuous functions, respectively. These are equivalences of categories, (quasi) inverse of each other. In fact the composition C Spec: A .....> A is just the Gelfand transform and by the CGNT we know that it is isomorphic to the identity functor. This is the hard part. To show that the functor Spec C: S .......>S is isomorhic to the identity functor is much easier and is elementary. You just have to show that the Spec (C(X))=X for any X.

Now this way of thinking about the CGNT makes it very similar to other duality theorems in mathematics that puts in duality a category of spaces with a category of commutative algebras. A grand example of this is Hilbert's Nullstellensatz which implies that: the category of affine algebraic varieties over an algebraically closed field is equivalent to the dual of the category of finitely generated commutative reduced algebras. (reduced means there are no nilpotent elements). I think, comparing the two theorems, this reduced condition should be compared with the C*-identity. Here is a question that has puzzled me for some time and is for experts in noncommutative algebraic geometry: what is the right notion of a noncommutative affine algebraic variety sugested by Nullstellensatz? We know that in NCG the category of C*-algebras is in many ways a good category of noncommutative spaces. In other words we keep the C*-identity. In the algebraic case shall we keep this reduced condition?

ii) The spectrum of a commutative C* -algebra may not be visible' at first sight, unless the algebra is already in the form C (X). This often leads to interesting results. For eaxmple, the algebra C_b (R) of bounded continuous functions on the real line R is a unital commutative
C*-algebra in a natural way. So, by CGNT , we know that C_b (R) = C ( X), where X is a compact Hausdorff space. What is X and how is it related to R? It is easy to see that $X= \beta \mathbb{R}$ is in fact the Stone-Cech compactification of R. More generally, for a locally compact Hausdorff space X the spectrum of C_b (X) can be shown to be homeomorphic to $\beta X$, the Stone-Cech compactification of X. For an example of a different flavour, let X be a topological space which is manifestly non-Hausdorff and let A=C (X). Then the spectrum of A has the effect of turning X into a Hausdorff space and is in some sense the Hausdorffization' of X. The reader should try to describe the spectrum of $L^{\infty} [0, \, 1]$.

Jamie Vicary said...

Dear Masoud,

Thanks for that interesting post. I think calling these two theorems the commutative and noncommutative Gelfand-Naimark theorem. Maybe somebody should try and get this change made on Wikipedia.

The result about involutive algebra maps being continuous is amazing, I hadn't come across that before! It's surprising just how powerful these involtions are... I wonder if there's anything profound that can be said about why this is.

I came across another result recently, which surprised me, and seems closely related to the one you just stated: every continuous linear homomorphism of commutative finite-dimensional C*-algebras is involution-preserving (and I expect this generalises to the infinite-dimensional case, although not the noncommutative case.) Are there any standard proofs of this result?

masoud said...

Dear Jamie,
Thanks for your comment. One thing that I did not mention, and I should have, is the spectral radius formula (Gelfand-Beurling) which is valid in any Banach algebra. Coupled with the C* identity it shows immediately that the norm of a C* algebra can be characterized algebraically as || x|| = square root of the spectral radius of x*x, and in particular is unique. The result on automatic continuity follow similarly. All this are pretty standard and well known of course. You can read about it in the book cited or any other books on operator algebras (there are many!), but I agree with you that one should strive for a deeper understanding here. As for finite dimensional commutative C*-algebras, since they are just a direct sum of a finite copies of C, what you said easily follows (just look at what happens to idempotents), and in fact as you said the result is not true in the NC case (take an inner automorphism of a matrix algebra by a non-unitary).

Anonymous said...

The theorem is a very nice one, of clear historic significance, and much worthy of discussion, but why is it at all important "how it should be called"?

Fabien Besnard said...

Thank you for this clarification Masoud. I personnaly use the name "Gelfand-Naimark theorem" for commutative GN theorem, and "Gelfand-Naimark-Segal" for noncommutative one. I think this is a rather common convention. Incidentally, it is quite easy to generalize the (commutative) GN theorem to the category of compact ordered spaces, and I've just put a paper on the archives about this !

masoud said...

Dear Fabien,
Thanks for your comment and for pointing to your paper in the ArXive. There is an old result of Stone which says that the category of sets is anti-equivalent to a certain sub category of the category of Boolean algebras. I wonder if this result can be derived from the CGNT? It is certainly in the same vein....

Fabien Besnard said...

Dear Masoud,
yes, it is certainly in the same vein. I don't know if one can recover Stone's representation theorem with the CGNT, but let's imagine we can do it. Then there should be a commutative C*-algebra canonically associated to a boolean algebra. The boolean algebra would then be recovered as the lattice of projections of the C*-algebra, I guess. We can even think that this boolean algebra/commutative C*-algebra association would survive in the noncommutative case, giving an orthomular lattice/C*-algebra correspondence. This would make orthomodular lattices a "noncommutative" generalization of boolean algebras. This is a bit daydreaming and I do not know if all this really works but this is certainly an interesting suggestion you made ! Is anyone aware of a way to canonically generate a C*-algebra from a boolean algebra ?

masoud said...

Dear Fabien,
As you know, one definition of a Boolean algebra (or Boolean ring ), the one that actually enters the Stone theorem I mentioned, is a ring in which every element is an idempotent. This of course automatically implies that the ring is commutative and allows no noncommutative generalization. The equivalent lattice theoretic version, however, lends itself to a noncommutative" or quantum" version as pioneered by von Neumann, but is there an algebraic formulations as an algebra with some extra properties? I believe there are some reconstruction theorems that show, under some conditions, the given lattice is the lattice of projections in a Hilbert space. To go beyond that, and I think that is what you are asking, looks like might be related to VN's continuous geometry.

Tom Leinster said...

The connections between the Gelfand-Naimark theorems, Stone's representation theorem, and many other kinds of duality are thoroughly explored in Peter Johnstone's excellent book "Stone Spaces".

Masoud, a question. If one wants to use the word "duality" rather than "theorem", how should one refer to the duality between compact Hausdorff spaces and commutative C*-algebras? "Commutative Gelfand-Naimark duality"? I think I've most often heard it referred to as simply "Gelfand duality", but perhaps this is inappropriate.

I'm also intrigued by your statement "These days everything must be categorical'". I wonder why you feel that way?

masoud said...

Dear Tom,
Thanks for your comments and for pointing out the book by Johnstone.
By the way, the NC Gelfand-Naimark theorem is not a duality result. Not clear to me at least how it could be. So if one just say G-N duality' would be clear and points to the right theorem I think. May be you have something else in mind... My comment about categories really just meant as a truism and nothing else. This is of course a very simple example of the power and beauty of categorical ideas and I think you agree with that.

Pedro Resende said...

Dear Masoud and Fabien,

There is a lattice theoretic (I should say locale theoretic) version of CGNT based on the idea that any Hausdorff space can be completely recovered up to homeomorphism from its locale of open sets (a locale is a complete lattice in which binary infs distribute over arbitrary sups); in particular, the points correspond, of course, to the maximal open sets. This means that one way to obtain the compact spectrum of a commutative unital C*-algebra is based on noticing that its closed ideals form a locale which, up to an isomorphism of locales, is already the topology of the spectrum.

Conversely, a direct construction of the commutative C*-algebra A from its locale of closed ideals I(A) is based on taking the algebra elements to be locale maps I(A)->C, where C is the locale of complex numbers (these locale maps are defined to be the locale homomorphisms C->I(A), that is, maps that preserve finite infs and arbitrary sups); the locale of complex numbers C is that of open sets of the complex plane, but it can be presented algebraically by generators and relations, hence without any reference to the actual complex numbers. As Tom says, Johnstone's "Stone Spaces" gives an excellent account of such things.

As regards noncommutative generalizations, there have in fact been such attempts, although not using orthomodular lattices but rather considering the complete lattice (no longer a locale) of closed right ideals of a C*-algebra A equipped with the additional structure of multiplication of closed right ideals, which plays the role of a "noncommutative intersection" of open sets. There is a 1989 paper by Borceux, Rosicky and Van Den Bossche showing that this structure classifies post-liminal C*-algebras.

For more general C*-algebras this is insufficient, and Mulvey has proposed considering the complete lattice of all the closed linear subspaces of a C*-algebra instead, also equipped with multiplication plus the operation of pointwise involution. Such a structure is what one calls a unital involutive quantale. At least for unital C*-algebras, it is remarkable that in this way one obtains a complete invariant. This follows from results of Mulvey and Pelletier and earlier work of Giles and Kummer, and Akemann, and it has been explicitly proved by myself and David Kruml (a preprint is available in the arXiv). However, in many ways this is still an unsatisfactory result. In particular this invariant, although functorial, is not an equivalence of categories and does not even restrict to one.

Fabien Besnard said...

Thanks a lot, Pedro, this seems very interesting. I've ordered Stone's book, and I'll try to understand all this better and read the references you give.

Masoud said...

Dear Pedro, Thanks a lot for your comments and references. It looks indeed things get fairly complicated as soon as one passes the commutative line'. This is, of course, a general principle.

Successful Researcher said...

I just learned that Israel Gelfand is no longer with us. R.I.P.