nLab
Gelfand duality

Context

Algebra

Geometry

Functional analysis

Duality

Noncommutative geometry

Contents

Idea

Gelfand duality is a duality between spaces and their algebras of functions for the case of compact topological spaces and commutative C-star algebras:

every (nonunital) C *C^\ast-algebra AA is equivalent to the C *C^\ast-algebra of continuous functions on the topological space called its Gelfand spectrum sp(A)sp(A).

This theorem is the basis for regarding non-commutative C *C^\ast-algebras as formal duals to spaces in noncommutative geometry.

Definitions

The statement of Gelfand duality involves the following categories and functors.

Definition

Write

  • C *AlgC^\ast Alg for the category of C-star algebras;

  • C *Alg nuC^\ast Alg_{nu} for the category of non-unital C *C^\ast-algebras;

  • C *Alg comC *AlgC^\ast Alg_{com} \subset C^\ast Alg for the full subcategory of commutative C *C^\ast-algebras;

  • C *Alg com,nuC *Alg nuC^\ast Alg_{com,nu} \subset C^\ast Alg_{nu} for the full subcategory of commutative non-unital C *C^\ast-algebras.

And

The duality itself is exhibited by the following functors

Definition

Write

C:Top cptC *Alg com op C : Top_{cpt} \to C^\ast Alg_{com}^{op}

for the functor which sends a compact topological space XX to the algebra of continuous functions C(X)={f:X|fcontinuous}C(X) = \{f : X \to \mathbb{C} | f \; continuous\}, equipped with the structure of a C *C^\ast-algebra in the evident way (…).

Write

C 0:*/Top cptC *Alg com,nu C_0 : */Top_{cpt} \to C^\ast Alg_{com,nu}

for the functor that sends (X,x 0)(X,x_0) to the algebra of continuous functions f:Xf : X \to \mathbb{C} for which f(x 0)=0f(x_0) = 0.

Definition

Write

sp:C *Alg com opTop cpt sp : C^\ast Alg_{com}^{op} \to Top_{cpt}

for the Gelfand spectrum functor: it sends a commutative C *C^\ast-algebra AA to the set of characters – non-vanishing C *C^\ast-algebra homomorphisms x:Ax : A \to \mathbb{C} – equipped with the spectral topology.

Similarly write

sp:C *Alg com,nu opTop lcpt. sp : C^\ast Alg_{com,nu}^{op} \to Top_{lcpt} \,.

Statement

Theorem

(Gelfand duality theorem)

The pairs of functors

C *Alg com opspCTop cpt C^\ast Alg_{com}^{op} \stackrel{\overset{C}{\leftarrow}}{\underset{sp}{\to}} Top_{cpt}

is an equivalence of categories.

Here C *Alg opC^\ast Alg^{op}_{\cdots} denotes the opposite category of C *Alg C^\ast Alg_{\cdots}.

Corollary

On non-unital C *C^\ast-algebras the above induces an equivalence of categories

C *Alg com,nu opspC 0*/Top cpt. C^\ast Alg_{com,nu}^{op} \stackrel{\overset{C_0}{\leftarrow}}{\underset{sp}{\to}} */Top_{cpt} \,.
Proof

The operation of unitalization () +(-)^+ constitutes an equivalence of categories

C *Alg nu() +kerC *Alg/ C^\ast Alg_{nu} \stackrel{\overset{ker}{\leftarrow}}{\underset{(-)^+}{\to}} C^\ast Alg / \mathbb{C}

between non-unital C *C^\ast-algebras and the over-category of C *C^\ast-algebras over the complex numbers \mathbb{C}.

Composed with the equivalence of theorem 1 this yields

C *Alg com,nu op() +(C *Alg com/) opC*/Top cpt. C^\ast Alg_{com,nu}^{op} \underoverset{\simeq}{(-)^+}{\to} (C^\ast Alg_{com}/\mathbb{C})^{op} \underoverset{\simeq}{C}{\to} * / Top_{cpt} \,.

The weak inverse of this is the composite functor

C 0:*/Top cptsp(C *Alg com/) opkerC *Alg com,nu op C_0 : */Top_{cpt} \underoverset{\simeq}{sp}{\to} (C^\ast Alg_{com}/\mathbb{C})^{op} \underoverset{\simeq}{ker}{\to} C^\ast Alg_{com,nu}^{op}

which sends (*x 0X)(* \stackrel{x_0}{\to} X) to ker(C(X)ev x 0)ker(C(X) \stackrel{ev_{x_0}}{\to} \mathbb{C}), hence to {fC(X)|f(x 0)=0}\{f \in C(X) | f(x_0) = 0\}. This is indeed C 0C_0 from def. 2.

Remark

With due care on defining the right morphisms, the duallity generalizes also to locally compact topological spaces. See for instance (Brandenburg 07).

For an overview of other generalizations see also this MO discussion.

Generalizations

In constructive mathematics

Gelfand duality makes sense in constructive mathematics hence internal to any topos: see constructive Gelfand duality theorem.

By horizontal categorification

Gelfand duality can be extended by horizontal categorification to define the notion of spaceoids as formal duals of commutative C *C^*-categories.

References

Textbook accounts include

  • N. P. Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics 1998. xx+529 pp. MR2000g:81081 doi

  • Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. gBooks

An exposition that explicitly gives Gelfand duality as an equivalence of categories and introduces all the notions of category theory necessary for this statement is in

  • Ivo Dell’Ambrogio, Categories of C *C^\ast-algebras (pdf)

Careful discussion of the duality for the more general case of locally compact topological spaces includes

Some other generalized contexts for Gelfand duality:

  • Hans Porst, Manfred B. Wischnewsky, Every topological category is convenient for Gelfand duality, Manuscripta mathematica 25:2, (1978) pp 169-204

  • H. Heunen, Klaas Landsman, Bas Spitters, S. Wolters, The Gelfand spectrum of a noncommutative C *C^\ast-algebra, J. Aust. Math. Soc. 90 (2011), 39–52 doi pdf

  • Christopher J. Mulvey, A generalisation of Gelfand duality, J. Algebra 56, n. 2, (1979) 499–505 doi

Revised on November 12, 2014 17:51:00 by Urs Schreiber (81.194.35.225)