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 (locally) compact topological spaces and commutative (nonunital) 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:Xfcontinuous}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.

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

  • 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)

Some other generalized contexts for Gelfand duality:

  • Hans-E. Porst, Manfred B. Wischnewsky, Every topological category is convenient for Gelfand duality, Manuscripta mathematica 25:2, (1978) pp 169-204
  • H. Heunen, N. P. 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 April 9, 2014 05:57:20 by Tim Porter (2.26.27.237)