nLab Gelfand duality

Context

Algebra

higher algebra

universal algebra

duality

In QFT and String theory

Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

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

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

Definitions

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

Definition

Write

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

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

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

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

And

The duality itself is exhibited by the following functors:

Definition

Write

$C \;\colon\; Top_{cpt} \to C^\ast Alg_{com}^{op}$

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

Write

$C_0 \;\colon\; */Top_{cpt} \to C^\ast Alg_{com,nu}$

for the functor that sends a pointed topological compact Hausdorff space $(X,x_0)$ to the algebra of continuous functions $f : X \to \mathbb{C}$ for which $f(x_0) = 0$.

Definition

Write

$sp : C^\ast Alg_{com}^{op} \to Top_{cpt}$

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

Similarly write

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

Statement

Theorem

(Gelfand duality theorem)

The pairs of functors

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

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

Corollary

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

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

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

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

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

Composed with the equivalence of theorem 1 this yields

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

The weak inverse of this is the composite functor

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

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

Remark

Since locally compact Hausdorff spaces are equivalently open subspaces of compact Hausdorff spaces, via the construction that sends a locally compact Hausdorff space $X$ to its one-point compactification, and since a continuous function on the compact Hausdorff spce $X^\ast$ which vanishes at the extra point is equivalently a continuous function on $X$ which vanishes at infinity, the above induces an equivalence between locally compact Hausdorff spaces and $C^\ast$-algebras of functions that vanish at infinity.

With due care on defining the right morphisms, the duality 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 $C^*$-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 $C^\ast$-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 $C^\ast$-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 June 3, 2017 06:25:25 by Urs Schreiber (178.6.236.87)