symmetric monoidal (∞,1)-category of spectra
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
(geometry $\leftarrow$ Isbell duality $\to$ algebra)
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.
The statement of Gelfand duality involves the following categories and functors.
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
Top${}_{Haus}$ for the category of Hausdorff topological spaces
$Top_{cpt}$ for the full subcategory of Top${}_{Haus}$ on the compact topological spaces;
$*/Top_{cpt}$ for the category of pointed topological? compact Hausdorff spaces, i.e. the pointed objects in $Top_{cpt}$;
$Top_{lcpt}$ for the category of Hausdorff and locally compact topological spaces with morphisms being the proper maps of topological spaces.
The duality itself is exhibited by the following functors:
Write
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
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 $f(x_0) = 0$.
Write
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
(Gelfand duality theorem)
The pairs of functors
is an equivalence of categories.
Here $C^\ast Alg^{op}_{\cdots}$ denotes the opposite category of $C^\ast Alg_{\cdots}$.
On non-unital $C^\ast$-algebras the above induces an equivalence of categories
The operation of unitalization $(-)^+$ constitutes an equivalence of categories
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
The weak inverse of this is the composite functor
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.
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.
Gelfand duality makes sense in constructive mathematics hence internal to any topos: see constructive Gelfand duality theorem.
Gelfand duality can be extended by horizontal categorification to define the notion of spaceoids as formal duals of commutative $C^*$-categories.
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
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