Algebras and modules
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Geometry on formal duals of algebras
Topics in Functional Analysis
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) -algebra is equivalent to the -algebra of continuous functions on the topological space called its Gelfand spectrum .
This theorem is the basis for regarding non-commutative -algebras as formal duals to spaces in noncommutative geometry.
The statement of Gelfand duality involves the following categories and functors.
for the category of C-star algebras;
for the category of non-unital -algebras;
for the full subcategory of commutative -algebras;
for the full subcategory of commutative non-unital -algebras.
The duality itself is exhibited by the following functors
for the functor which sends a compact topological space to the algebra of continuous functions , equipped with the structure of a -algebra in the evident way (…).
for the functor that sends to the algebra of continuous functions for which .
for the Gelfand spectrum functor: it sends a commutative -algebra to the set of characters – non-vanishing -algebra homomorphisms – equipped with the spectral topology.
(Gelfand duality theorem)
The pairs of functors
is an equivalence of categories.
Here denotes the opposite category of .
On non-unital -algebras the above induces an equivalence of categories
The operation of unitalization constitutes an equivalence of categories
between non-unital -algebras and the over-category of -algebras over the complex numbers .
Composed with the equivalence of theorem 1 this yields
The weak inverse of this is the composite functor
which sends to , hence to . This is indeed from def. 2.
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 -categories.
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 -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 -algebra, J. Aust. Math. Soc. 90 (2011), 39–52 doi pdf