A horizontal categorification of Gel'fand duality
Copyright policy )A horizontal categorification of Gel'fand duality
In the setting of C*-categories, we provide a definition of "spectrum" of a commutative full C*-category as a one-dimensional unital saturated Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gelfand duality theorem generalizing the usual Gelfand duality between the categories of commutative unital C*-algebras and compact Hausdorff spaces. Although many of the individual ingredients that appear along the way are well-known, the somehow unconventional way we "glue" them together seems to shed some new light on the subject.
22 pages, AMS-LaTeX2e, results unchanged, several improvements in the exposition, one section added, to appear in Advances in Mathematics
- University of Rome Tor Vergata Italy
- Sapienza University of Rome Italy
- University of Newcastle Australia Australia
- Chulalongkorn University Thailand
- Thammasat University Thailand
Mathematics(all), Duality, non-commutative geometry, Fell bundle, Mathematics - Operator Algebras, 46L87 (Primary) 46M15, 46L08, 46M20, 16D90, 18F99 (Secondary), C⁎-category, Mathematics - Category Theory, Categories, functors in functional analysis, Noncommutative topology, \(C^*\)-category, C*-category; Duality; Fell bundle; Non-commutative geometry; Mathematics (all), Non-commutative geometry, Categories in geometry and topology, FOS: Mathematics, duality, Category Theory (math.CT), noncommutative geometry, C*-category, Operator Algebras (math.OA)
Mathematics(all), Duality, non-commutative geometry, Fell bundle, Mathematics - Operator Algebras, 46L87 (Primary) 46M15, 46L08, 46M20, 16D90, 18F99 (Secondary), C⁎-category, Mathematics - Category Theory, Categories, functors in functional analysis, Noncommutative topology, \(C^*\)-category, C*-category; Duality; Fell bundle; Non-commutative geometry; Mathematics (all), Non-commutative geometry, Categories in geometry and topology, FOS: Mathematics, duality, Category Theory (math.CT), noncommutative geometry, C*-category, Operator Algebras (math.OA)
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