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\(C^\ast\)-algebras of 2-groupoids. (English) Zbl 1346.18007

Given a group one can construct its full or reduced group \(C^\ast\)-algebra. This construction has been extended by J. Renault [A groupoid approach to \(C^*\)-algebras. York: Springer-Verlag (1980; Zbl 0433.46049)] and others to groupoids: categories in which all morphisms are invertible. This paper extends it further to 2-groupoids: (strict) 2-categories in which all 1-cells and 2-cells are invertible.
To be precise, the 2-groupoid G needs to be locally compact, and have a 2-Haar system, which consists of two families of measures on 2-cells, indexed by 0-cells and 1-cells. Invariance of 2-Haar systems corresponds to horizontal/vertical composition of 2-cells. The 2-Haar system provides the involution on the topological algebra \(C_c(G)\). Similar 2-Haar systems give strongly Morita equivalent \(C^\ast\)-algebras.
To construct the reduced \(C^\ast\)-algebra, the author first finds a natural vertical/horizontal representation for 2-groupoid on Hilbert bundles, and a \(C^\ast\)-correspondence between them. Certain closed 2-subgroupoids then induce representions, and allow one to construct the vertical/horizontal reduced \(C^\ast\)-algebra.
Finally, reduced \(C^\ast\)-algebra of \(r\)-discrete principal 2-groupoids are studied in more detail; find ideals and masas. Here, a 2-groupoid is principal when the map (codomain,domain) is injective on both 1- and 2-cells, and it is r-discrete when 0-cells form an open subset of 1-cells, and 1-cells form an open subset of 2-cells.

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)

Citations:

Zbl 0433.46049
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References:

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