2020/02/14 by Rohit Dilip Holkar, Holkar, Rohit Dilip
Mathematics · #18D05 #22A22 #22D25 #46L08 #46L89 #47L30 #58B30 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #Category Theory (math.CT) #Combinatorics #Countable set #Equivariant map #FOS: Mathematics #Hausdorff space #Locally compact space #Mathematics #Operator Algebras (math.OA) #Pure mathematics #Second-countable space #Topology (electrical circuits) #math.CT #math.OA #msc:18D05 #msc:22A22 #msc:22D25 #msc:46L08 #msc:46L89 #msc:47L30 #msc:58B30
paper · pdf · doi:10.48550/arxiv.2002.05881
Submitted
arxiv created 2020/02/14 · openalex publication_date 2020/02/14 · arxiv updated 2020/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
It is known that a topological correspondence \((X,λ)\) from a locally compact groupoid with a Haar system \((G,α)\) to another one, \((H,β)\), produces a \(\textrmC^*\)-correspondence \(H(X,λ)\) from \(\textrmC^*(G,α)\) to \(\textrmC^*(H,β)\). In one of our earlier article we described composition two topological correspondences. In the present article, we prove that second countable locally compact Hausdorff topological groupoids with Haar systems form a bicategory \(\mathfrakT\) when equipped with a topological correspondences as 1-arrows. The equivariant homeomorphisms of topological correspondences preserving the families of measures are the 2-arrows in~\(\mathfrakT\). One the other hand, it well-known that \(\textrmC^*\)-algebras form a bicateogry \(\mathfrakC\) with \(\textrmC^*\)-correspondences as 1-arrows. The 2-arrows in \(\mathfrakC\) are unitaries of Hilbert \(\textrmC^*\)-modules that intertwine the representations. In this article, we show that a topological correspondence going to a \(\textrmC^*\)-one is a bifunctor~\(\mathfrakT→\mathfrakC\).