New York Journal of Mathematics
Volume 27 (2021), 740-775


Kyle Austin and Atish Mitra

Groupoid models of C*-algebras and the Gelfand functor

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Published: April 25, 2021.
Keywords: Groupoid models, Gelfand functor, Jiang-Su algebra, Razak-Jacelon algebra.
Subject: Primary 54F45; Secondary 55M10.

We construct a large class of morphisms, which we call partial morphisms, of groupoids that induce *-morphisms of maximal and minimal groupoid C*-algebras. We show that the assignment of a groupoid to its maximal (minimal) groupoid C*-algebra and the assignment of a partial morphism to its induced morphism are functors (both of which extend the Gelfand functor). We show how to geometrically visualize lots of *-morphisms between groupoid C*-algebras. As an application, we construct, without any use of the classification theory, groupoid models of the entire inductive systems used in the original constructions of the Jiang-Su algebra Z and the Razak-Jacelon algebra W. Consequently, the inverse limit of the groupoid models for the aforementioned systems are models for Z and W, respectively.


The first named author was funded by the Israel Science Foundation (grant No. 522/14).

Author information

Kyle Austin:
Ben-Gurion University of the Negev
P.O.B. 653
Beer-Sheva 8410501, Israel


Atish Mitra:
Montana Technological University
1300 West Park Street
Butte, MT 59701, USA