An equivariant pullback structure of trimmable graph $C^*$-algebras

Journal of Noncommutative Geometry, vol. 16, pp. 761–785

Abstract

To unravel the structure of fundamental examples studied in noncommutative topology, we prove that the graph C^* -algebra C^*(E) of a trimmable graph E is U(1) -equivariantly isomorphic to a pullback C^* -algebra of a subgraph C^* -algebra C^*(E'') and the C^* -algebra of functions on a circle tensored with another subgraph C^* -algebra C^*(E') . This allows us to approach the structure and K-theory of the fixed-point subalgebra C^*(E)^{U(1)} through the (typically simpler) C^* -algebras C^*(E') , C^*(E'') and C^*(E'')^{U(1)} . As examples of trimmable graphs, we consider one-loop extensions of the standard graphs encoding respectively the Cuntz algebra \mathcal{O}_2 and the Toeplitz algebra \mathcal{T} . Then we analyze equivariant pullback structures of trimmable graphs yielding the C^* -algebras of the Vaksman–Soibelman quantum sphere S^{2n+1}_q and the quantum lens space L_q^3(l;1,l) , respectively.

Authors 4

1. Francesca Arici Aachen

   Leiden University

   Affiliation as printed

   Leiden University, Netherlands

2. Francesco D’Andrea

   Istituto Nazionale di Fisica Nucleare, Sezione di Napoli · University of Naples Federico II

   Affiliation as printed

   Università di Napoli Federico II and I.N.F.N. Sezione di Napoli, Italy

3. Piotr M. Hajac

   Polish Academy of Sciences

   Affiliation as printed

   Polish Academy of Sciences, Warsaw, Poland

4. Mariusz Tobolski

   University of Wrocław

   Affiliation as printed

   Uniwersytet Wrocławski, Poland