Preferred traces on C*-algebras of self-similar groupoids arising as fixed points

Recent results of Laca, Raeburn, Ramagge and Whittaker show that any self-similar action of a groupoid on a graph determines a 1-parameter family of self-mappings of the trace space of the groupoid C*-algebra. We investigate the fixed points for these self-mappings, under the same hypotheses that Laca et al. used to prove that the C*-algebra of the self-similar action admits a unique KMS state. We prove that for any value of the parameter, the associated self-mapping admits a unique fixed point, which is a universal attractor. This fixed point is precisely the trace that extends to a KMS state on the C*-algebra of the self-similar action.

c*-algebra; self-similar group; self-similar groupoid; kms state; trace; fixed point

Preferred traces on C*-algebras of self-similar groupoids arising as fixed points Articles