arXiv subjects
Publications and source records attributed to Tyrone Crisp.
We study some aspects of the functor of parabolic induction within the context of reduced group C*-algebras and related operator algebras. We explain how Frobenius reciprocity fits naturally within the context of operator modules, and examine the prospects for an operator algebraic formulation of Bernstein's reciprocity theorem (his second adjoint theorem).
We present two applications of explicit formulas, due to Cuntz and Krieger, for computations in K-homology of graph C*-algebras. We prove that every K-homology class for such an algebra is represented by a Fredholm module having finite-rank commutators; and we exhibit generating Fredholm modules for the K-homology of quantum lens spaces.
Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.
We consider the special linear group G=SL2 over a p-adic field, and its diagonal subgroup M=GL1. Parabolic induction of representations from M to G induces a map in equivariant homology, from the Bruhat-Tits building of M to that of G. We compute this map at the level of chain complexes, and show that it is given by parahoric induction (as defined by J.-F. Dat).
It is known that given a directed graph E and a subset X of vertices, the sum of the projections associated to the vertices in X in the C*-algebra of E converges strictly in the multiplier algebra to a projection P. Here we give a construction which, in certain cases, produces a directed graph F whose C*-algebra is isomorphic to the corner P(C*(E))P. Corners of this type arise naturally as the fixed point algebras of discrete coactions on graph algebras related to labellings. We prove this fact, and show that our construction is applicable to such a case whenever the labelling satisfies an analogue of Kirchhoff's voltage law.
In this paper we describe an operation on directed graphs which produces a graph with fewer vertices, such that the C*-algebra of the new graph is Morita equivalent to that of the original graph. We unify and generalize several related constructions, notably delays and desingularizations of directed graphs.