Stegel, Giovanni (2004) Inductive algebras for trees. Pacific Journal of Mathematics, Vol. 216 (1), p. 177-200. ISSN 0030-8730. Article.
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Let G be a locally compact group and π : G →U(H) a unitary representation of G. A commutative subalgebra of BH is called π-inductive when it is stable through conjugation by every operator in the range of π. This concept generalizes Mackey’s definition of a system of imprimitivity for π; it is expected that studying inductive algebras will lead to progress in the classification of realizations of representations on function spaces. In this paper we take as G the automorphism group of a locally finite homogeneous tree; we consider the principal spherical representations of G, which act on a Hilbert space of functions on the boundary of the tree, and classify the maximal inductive algebras of such representations. We prove that, in most cases, there exist exactly two such algebras.
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