We generalize the notion of a Hecke pair of groups to the case of an inclusion of ergodic discrete measured equivalence relations. A key ingredient in dening this new concept is a commensurability subrelation introduced and discussed in . As in the group case, with each such Hecke pair, we associate a von Neumann algebra which we call the Hecke von Neumann algebra of the pair. It is shown that the Hecke von Neumann algebra thus defined is realized as one of the relative commutants of the tower of the corresponding inclusion of factors.
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Hisashi Aoi, Takehiko Yamanouchi, Hecke von Neumann Algebra of Ergodic Discrete Measured Equivalence Relations. Publ. Res. Inst. Math. Sci. 46 (2010), no. 3, pp. 607–667DOI 10.2977/PRIMS/20