TAILIEUCHUNG - Báo cáo toán học: "The type of the regular representation of certain transitive "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài:Các loại của các đại diện thường xuyên của groupoids bắc. | OPERATOR THEORY 114 1985 249-261 Copyright by INCREST 1985 THE TYPE OF THE REGULAR REPRESENTATION OF CERTAIN TRANSITIVE GROUPOIDS SHIGERU YAMAGAMI 1. INTRODUCTION Let G be a not necessarily connected Lie group. If GxG is provided with a groupoid structure in a trivial manner . for gj g2 gí gá G Gx G gi g2 -1 g2 gj. and gi g2 is composable with g g2 if and only if g2 g with the resulting composition equal to gi g2 we can use Haar measure to form a convolution algebra C G X G and its regular representation on L2 Gx G . Then the generated von Neumann algebra is a type I factor and acts on L2 GxG as a standard representation. Although these facts are all trivial if we take a quotient of Gx G by a suitable subgroup of G X G the situation becomes rather complicated and the analysis of its regular representation becomes an interesting problem. In this paper we will carry out the type analysis of the von Neumann algebra associated with a groupoid of this type. More precisely let H be a closed subgroup of G and let H be a normal subgroup of H which contains the connected component of H. We form a closed subgroup D of Gx G 1 D QhJt eHxH Then r GxGỊD has the structure of groupoid induced from that of GxG which is transitive in the sense that the canonical equivalence relation of the grou-poid is transitive see 10 for example . We assume that H has a unitary character z which is invariant under the action of H . ZƠ A -1 z Ậ for h G H Ịie H. We obtain a character zd of D defined by 2 zd a b a- b for a b G D. Let tp be a C -function on G X G which satisfies 3 z gl l g2 2 G l 2 1 2Xd a2 2 250 SHIGERƯ YAMAGAM1 for gi g2 e G X G ứi ứ2 e D and has a compact support modulo D. Here dCiH a 5 and Ộ denote Lie algebras of G and H respectively . We denote the set of all these p s by 91 and equip it with the following e-algebra structure For p pỵ p2 e 91 p2 e 91 and pữ e 91 are defined by 4 5 P1 2 ểi gỉ ỷ dgợựgi. g Pỉ g g GfH p gl gì p gz gl see 2 for example for the .

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