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On a Conjecture of Montgomery

George D. MostowGerhard Paul Hochschild

出版	Mathematics Division, Air Force Office of Scientific Research, 1957
URL	http://books.google.com.hk/books?id=dPxxzNubTxUC&hl=&source=gbs_api
EBook	FULL_PUBLIC_DOMAIN

註釋

Let G be a Lie group with a finite number of connected components and let R(b) denote the ring of complex-valued continuous functions no G whose translates are finite dimensional. We investigate 1) the algebraic structure of R(b), 2) the group A of automorphisms of R(b) regarded as G-module, 3) the relation between G and A. A case of central interest is the one in which R(b) is a finitely generated ring. Under this hypothesis, A turns out to be the "universal complexification" of G. This result can be regarded as a direct generalization of Tannaka's duality theorem for compact Lie groups and Harish-Chandra's analogue for connected semi-simple groups. The hypothesis that R(b) be finitely generated is equivalent to the condition that G modulo the topological closure of the commutator subgroup of the connected component of the identity be compact. Conversely, if A is the universal complexification of G, then R(b) is finitely generated. Thus the class of Lie groups with R(b) finitely generated is the precise class for which Tannaka's duality holds.