From: israel@math.ubc.ca (Robert Israel) Subject: Re: Q: Reference for a general Stone's theorem? Date: 12 Oct 1999 18:11:04 GMT Newsgroups: sci.math.research Keywords: SNAG (Stone-Naimark-Ambrose-Godement) Theorem In article <38032DD7.A27A25CA@uni-tuebingen.de>, Martin Schlottmann writes: > In "An Invitation to von Neumann Algebras", V. S. Sunder > mentions without proof or reference the following under > the name of "Stone's theorem": > Given a strongly continuous unitary representation t |-> u_t > of a locally compact abelian group G in a Hilbert space H, > there is a projection-valued measure E in H on the dual G^ > such that, for every vectors x, y from H, the function > t |-> < x, u_t y > is the Fourier transform of the measure > < x, E(.) y > (identifying G and G^^ according to Pontryagin). > Whereas a proof is easily reconstructed with the help of > Bochner's theorem and the usual spectral theorem, I seem > to have a hard time localizing a proper reference for this > general theorem in the literature. This version of Stone's Theorem is sometimes called the SNAG theorem, for Stone-Naimark-Ambrose-Godement. References: Lynn H. Loomis, An Introduction to Abstract Harmonic Analysis, Van Nostrand 1953, section 36. F. Riesz and B. Sz-Nagy, Functional Analysis, Ungar 1955, sec. 140. The original papers are: M.H. Stone, Ann. Math. 33 (1932) 643-648 M.A. Naimark, Izv. Akad. Nauk SSSR. Ser. Mat. 7 (1943) 237-244 W. Ambrose, Duke Math. J. 11 (1944) 589-595 R. Godement, C.R. Acad. Sci. Paris 218 (1944) 901-903 Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2