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 <martin.schlottmann@uni-tuebingen.de> 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