Multipliers on Locally Compact Groups
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93
K. R. Parthasarathy University of Manchester, Manchester, England
Multipliers on Locally Compact Groups 1969
Springer-Verlag Berlin · Heidelberg· New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer- Verlag Berlin· Heidelberg 1969 Library of Congress Catalog Card Number 71- 84142 Printed in Germany. Title No. 3699
CON TEN T S
1.
Introduction
1
2.
Standard groups with a right invariant measure
3
Borel multipliers on a locally compact group
11
4.
Multipliers on Lie groups
22
5.
Multipliers on some special groups
28
3.
- 1 -
MULTIPLIERS IN LOCALLY COMPACT GROUPS
1.
INTRODUCTION In the mathematical formulation of quantum mechanics proposed by Von Neumann (cf[9], [14], [15]) the set of all propositions concerning a quantum mechanical system is an ortho-complemented lattice •
..
The simplest example of such a lattice is the lattice ;t closed subspaces of a separable Hilbert space
of all
The observables
in such a system turn out to be self adjoint operators
In
order to construct the standard observables like energy, linear, angular and spin angular momenta, etc., in such a system it is necessary to study the effect of coordinate transformations by a group G of symmetries.
In this context the study of representations of G in
the group of automorphisms of the lattice
;t
(f}) is of great import-
ance. By a theorem of Wigner (cf [14] Theorem 7.27, page 167) it is known that every automorphism T of T . . ant1un1tary operator U •
is induced by a unitary or
T The operator U is determined uniquely upto
a scalar multiple of modulus unity. Let
be a representation of G in the group of automorphisms
- 2 of
:t ( Q.? ) •
Suppose
T
g
is induced by the opera tor Ug which may be
unitary or auniunitary.
If the group G is sufficiently regular (as
for example a connected Lie group) we can take U
g
all gEG.
Since
T
e
to be unitary for
is the identity automorphism we can take U to be e induce the same Since U U and U gl g2 glg2 it follows that there exists a constant 0(gl,g2)
the identity operator. automorphism
T
glg2 of modulus unity such that
•
for all
Since
• we have
•
0(gl,g2) o(gl g2,g3)
(1.1)
for all gl,g2,g3 EG Since
U -I e
'
o(g, e)
•
o(e, g)
•
1
for all
ge:G.
Any function 0 defined on G x G, taking values in the multiplicative group of all complex numbers of modulus unity and satisfying equations (1.1) and (1.2) is called a multiplier
on G x G.
The purpose of this article is to make a systematic analysis of the multipliers on certain locally compact groups. analysis is, of course, not new.
Such an
It was originally done by Bargmann
(1.2)
- 3 -
[lJ, and then by Mackey L8], [101, Varadarajan[13], Simms [12J and many others.
The presentation given here is probably more unified
and self-contained, many of the proofs are simpler and some of the results are more general.
The main ideas are contained in [lJ, [8],
[10] and [13J. Keeping the math
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