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