locally compact quantum groups from von Neumann/C^*- algebras with Haar measures

locally compact quantum groups from von Neumann/

C

*

- algebras with Haar measures

0.1

Hilbert spaces, Von Neumann algebras and Quantum Groups

John von Neumann

introduced a

mathematical foundation

for Quantum Mechanics in the form of

W

*

-

algebras

(

http://

planetmath

.org/WeakHopfCAlgebra2

)
of (quantum)

bounded operators

in a (quantum:= presumed

separable

, i.e. with a countable basis)

Hilbert space

H

S

. Recently, such

von Neumann algebras

,

W

*

(

http://planetmath.org/WeakHopfCAlgebra2

) and/or (more generally) C*-algebras are, for example, employed to define

locally compact quantum groups

C

⁢

Q

⁢

G

l

⁢

c

(

http://planetmath.org/LocallyCompactQuantumGroup

) by equipping such
algebras with a co-associative

multiplication

(

http://planetmath.org/WeakHopfCAlgebra2

)
and also with associated, both left– and right– Haar measures, defined by two semi-finite normal weights

[

1

]

.

0.1.1

Remark on Jordan-Banach-von Neumann (JBW) algebras,

J

⁢

B

⁢

W

⁢

A

A

Jordan–

Banach algebra

(a JB–algebra for short) is both a real

Jordan algebra

and a

Banach space

, where for all

S

,

T

∈

𝔄

ℝ

, we have the following.

A

JLB–algebra

is a

J

⁢

B

–algebra

𝔄

ℝ

together with a

Poisson bracket

for
which it becomes a Jordan–

Lie algebra

J

⁢

L

for some

ℏ

2

≥

0

. Such JLB–algebras often
constitute the real part of several widely studied complex associative algebras.
For the purpose of

quantization

, there are fundamental

relations

between

𝔄

s

⁢

a

, JLB and

Poisson algebras

(

http://planetmath.org/JordanBanachAndJordanLieAlgebras

).

Definition 0.1

.

A JB–algebra which is monotone

complete

and admits a separating set of normal sets is
called a

JBW-algebra

.

These appeared in the work of von Neumann who developed an

orthomodular lattice

theory of projections on

L

⁢

(

H

)

on which to study

quantum logic

. BW-algebras have the following

property

: whereas

𝔄

s

⁢

a

is a J(L)B–algebra, the self-adjoint part of a von Neumann algebra is a JBW–algebra.

References

1

Leonid Vainerman. 2003.

http://planetmath.org/?op=getobj&from=books&id=160

“Locally Compact Quantum Groups and

Groupoids

”:

Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21-23, 2002., Walter de Gruyter Gmbh & Co: Berlin.

2

Von Neumann and the

http://plato.stanford.edu/entries/qt-nvd/

Foundations

of

Quantum Theory

.

3

B

ö

hm, A., 1966,

Rigged Hilbert Space

and

Mathematical Description

of Physical Systems,

Physica A

, 236: 485-549.

4

B

ö

hm, A. and Gadella, M., 1989,

Dirac Kets, Gamow Vectors and Gel’fand

Triplets

, New York: Springer-Verlag.

5

Dixmier, J., 1981,

Von Neumann Algebras

, Amsterdam: North-Holland Publishing Company. [First published in French in 1957:

Les Alge’bres d’Ope’rateurs dans l’Espace Hilbertien

, Paris: Gauthier-Villars.]

6

Gelfand, I. and Neumark, M., 1943, On the Imbedding of

Normed Rings

into the Ring of

Operators

in Hilbert Space,

Recueil Mathe’matique

[Matematicheskii Sbornik] Nouvelle Se’rie, 12 [54]: 197-213. [Reprinted in C*-algebras: 1943-1993, in the series Contemporary Mathematics, 167, Providence, R.I. :

American Mathematical Society

, 1994.]

7

Grothendieck, A., 1955, Produits Tensoriels Topologiques et Espaces Nucl

é

aires,

Memoirs of the American Mathematical Society

, 16: 1-140.

8

Horuzhy, S. S., 1990,

Introduction

to Algebraic Quantum Field Theory

, Dordrecht: Kluwer Academic Publishers.

9

J. von Neumann.,1955,

Mathematical Foundations of Quantum Mechanics.

, Princeton, NJ: Princeton University Press. [First published in German in 1932:

Mathematische Grundlagen der Quantenmechanik

, Berlin: Springer.]

10

J. von Neumann, 1937,

Quantum Mechanics of

Infinite

Systems

, first published in (Radei and Statzner 2001, 249-268). [A mimeographed version of a lecture given at Pauli’s seminar held at the Institute for Advanced Study in 1937, John von Neumann Archive, Library of Congress, Washington, D.C.]

Title

locally compact quantum groups from von Neumann/

C

*

- algebras with Haar measures

Canonical

name

LocallyCompactQuantumGroupsFromVonNeumannCAlgebrasWithHaarMeasures

Date of creation

2013-03-22 18:24:28

Last modified on

2013-03-22 18:24:28

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

16

Author

bci1 (20947)

Entry type

Topic

Classification

msc 47A70

Classification

msc 46N50

Classification

msc 47L30

Classification

msc 47N50

Classification

msc 81P15

Classification

msc 46C05

Synonym

locally compact quantum groups

Synonym

quantum groupoids

Synonym

Hopf and weak Hopf algebras

Related topic

HilbertSpace

Related topic

QuantumSpaceTimes

Related topic

VonNeumannAlgebra

Related topic

WeakHopfCAlgebra2

Related topic

ClassificationOfHilbertSpaces

Related topic

QuantumSpaceTimes

Related topic

VonNeumannAlgebra

Related topic

WeakHopfCAlgebra2

Related topic

JordanBanachAndJordanLieAlgebras

Related topic

QuantumLogic

Related topic

Distribution4

Defines

JBW-algebras

Defines

J

⁢

B

⁢

W

algebras

Defines

J

⁢

B

–algebra

Defines

J

⁢

B

⁢

W

⁢

A

Defines

J

⁢

L

Defines

Jordan-Banach-von Neumann algebras

Defines

C

⁢

Q

⁢

G

l

⁢

c

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