A von Neumann algebra approach to quantum metrics. Quantum relations

153 Pages·2012·1.124 MB·English

by  Greg Kuperberg, Nik Weaver

MEMOIRS
of the
American Mathematical Society
Number 1010
A von Neumann Algebra Approach
to Quantum Metrics
Greg Kuperberg
Nik Weaver
Quantum Relations
Nik Weaver
January 2012 • Volume 215 • Number 1010 (first of 5 numbers) • ISSN 0065-9266
American Mathematical Society
Number 1010
A von Neumann Algebra Approach
to Quantum Metrics
Greg Kuperberg
Nik Weaver
Quantum Relations
Nik Weaver
January2012 • Volume215 • Number1010(firstof5numbers) • ISSN0065-9266
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Contents
A von Neumann Algebra Approach to Quantum Metrics
by Greg Kuperberg and Nik Weaver 1
Introduction 3
Chapter 1. Measurable and quantum relations 7
Chapter 2. Quantum metrics 11
2.1. Basic definitions 11
2.2. More definitions 14
2.3. The abelian case 21
2.4. Reflexivity and stabilization 24
2.5. Constructions with quantum metrics 25
2.6. Intrinsic characterization 33
Chapter 3. Examples 37
3.1. Operator systems 37
3.2. Graph metrics 39
3.3. Quantum metrics on M (C) 39
2
3.4. Quantum Hamming distance 40
3.5. Quantum tori 43
3.6. Ho¨lder metrics 47
3.7. Spectral triples 48
Chapter 4. Lipschitz operators 53
4.1. The abelian case 53
4.2. Spectral Lipschitz numbers 56
4.3. Commutation Lipschitz numbers 64
4.4. Little Lipschitz spaces 69
Chapter 5. Quantum uniformities 73
5.1. Basic results 73
5.2. Uniform continuity 75
Bibliography 79
iii
iv CONTENTS
Quantum Relations by Nik Weaver 81
Introduction 83
Chapter 1. Measurable relations 85
1.1. Basic definitions 85
1.2. Constructions with measurable relations 88
1.3. Conversion to classical relations 91
1.4. Basic results 94
1.5. Measurable metrics 97
Chapter 2. Quantum relations 103
2.1. Basic definitions 103
2.2. Constructions with quantum relations 104
2.3. Basic results 106
2.4. The abelian case 108
2.5. Operator reflexivity 112
2.6. Intrinsic characterization 117
2.7. Quantum tori 126
Bibliography 131
Notation Index 133
Subject Index 135
Abstract
A von Neumann Algebra Approach to Quantum Metrics by Greg Ku-
perberg and Nik Weaver
We propose a new definition of quantum metric spaces, or W*-metric spaces,
in the setting of von Neumann algebras. Our definition effectively reduces to the
classical notion in the atomic abelian case, has both concrete and intrinsic charac-
terizations, andadmitsawidevarietyoftractableexamples. Anatural application
and motivation of our theory is a mutual generalization of the standard models of
classical and quantum error correction.
Quantum Relations by Nik Weaver
Wedefinea“quantumrelation”onavonNeumannalgebraM⊆B(H)tobea
weak*closedoperatorbimoduleoveritscommutantM(cid:2). Althoughthisdefinitionis
framed in terms of a particular representation of M, it is effectively representation
independent. Quantum relations on l∞(X) exactly correspond to subsets of X2,
i.e., relations on X. There is also a gooddefinition of a “measurable relation” on a
measure space, to which quantum relations partially reduce in the general abelian
case.
By analogy with the classical setting, we can identify structures such as quan-
tum equivalence relations, quantum partial orders, and quantum graphs, and we
cangeneralize Arveson’sfundamental work onweak*closedoperator algebras con-
taining a masa to these cases. We are also able to intrinsically characterize the
quantum relations on M in terms of families of projections in M⊗B(l2).
Received by the editor [A von Neumann Algebra Approach to Quantum Metrics] May 26,
2010,andinrevisedformSeptember29,2010;[QuantumRelations]May25,2010,andinrevised
formOctober25,2010.
ArticleelectronicallypublishedonMarch29,2011;S0065-9266(2011)00637-4.
2010 Mathematics Subject Classification. Primary 46L89,28A99; Secondary 46L10, 54E35,
81P70.
Keywordsandphrases. [AvonNeumannAlgebraApproachtoQuantumMetrics]Quantum
errorcorrection,quantummetrics,quantumtori,spectraltriples,vonNeumannalgebras;[Quan-
tumRelations]Measurablemetrics,measurablerelations,operatorreflexivity,quantumrelations,
quantumtori.
(cid:2)c2011 American Mathematical Society
v

A von Neumann Algebra
Approach to Quantum Metrics
by Greg Kuperberg and Nik Weaver

Introduction
It has proven to be fruitful in abstract analysis to think of various structures
connectedtoHilbertspaceas“noncommutative”or“quantum”versionsofclassical
mathematicalobjects. Thispointofviewhasbeenemphasizedin[8](seealso[34]).
Forinstance, itiswellestablishedthatC*-algebrasandvonNeumannalgebrascan
profitably be thought of as quantum topological and measure spaces, respectively.
The use of the word “quantum” is called for if, for example, the structures in
questionplayaroleinmodellingquantummechanicalsystemsanalogoustotherole
played by the corresponding classical mathematical structures in classical physics.
Basic examples in noncommutative geometry such as the quantum tori [21]
clearlyexhibitametricaspectinthattheycarryanaturalnoncommutativeanalog
ofthealgebraofboundedscalar-valuedLipschitzfunctionsonametricspace. How-
ever, the general notion of a quantum metric has been elusive. Possible definitions
havebeenproposedbyConnes[9],Rieffel[22],andWeaver[31]. Connes’definition
involveshisnotionofspectraltriplesandispatternedaftertheDiracoperatorona
Riemannian manifold. Possibly the right interpretation of this definition is “quan-
tumRiemannianmanifold”ratherthan“quantummetricspace”. Weaverproposed
a definition involving unbounded derivations of von Neumann algebras into dual
operator bimodules. This definition neatly recovers classical Lipschitz algebras in
the abelian case, but it has not led to a deeper structure theory. Rieffel’s defini-
tion, which is also called a C*-metric space [23], generalizes the classical Lipschitz
seminorm on functions on a metric space. This definition has attracted the most
interest recently; among other interesting properties, it leads to a useful model of
Gromov-Hausdorff convergence.
We introduce a new definition of a quantum metric space. To distinguish
between our model and that of Rieffel, it can also be called a W*-metric space.
Recall that an operator system is a linear subspace of B(H) that is self-adjoint
and contains the identity operator. We say that a W*-filtration of B(H) is a one-
parameter family of weak* closed operator systems V , t∈[0,∞), such that
t
(i) VsVt ⊆(cid:2)Vs+t for all s,t≥0
(ii) V = V for all t≥0.
t s>t s
Notice that V is automatically a von Neumann algebra, since the filtration con-
0
dition (i) implies that it is stable under products. We define a W*-metric on a
von Neumann algebra M ⊆ B(H) to be a W*-filtration {V } such that V is the
t 0
commutant of M. Since we interpret a W*-metric as a type of quantum metric,
and since it is the main type that we will consider in this article, we will also just
call it a quantum metric.
We will justify this definition with various constructions and results. We can
begin with a correspondence table between the usual axioms of a metric space and
3