Modular Theory in Operator Algebras

The first edition of this book appeared in 1981 as a direct continuation of Lectures of von NeumannAlgebras (Ş. V. Strătilă and L. Zsidó) and, until 2003, was the only comprehensive monographon the subject. Addressing students of mathematics and physics and researchers interested inoperator algebras, noncommutative geometry and free probability, this revised edition covers thefundamentals and some latest developments in the field of operator algebras. The intent is to makemodular theory accessible, with complete proofs, to readers having elementary training in operatoralgebras

The text provides detailed discussion of normal weights, conditional expectations andU. Haagerup’s operator-valued weights, groups of automorphisms and their spectral theory,duality theory for noncommutative groups, and crossed products of von Neumann algebras byactions of groups and duals of groups, which enables the extension of M. Takesaki duality fornoncommutative groups of automorphisms. It also contains detailed discussion on the group-measure space construction of factors and information about ITPFI factors and Krieger factors.

The core of the book is the continuous decomposition of A. Connes and M. Takesaki and discretedecomposition of A. Connes for type III factors.

It also explores new results, such as the A. Ocneanu’s result on the actions of amenable groupson the hyperfinite factor, H. Kosaki’s extension of the V. Jones index to arbitrary factors andF. Rădulescu’s examples of non-hyperfinite factors of type III𝜆, 𝜆 ∈ (0, 1) and of type III1.

Şerban Strătilă is Professor at the Institute of Mathematics, Romanian Academy, and at theDepartment of Mathematics, University of Bucharest, Romania. His current research areas includeoperator algebras and representation theory. He received the 1975 Simion Stoilow Prize forMathematics from the Romanian Academy. He has published Lectures on von Neumann Algebras,2nd edition (2019) with Cambridge University Press.

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Cambridge–IISc Series

Modular Theory in OperatorAlgebrasSecond Edition

Şerban Valentin Strătilă

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University Press, 2020. | Includes bibliographical references and index.

touch / A.K. Nandakumaran, P.S. Datti.

ISBN 978-1-108-83980-8 Hardback

University Printing House, Cambridge CB2 8BS, United Kingdom

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477 Williamstown Road, Port Melbourne, VIC 3207, Australia

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79 Anson Road, #06�04/06, Singapore 079906

Cambridge University Press is part of the University of Cambridge.

It furthers the University’s mission by disseminating knowledge in the pursuit ofeducation, learning and research at the highest international levels of excellence.

www.cambridge.orgInformation on this title: www.cambridge.org/9781108489607

© Şerban Valentin Strătilă 2020

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.

First edition published by Editura Academiei and Abacus Press, 1981Second edition published by Cambridge University Press, 2020

Printed in India

A catalogue record for this publication is available from the British Library

ISBN 978-1-108-48960-7 Hardback

Cambridge University Press has no responsibility for the persistence or accuracyof URLs for external or third party internet websites referred to in this publication,and does not guarantee that any content on such websites is, or will remain,accurate or appropriate.

To the memory of my wife Sanda Strătilă,the writer Alexandra Stănescu (1944–2011)

Contents

Preface to the second edition ixPreface to the first edition xi

Chapter I Normal Weights 11 Characterizations of Normality 12 The Standard Representation 103 The Balanced Weight 334 The Pedersen–Takesaki Construction 495 The Converse of the Connes Theorem 636 Equality and Majorization of Weights 677 The Spatial Derivative 778 Tensor Products 93

Chapter II Conditional Expectations and Operator-Valued Weights 999 Conditional Expectations 9910 Existence and Uniqueness of Conditional Expectations 11111 Operator-Valued Weights 12912 Existence and Uniqueness of Operator-Valued Weights 142

Chapter III Groups of Automorphisms 15913 Groups of Isometries on Banach Spaces 15914 Spectra and Spectral Subspaces 16515 Continuous Actions on W∗-algebras 18116 The Connes Invariant Γ(𝜎) 19617 Outer Automorphisms 208

Chapter IV Crossed Products 22918 Hopf–von Neumann Algebras 22919 Crossed Products 26120 Comparison of cocycles 28821 Abelian Groups 30222 Discrete Groups 313

vii

viii Contents

Chapter V Continuous Decompositions 33123 Dominant Weights and Continuous Decompositions 33124 The Flow of Weights 34825 The Fundamental Homomorphism 35626 The Extension of the Modular Automorphism Group 360

Chapter VI Discrete Decompositions 36927 The Connes Invariant T(ℳ) 36928 The Connes Invariant S(ℳ) 37429 Factors of Type III𝜆 (0 ≤ 𝜆 < 1) 38330 The Discrete Decomposition of Factors of Type III𝜆 (0 ≤ 𝜆 < 1) 396

Appendix 415References 429Notation Index 443Subject Index 445

Preface to the Second Edition

The first edition of this book appeared in 1981 as a direct continuation of Lectures of von NeumannAlgebras by Ş.V. Strătilă and L. Zsidó and, until 2003, was the only comprehensive monograph onthe subject.

The book Lectures on von Neumann Algebras, 2nd Edition, Cambridge University Press, 2019,will be always referred to as [L]. It is assumed that the reader is familiar with the material containedin this book, including the terminology and notation.

The present book contains the continuous decomposition and the discrete decomposition forfactors of type III and all the necessary results such as the extensive theory of normal weightsincluding the U. Haagerup characterization of normality, the A. Connes theorem of Radon–Nikodymtype and the Pedersen–Takesaki construction, the conditional expectations and the operator-valuedweights, a detailed consideration of groups of automorphisms and their spectral theory, and thetheory of crossed products. In order to include our extension of the Takesaki duality theorem tononcommutative groups of automorphisms, we considered a simultaneous generalization of groupsand duals of groups, namely the Kac algebras, and their actions on von Neumann algebras, as well asthe corresponding crossed products. Instead of Kac algebras we can also consider quantum groups,the arguments being exactly the same as for Kac algebras.

In this second edition we added information and references for several results, which appearedafter 1981, untill now, including A. Ocneanu’s theorem concerning actions of amenable groups andits extensions, H. Kosaki’s extension of the index to arbitrary factors and F. Rădulescu’s examplesof non-hyperfinite factors of type III𝜆, 𝜆 ∈ (0, 1) and of type III1 relying on D.-V. Voiculescu’stheory of free probability.

For subjects such as the equivalence of injectivity and hyperfiniteness (originally due toA. Connes) we indicate references for more recent shorter proofs. This guarantees the uniquenessof the hyperfinite factor of type II∞ and of type III𝜆 factors, 𝜆 ∈ (0, 1). Another important resultwhich could not be considered in detail, due to its length, is the proof of U. Haagerup of theA. Connes conjecture that the hyperfinite factor of type III1 is unique. For the same reason werestricted ourselves to only a few results concerning Krieger factors and ITPFI factors. However,we considered in detail the “group-measure space construction” which produced many examplesof factors, among them the first example of a type III factor by Murray and von Neumann, thePukanszky non-hyperfinite factor of type III and also the famous Powers factors which are actuallythe only hyperfinite factors of type III𝜆, 𝜆 ∈ (0, 1).

I am grateful to Gadadhar Misra and V. Sunder for proposing this second edition, to GadadharMisra for his considerable help with the Latex conversion of the book, to Ms Rajitha Reddy for herexcellent typing, to Florin Rădulescu for his help for Ocneanu’s theorem and his own theorems, andto Alexandru Negrescu for his help in typing the new material for the second edition.

I am also grateful to Cambridge University Press and to Ms Taranpreet Kaur for editing thissecond edition.

Şerban Strătilă Bucharest, January 20, 2020

Preface to the First Edition

The discovery of the modular operator and the modular automorphism group associated with anormal semifinite faithful weight has led to a powerful theory – the modular theory – which isnowadays essential to the consideration of many problems concerning operator algebras. This theoryhas been developed in close association with the effort to understand the structure and produceexamples and refined classifications of factors. Thus, the crossed product construction, which gaverise to the first non-trivial examples of factors, has been shown to play a fundamental role in thestructure theory as well, by reducing the study of the purely infinite algebras to the study of themore familiar semifinite algebras and their automorphisms. Moreover, several algebraic invariants,previously defined only in some special cases, have been introduced via the modular theory forarbitrary factors and the corresponding classification has been proved to be almost complete forapproximately finite dimensional factors.

The present book is a unified exposition of the technical tools of the modular theory and ofits applications to the structure and classification of factors. It is based on several works recentlypublished in periodicals or just circulated as preprints. The main sources used in writing this bookare the works of W. B. Arveson, A. Connes, U. Haagerup, M. Landstad, G. K. Pedersen, M. Takesaki,and J. Tomiyama. The general treatment of crossed products follows an article by Ş. Strătilă,D. Voiculescu and L. Zsidó. Due to the wealth and variety of results recently obtained, it has notbeen possible to include here a detailed exposition of the classification of injective factors and theirautomorphisms; these topics and several others are just mentioned in the Notes sections, togetherwith appropriate references.

The reader is assumed to have a good knowledge of the general theory of von Neumannalgebras, including the standard forms. Actually, the present book can be viewed as a sequel toa previous book, Ş. Strătilă and L. Zsidó – Lectures on von Neumann algebras, Editura Academiei& Abacus Press, 1979, which is often quoted here and referred to as [L]. There is also an Appendixwhich contains some supplementary results on positive self-adjoint operators and introduces theterminology connected with W∗-algebras.

The list of references in the present book contains only those items which have been used,quoted or consulted. A more extensive bibliography is contained in [L] (and in the Preprint Series,INCREST, Bucharest) and the new preprints are periodically recorded in C∗–News (issued byCPT/CNRS, Marseille).

I am very indebted to Zoia Ceauşescu, Alain Connes, Sanda Strătilă and Dan Voiculescu forthe moral support they offered me in writing this book. I am grateful to my colleagues ConstantinApostol, Grigore Arsene, Zoia Ceauşescu, Radu Gologan, Adrian Ocneanu, Cornel Pasnicu, MihaiPimsner, Sorin Popa and Dan Timotin for several useful discussions and a critical reading of variousparts of the manuscript. During his short visit in Romania, Alain Connes kindly informed me of themost recent developments of the theory. Thanks are also due to the National Institute for Scientificand Technical Creation, for the technical assistance.

xii Preface to the First Edition

It is a pleasure for me to acknowledge the most efficient and understanding cooperation of thePublishing House of the Romanian Academy (Editura Academiei) and Abacus Press, especiallyMrs Sorana Gorjan, who edited this book, and Dr Simon Wassermann of Glasgow University, whosecomments on the original translation were most helpful.

Şerban Strătilă Bucureşti, Romania, October 1979

CHAPTER I

Normal Weights

1 Characterizations of Normality

In this section, we prove the Theorem of Haagerup asserting that every normal weight on aW ∗-algebra is the pointwise least upper bound of the normal positive forms it majorizes.

1.1. Let 𝒜 be a C∗-algebra. A weight on 𝒜 is a mapping 𝜑 ∶ 𝒜+ → [0,+∞] with the properties

𝜑(x + y) = 𝜑(x) + 𝜑( y), 𝜑(𝜆x) = 𝜆𝜑(x) (x, y ∈ 𝒜+, 𝜆 ∈ ℝ+).

The set

𝔉𝜑 = {x ∈ 𝒜+;𝜑(x) < +∞}

is a face of 𝒜+, the set

𝔑𝜑 = {x ∈ 𝒜 ;𝜑(x∗x) < +∞}

is a left ideal of 𝒜 , and the set

𝔐𝜑 = 𝔑∗𝜑𝔑𝜑 = lin 𝔉𝜑

is a facial subalgebra of 𝒜 with 𝔐𝜑 ∩𝒜+ = 𝔉𝜑([L], 3.21), hence 𝜑 can be extended uniquely to apositive linear form, still denoted by 𝜑, on the *-algebra 𝔐𝜑.

A family ℱ of weights on 𝒜 is called sufficient if

x ∈ 𝒜 and 𝜑(a∗x∗xa) = 0 for all 𝜑 ∈ ℱ , a ∈ 𝔑𝜑 ⇒ x = 0

and is called separating if

x ∈ 𝒜 and 𝜑(x∗x) = 0 for all 𝜑 ∈ ℱ ⇒ x = 0.

In particular, the weight 𝜑 is called faithful if

x ∈ 𝒜 and 𝜑(x∗x) = 0 ⇒ x = 0.

1

2 Normal Weights

1.2. Let 𝜑 be a weight on the C∗-algebra 𝒜 . The formula

(a|b)𝜑 = 𝜑(b∗a) (a, b ∈ 𝔑𝜑)defines a prescalar product on 𝔑𝜑 with the properties:

(xa|xa)𝜑 ≤ ‖x‖2(a|a)𝜑 (x ∈ 𝒜 , a ∈ 𝔑𝜑),(xa|b)𝜑 = (a|x∗b)𝜑 (x ∈ 𝒜 , a, b ∈ 𝔑𝜑).

Let ℋ𝜑 be the Hilbert space associated with 𝔑𝜑 with the scalar product (⋅|⋅)𝜑. It follows that thereexists a *-representation 𝜋𝜑 ∶ 𝒜 → ℬ(ℋ𝜑), uniquely determined, such that

(𝜋𝜑(x)a𝜑|b𝜑)𝜑 = 𝜑(b∗xa) (x ∈ 𝒜 , a, b ∈ 𝔑𝜑), (1)where 𝔑𝜑 ∋ a ↦ a𝜑 ∈ ℋ𝜑 denotes the canonical mapping. The *-representation 𝜋𝜑 is called theGNS representation or the standard representation associated with 𝜑. We remark that

𝜑(x∗) = 𝜑(x) (x ∈ 𝔐𝜑), (2)|𝜑(b∗a)|2 ≤ 𝜑(a∗a)𝜑(b∗b) (a, b ∈ 𝔑𝜑). (3)1.3. Let ℳ be a W ∗-algebra. A weight 𝜑 on ℳ is called normal if

𝜑(supi

xi) = supi𝜑(xi)

for every norm-bounded increasing net {xi}i ⊂ℳ+, and lower w-semicontinuous if the convex sets

{x ∈ ℳ+;𝜑(x) ≤ 𝜆} (𝜆 ∈ ℝ+)are w-closed. An important result concerning weights on W ∗-algebras is the following characteriza-tion of normality:

Theorem (U. Haagerup). Let 𝜑 be a weight on the W ∗-algebra ℳ. The following statements areequivalent:

(i) 𝜑 is normal;(ii) 𝜑 is lower w-semicontinuous;

(iii) 𝜑(x) = sup{f (x); f ∈ ℳ+∗ , f ≤ 𝜑} for all x ∈ ℳ+.Later (2.10, 5.8) we shall see that 𝜑 is normal if and only if it is a sum of normal positive forms,

in accordance with the definition used in ([L], 10.14).In Sections 1.4–1.7, we present some general results that will be used in the proof of the theorem;

Sections 1.6–1.12 contain the main steps of the proof.

1.4 Proposition. If x, x1,… , xn ∈ ℬ(ℋ ) and x∗x = x∗1x1 +… + x∗nxn, then there exist z1,… , zn ∈

ℛ{x, x1,… , xn} such that z∗1z1 +…+ z∗nzn = s(xx

∗) and xk = zkx for all 1 ≤ k ≤ n.Proof. The equations zk(x𝜉) = xk𝜉(𝜉 ∈ ℋ ) and zk𝜂 = 0(𝜂 ∈ ℋ

⨁xℋ ) define operators zk ∈

ℬ(ℋ ), ‖zk‖ ≤ 1, with xk = zkx and zk(ℋ ⊖ xℋ ) = 0. Using the double commutant theorem

Characterizations of Normality 3

([L], 3.2) it is easy to check that zk ∈ ℛ{x, xk}. Also the relation∑

k z∗kzk = s(xx

∗) follows, since thepositive operator (

∑k z

∗kzk)

1∕2 vanishes on ℋ ⊖ xℋ and x∗(∑

k z∗kzk)x = x

∗x.In particular if x, y ∈ ℬ(ℋ ) and y∗y ≤ x∗x, there exists z ∈ ℛ{x, y} such that z∗z ≤ s(xx∗) and

y = zx.

1.5. For each 𝛼 > 0 we shall consider the function

f𝛼 ∶ (−𝛼−1,+∞) → ℝ

defined by f𝛼(t) = t(1 + 𝛼t)−1 = a−1(1 − (1 + 𝛼t)−1). These functions have the following properties:

f𝛼(t) ≤ min{t, 𝛼−1} (t ∈ (−𝛼−1,+∞)) (1)𝛼 ≤ 𝛽 ⇒ f𝛼(t) ≥ f𝛽(t) (t ∈ (−𝛼−1,+∞)) (2)𝛼 ≤ 𝛽 ⇒ 𝛼f𝛼(t) ≤ 𝛽f𝛽(t) (t ∈ (0,+∞)) (3)f𝛼( f𝛽(t)) = f𝛼+𝛽(t) (t ∈ (−(𝛼 + 𝛽)−1,+∞)) (4)lim𝛼→0

f𝛼(t) = t uniformly on compact subsets of ℝ (5)

lim𝛼→∞

𝛼f𝛼(t) = 1 uniformly on compact subsets of ℝ+. (6)

A continuous function f ∶ I → ℝ is called operator monotone on the interval I ⊂ ℝ if for everyx, y ∈ ℬ(ℋ ), x = x∗, y = y∗, with Sp(x) ⊂ I, Sp( y) ⊂ I, we have x ≤ y ⇒ f (x) ≤ f ( y). For instance,it is easy to see that

the functions f𝛼 are operator monotone (𝛼 ∈ ℝ+). (7)

Also, we recall that (see Pedersen, 1973–1977 or Strătilă & Zsidó, 1977, 1979)

the functions t → t𝛾 are operator monotone (0 < 𝛾 < 1). (8)

On the other hand, using (A.2) we see that

the functions f𝛼 are operator continuous (𝛼 ∈ ℝ+). (9)

1.6. Let 𝒳 be a locally convex Hausdorff real vector space with a partial ordering defined by aconvex cone 𝒳+ ⊂ 𝒳 such that 𝒳+ ∩ (−𝒳+) = {0} and 𝒳 = (𝒳+ −𝒳+). The dual cone 𝒳 ∗+ ={f ∈ 𝒳 ∗; f (x) ≥ 0 for all x ∈ 𝒳+} defines a partial ordering on 𝒳 ∗. A subset ℰ of 𝒳+ is calledhereditary if

x ∈ ℰ , y ∈ 𝒳+, x − y ∈ 𝒳+ ⇒ y ∈ ℰ .

For ℰ ⊂ 𝒳+ and ℱ ⊂ 𝒳 ∗+, we define ℰ∧ and ℱ ∧ by

ℰ∧ = {f ∈ 𝒳 ∗+; f (x) ≤ 1 for all x ∈ ℰ},ℱ ∧ = {x ∈ 𝒳+; f (x) ≤ 1 for all f ∈ ℱ }.

4 Normal Weights

Proposition. For 𝒳 as above the following statements are equivalent:

(i) ℰ = (ℰ −𝒳+) ∩𝒳+ for every closed hereditary convex subset ℰ of 𝒳+;(ii) ℰ = ℰ∧∧ for every closed hereditary convex subset ℰ of 𝒳+;

(iii) every subadditive, positively homogeneous, increasing and lower semicontinuous function 𝜑 ∶𝒳+ → [0,+∞] has the property

𝜑(x) = sup{f (x); f ∈ 𝒳 ∗+, f ≤ 𝜑} (x ∈ 𝒳+).Proof. We shall denote by 𝒮 0 the polar of a subset 𝒮 of 𝒳 or 𝒳 ∗.

(i) ⇒ (ii). The sets ℱ = ℰ∧ and ℱ ′ = −(ℰ −𝒳+)0 = {f ∈ 𝒳 ∗; f (x) ≤ 1 for all x ∈ ℰ −𝒳+} areequal. Indeed, it is clear that ℱ ⊂ ℱ ′. Let f ∈ ℱ ′ and x ∈ 𝒳+. Since 0 ∈ ℰ , we have f (−𝜆x) ≤ 1for all 𝜆 ≥ 0, whence f (x) ≥ 0. Thus ℱ ′ ⊂ 𝒳 ∗+ and so ℱ ′ ⊂ ℱ .

By the bipolar theorem it follows that (ℰ −𝒳+) = (ℰ − 𝒳+)00 = (−ℱ )0 = {x ∈ 𝒳 ;f (x) ≤ 1 for allf ∈ ℱ } and, using (i), we get ℰ = (ℰ −𝒳+) ∩𝒳+ = {x ∈ 𝒳+; f (x) ≤ 1 for all f ∈ℱ } = ℰ∧∧.

(ii) ⇒ (iii). If 𝜑 satisfies the conditions required in (iii), then the set ℰ = {x ∈ 𝒳+;𝜑(x) ≤ 1} isclosed, hereditary and convex. Also, ℱ = ℰ∧ = {f ∈ 𝒳 ∗+; f (x) ≤ 𝜑(x) for all x ∈ 𝒳+} and, by (ii),{x ∈ 𝒳+;𝜑(x) ≤ 1} = ℰ = ℱ ∧ = {x ∈ 𝒳+; supf∈ℱ f (x) ≤ 1}. It follows that 𝜑(x) = sup{f (x);f ∈ ℱ }, for all x ∈ 𝒳+.

(iii) ⇒ (i). Let ℰ ⊂ 𝒳+ be closed, hereditary and convex. Define 𝜑(x) = inf{𝜆 > 0; x ∈ 𝜆ℰ} ifx ∈

⋃𝜆>0 𝜆ℰ and 𝜑(x) = +∞ otherwise. Then 𝜑 satisfies the hypotheses in (iii) and therefore

𝜑(x) = sup{f (x); f ∈ ℱ }(x ∈ 𝒳+), where ℱ = {f ∈ 𝒳 ∗+; f (x) ≤ 𝜑(x) for all x ∈ 𝒳+}. Itfollows that ℰ − 𝒳+ ⊂ {x ∈ 𝒳 ; f (x) ≤ 1 for all f ∈ ℱ } and, since the latter set is closed, weget (ℰ −𝒳+) ∩𝒳+ ⊂ {x ∈ 𝒳+; f (x) ≤ 1 for all f ∈ ℱ } ⊂ ℰ , hence (ℰ −𝒳+) ∩𝒳+ = ℰ .1.7 Proposition. Let ℳ be a W ∗-algebra and ℰ ⊂ ℳ+ a w-closed hereditary convex set. Thenℰ = (ℰ −ℳ+)w ∩ℳ+.

Proof. We shall use the properties of the functions f𝛼 from 1.5. For x ∈ ℳh let 𝛼x = sup{𝛼 > 0;−𝛼−1 ≤ x}. Consider the set

𝒮 = {x ∈ ℳh; f𝛼(x) ∈ ℰ −ℳ+ for all 𝛼 ∈ (0, 𝛼x)},

and let ℳ𝜆; . = {x ∈ ℳ; ‖x‖ ≤ 𝜆}.We first show that for every 𝜆 > 0 the set 𝒮 ∩ℳ𝜆 is s-closed.Indeed, let x ∈ 𝒮 ∩ℳ𝜆

s. There is a net {xi}i∈I ⊂ 𝒮 such that ‖xi‖ ≤ 𝜆 and xi s→ x. Then

𝛼xi ≥ 1∕𝜆, hence f𝛼(xi) ∈ ℰ − ℳ+ for every 𝛼 ∈ (0, 1∕2𝜆) and every i ∈ I. Let 𝛼 ∈ (0, 1∕2𝜆) befixed. There is a net {yi}i∈I ⊂ ℰ such that

f𝛼(xi) ≤ yi (i ∈ I).Since f𝛼 is operator monotone,

f2𝛼(xi) = f𝛼( f𝛼(xi)) ≤ f𝛼( yi) (i ∈ I).

Characterizations of Normality 5

Since f2𝛼 is operator continuous on [−𝜆,+𝜆],

f2𝛼(xi)s→ f2𝛼(x).

Since 0 ≤ f𝛼( yi) ≤ 𝛼−1 and ℳ1 is w-compact, we may assume that there is y ∈ ℳ such thatf𝛼( yi)

w→ y.

Since 0 ≤ f𝛼( yi) ≤ yi ∈ ℰ and ℰ is hereditary, f𝛼( yi) ∈ ℰ and, since ℰ is w-closed, it follows thaty ∈ ℰ . Then

y − f2𝛼(x) = w- limi ( f𝛼( yi) − f2𝛼(xi)) ≥ 0,hence f2𝛼(x) ∈ ℰ −ℳ+. We have thus proved that

f𝛼(x) ∈ ℰ −ℳ+ for every 𝛼 ∈ (0, 1∕𝜆).

Consider now 𝛼 ∈ [1∕𝜆, 𝛼x) and 𝛽 ∈ (0, 1∕𝜆). Then f𝛼(x) ≤ f𝛽(x), hence f𝛼(x) ∈ (ℰ −ℳ+) −ℳ+ =ℰ −ℳ+. We conclude that x ∈ 𝒮 ∩ℳ𝜆.

We now show that 𝒮 is convex.Indeed, it is sufficient to show that each 𝒮 ∩ℳ𝜆 is convex, and this will follow from the equality

𝒮 ∩ℳ𝜆 = ((ℰ −ℳ+) ∩ℳ𝜇)s ∩ℳ𝜆 for 𝜇 > 𝜆.

If x ∈ 𝒮 ∩ℳ𝜆, then f𝛼(x) ∈ ℰ −ℳ+ for 𝛼 ∈ (0, 𝛼x) and f𝛼(x) ∈ ℳ𝜇 for small 𝛼 > 0, hence

x = s- lim𝛼→0

f𝛼(x) ∈ ((ℰ −ℳ+) ∩ℳ𝜇)s ∩ℳ𝜆.

Conversely, since ℰ is hereditary and f𝛼(x) ≤ x for all 𝛼 ∈ (0, 𝛼x), we have ℰ − ℳ+ ⊂ 𝒮 , hence(ℰ −ℳ+) ∩ℳ𝜇 ⊂ 𝒮 ∩ℳ𝜇. Using the first part of the proof we get ((ℰ −ℳ+) ∩ℳ𝜇)

s⊂ 𝒮 ∩ℳ𝜇,

and the desired inclusion follows.Using the Krein-S̆mulian theorem ([L], C.1.1; Dunford & Schwartz, 1958, 1963, V.5.7), from the

earlier it follows that 𝒮 is w-closed. We have seen that ℰ −ℳ+ ⊂ 𝒮 . Since x = w− lim𝛼→0 f𝛼(x),we obtain 𝒮 ⊂ (ℰ −ℳ+)

w. Consequently, 𝒮 = (ℰ −ℳ+)

w.

Finally, let x ∈ (ℰ −ℳ+)w∩ℳ+ = 𝒮 ∩ℳ+. For every 𝛼 > 0 we have f𝛼(x) ∈ (ℰ −ℳ+) ∩ℳ+,

hence f𝛼(x) ∈ ℰ , as ℰ is hereditary. It follows that x = w- lim𝛼→0 f𝛼(x) ∈ ℰ .From Propositions 1.6 and 1.7, we obtain the equivalence (ii) ⇒ (iii) in Theorem 1.3, as the

implication (iii) ⇒ (ii) is obvious.In Sections 1.8–1.12, we assume that 𝜑 is a fixed normal weight on the W ∗-algebra ℳ.

1.8 Lemma. There exists a linear mapping Φ ∶ 𝔐𝜑 → 𝜋𝜑(ℳ)′∗, uniquely determined, such that

Φ(b∗a)(T ′) = (T ′a𝜑|b𝜑)𝜑 (T ′ ∈ 𝜋𝜑(ℳ)′, a, b ∈ 𝔑𝜑). (1)

6 Normal Weights

Moreover, for every x ∈ 𝔐𝜑 ∩ℳh we have

‖Φ(x)‖ = inf{𝜑( y) + 𝜑(z); y, z ∈ 𝔐𝜑 ∩ℳ+, x = y − z}. (2)Proof. The uniqueness of Φ follows from the relation 𝔐𝜑 = 𝔑∗𝜑𝔑𝜑.

If a, b, c ∈ 𝔑𝜑, c∗ = c and c∗c = a∗a + b∗b, then, by Proposition 1.4, there exist x, y ∈ ℳ suchthat a = xc, b = yc and x∗x + y∗y = s(cc∗) = s(c), and for every T ′ ∈ 𝜋𝜑(ℳ)′ we have

(T ′c𝜑|c𝜑)𝜑 = (T ′𝜋𝜑(x∗x + y∗y)c𝜑|c𝜑)𝜑= (T ′𝜋𝜑(x)c𝜑|𝜋𝜑(x)c𝜑)𝜑 + (T ′𝜋𝜑( y)c𝜑|𝜋𝜑( y)c𝜑)𝜑= (T ′a𝜑|a𝜑)𝜑 + (T ′b𝜑|b𝜑)𝜑.

It follows that the mapping

Φ0 ∶ 𝔐𝜑 ∩ℳ+ ∋ a∗a ↦ 𝜔a𝜑 |𝜋𝜑(ℳ)′ ∈ 𝜋𝜑(ℳ)′∗.is well defined and additive. Clearly, Φ0 is positively homogeneous. Since 𝔐𝜑 = lin(𝔐𝜑∩ℳ+),Φ0has a unique linear extension Φ to 𝔐𝜑 and (1) follows using the polarization relation ([L], p. 75).

The function 𝜌 defined on 𝔐𝜑 ∩ℳh by the right-hand side of (2) is a semi-norm on 𝔐𝜑 ∩ℳh.If x ∈ 𝔐𝜑 ∩ ℳ+, then ‖Φ(x)‖ = Φ(x)(1) = ((x1∕2)𝜑|(x1∕2)𝜑)𝜑 = 𝜑(x) = 𝜌(x). Consequently, forx = y − z, with y, z ∈ 𝔐𝜑 ∩ ℳ+, we have ‖Φ(x)‖ ≤ ‖Φ( y)‖ + ‖Φ(z)‖ = 𝜑( y) + 𝜑(z). Hence‖Φ(x)‖ ≤ 𝜌(x) for all x ∈ 𝔐𝜑 ∩ℳh.

Let x0 ∈ 𝔐𝜑 ∩ℳh. By the Hahn–Banach theorem, there exists a real linear form f on 𝔐𝜑 ∩ℳhsuch that f (x0) = 𝜌(x0) and |f (x)| ≤ 𝜌(x) for every x ∈ 𝔐𝜑 ∩ ℳh. Then f can be extended to acomplex linear form, still denoted by f, on 𝔐𝜑. Since −𝜑(x) ≤ f (x) ≤ 𝜑(x) for any x ∈ 𝔐𝜑 ∩ℳ+,we may consider 𝜑 + f and 𝜑 − f as weights on ℳ. Consequently, using the Schwarz inequality1.2.(3), for a, b ∈ 𝔑𝜑 we obtain

f (b∗a) ≤ 2−1[|(𝜑 + f )(b∗a)| + |(𝜑 − f )(b∗a)|]≤ 2−1[(𝜑 + f )(a∗a)1∕2(𝜑 + f )(b∗b)1∕2 + (𝜑 − f )(a∗a)1∕2(𝜑 − f )(b∗b)1∕2]≤ 2−1[(𝜑 + f )(a∗a) + (𝜑 − f )(a∗a)]1∕2[(𝜑 + f )(b∗b) + (𝜑 − f )(b∗b)]1∕2= 𝜑(a∗a)1∕2𝜑(b∗b)1∕2 = ‖a𝜑‖𝜑‖b𝜑‖𝜑.

Thus, there exists an operator T ′ ∈ ℬ(ℋ𝜑), ‖T ′‖ ≤ 1, such that f (b∗a) = (T ′a𝜑∕b𝜑)𝜑 for alla, b ∈ 𝔑𝜑. Moreover T ′ ∈ 𝜋𝜑(ℳ)′, since for every x ∈ ℳ and every a, b ∈ 𝔑𝜑 we have

(T ′𝜋𝜑(x)a𝜑|b𝜑)𝜑 = f (b∗xa) = (𝜋𝜑(x)T ′a𝜑|b𝜑)𝜑.It follows that 𝜌(x0) = |f (x0)| = |Φ(x0)(T ′)| ≤ ‖Φ(x0)‖‖T ′‖ ≤ ‖Φ(x0)‖.1.9 Lemma. Let {xn} be a norm-bounded sequence in 𝔐𝜑 ∩ℳ+ such that the sequence {Φ(xn)} isnorm-convergent in 𝜋𝜑(ℳ)′∗. Then

xns→ x ∈ ℳ ⇒ x ∈ 𝔐𝜑 ∩ℳ+, (1)

xns→ 0 ⇒ ‖Φ(xn)‖ → 0. (2)

Characterizations of Normality 7

Proof. Let 𝜀 > 0 and 𝜓 = limn Φ(xn) ∈ 𝜋𝜑(ℳ)′∗. Without loss of generality we may assume that‖Φ(xn) − 𝜓‖ < 𝜀∕2n, so that ‖Φ(xn+1 − xn)‖ < 𝜀∕2n−1 for all n ∈ ℕ. By Lemma 1.8, there existsequences {yn} and {zn} in 𝔐𝜑 ∩ℳ+ such that

xn+1 − xn = yn − zn and 𝜑( yn) + 𝜑(zn) < 𝜀∕2n−1 (n ∈ ℕ).

We shall again use the functions f𝛼 from Section 1.5.

(1) Since xn+1 ≤ x1 +∑nk=1 yk and xn+1 s→ x, we obtainf𝛼(x) = s- limn f𝛼(xn+1) ≤ supn f𝛼

(x1 +

n∑k=1

yk

)

and then, using the normality of 𝜑,

𝜑( f𝛼(x)) ≤ supn𝜑(

f𝛼

(x1 +

n∑k=1

yk

))≤ supn𝜑

(x1 +

n∑k=1

yk

)

≤ 𝜑(x1) +∞∑

k=1𝜑( yk) ≤ 𝜑(x1) +

∞∑k=1

𝜀∕2k−1 = 𝜑(x1) + 2𝜀.

Since f𝛼(x) ↑ x, again using the normality of 𝜑 we get

𝜑(x) = sup𝛼>0

𝜑( f𝛼(x)) ≤ 𝜑(x1) + 2𝜀 < +∞,hence x ∈ 𝔐𝜑 ∩ℳ+.

(2) Since − supn ‖xn‖ ≤ x1 − xn+1 ≤ ∑nk=1 zk, for 𝛼 > (supn ‖xn‖)−1 we obtainf𝛼(x1 − xn+1) ≤ sup

nfn

(n∑

k=1zk

).

Since x1 − xn+1s→ x1, it follows that

f𝛼(x1) = s- limn f𝛼(x1 − xn+1) ≤ supn f𝛼(

n∑k=1

zk

).

Using the normality of 𝜑 we infer that

𝜑(x1) = sup𝛼>0

𝜑( f𝛼(x1)) ≤ sup𝛼>0

supn𝜑

(f𝛼

(n∑

k=1zk

))

≤ supn𝜑

(n∑

k=1zk

)≤ ∞∑

k=1𝜀∕2k−1 = 2𝜀.

Consequently, ‖𝜓‖ ≤ ‖𝜓 − Φ(x1)‖ + ‖Φ(x1)‖ ≤ 𝜀∕2 + 2𝜀 = 3𝜀∕2. We conclude that 𝜓 = 0.

8 Normal Weights

1.10. Let 𝒢𝜑 = {(x, x𝜑); x ∈ 𝔑𝜑} ⊂ℳ×ℋ𝜑. Since every Hilbert space is a reflexive Banach space,ℳ ×ℋ𝜑 is the dual of the Banach space ℳ∗ ×ℋ𝜑. For 𝜆, 𝜇 > 0, let ℳ𝜆 = {x ∈ ℳ; ‖x‖ ≤ 𝜆} and(ℋ𝜑)𝜇 = {𝜉 ∈ ℋ𝜑; ‖𝜉‖ ≤ 𝜇}.Lemma. If ℳ is countably decomposable, then 𝒢𝜑 ∩ (ℳ𝜆 × (ℋ𝜑)𝜇) is 𝜎(ℳ × ℋ𝜑,ℳ∗ × ℋ𝜑)-compact, for ever 𝜇 > 0.

Proof. Since 𝒢𝜑∩(ℳ𝜆×(ℋ𝜑)𝜇) is convex and bounded, it is sufficient to prove that it is closed withrespect to the product topology 𝜏 on ℳ × (ℋ𝜑) of the s∗-topology on ℳ and the norm-topology onℋ𝜑. Since ℳ is countably decomposable, ℳ𝜆 is s∗-metrizable ([L], E.5.7; Strătilă & Zsidó, 1977,1979, 8.12).

If (x, 𝜉) ∈ ℳ ×ℋ𝜑 is 𝜏-adherent to 𝒢𝜑 ∩ (ℳ𝜆 × (ℋ𝜑)𝜇), then there exists a sequence {xn} in ℳ𝜆such that xn

s∗→ x, ‖(xn)𝜑‖𝜑 ≤ 𝜇 and ‖(xn)𝜑−𝜉‖ → 0. Then x∗nxn s∗→ x∗x and Φ(x∗nxn) = 𝜔(xn)𝜑 is norm

convergent to 𝜔𝜉 , whence x ∈ 𝔑𝜑 by Lemma 1.8.(1). On the other hand, (xn − x)∗(xn − x)s→ 0 and

Φ((xn − x)∗(xn − x)) = 𝜔(xn)𝜑−x𝜑 → 𝜔𝜉−x𝜑

so that 𝜔𝜉−x𝜑 = 0 by Lemma 1.8.(2). Thus 𝜉 = x𝜑 and (x, 𝜉) ∈ 𝒢𝜑.

1.11. If ℳ is not countably decomposable, we consider the set 𝒫0 of all countably decomposableprojections of ℳ and put

ℳ0 =⋃

p∈𝒫0

pℳp.

It is easy to check that ℳ0 is a self-adjoint ideal in ℳ.

Lemma. Let ℰ be a hereditary convex subset of ℳ0 ∩ℳ+. Then ℰ is w-closed in ℳ0 if and onlyif ℰ ∩ pℳp is w-closed for every p ∈ 𝒫0.

Proof. Assume that ℰ ∩ pℳp is w-closed for every p ∈ 𝒫0. The set ℱ = {x ∈ ℳ; x∗x ∈ ℰ} isconvex and aℱ ⊂ ℱ for all a ∈ ℳ1.

We first show that pℱ , or equivalently ℱ ∗p, is w-closed for any p ∈ 𝒫0. Using the Krein-S̆muliantheorem and the fact that any s-closed convex set is also w-closed, it is sufficient to show that ℱ ∗p∩ℳ𝜆 is s-closed for every 𝜆 > 0. Let x ∈ ℳ be such that x∗ is s-adherent to ℱ ∗p ∩ ℳ𝜆. Sinceℳp∩ℳ𝜆 is s-metrizable, there exists a sequence {xn} in pℱ , ‖xn‖ ≤ 𝜆, with x∗n s→ x∗. There existsa projection q ∈ 𝒫0 such that xn ∈ qℳq for all n ∈ ℕ. Thus

xn ∈ ℱ ∩ qℳq = {y ∈ qℳq; y∗y ∈ ℰ ∩ qℳq} (n ∈ ℕ).

By assumption, ℰ ∩ qℳq is w-closed, hence ℱ ∩ qℳq is s-closed. It follows that ℱ ∩ qℳq isw-closed and, consequently, x ∈ ℱ ∩ qℳq. Since px = x and ‖x‖ ≤ 𝜆, we get x∗ ∈ ℱ ∗p ∩ ℳ𝜆.Hence pℱ is w-closed.

Characterizations of Normality 9

Let x ∈ ℳ0 be w-adherent to ℰ . There exists a net {xi}i∈I ⊂ ℰ such that xis→ x. Then

p = l(x) ∈ 𝒫0 and px1∕2i

s→ px1∕2 = x1∕2. By the earlier paragraph, we know that pℱ is s-closed,

hence x1∕2 ∈ pℱ ⊂ ℱ , that is, x ∈ ℰ . Hence ℰ is w-closed in ℳ0.The converse is obvious.

1.12. Proof of Theorem 1.3. As we have already seen (1.7), (ii) ⇔ (iii). The implication (ii) ⇒ (i) isobvious. To show that (i) ⇒ (ii), we have to prove that the set

ℰ = {x ∈ ℳ+;𝜑(x) ≤ 1}is w-closed. Clearly, ℰ is hereditary and convex.

Assume first that ℳ is countably decomposable. As in the last part of the proof of Lemma 1.11, itis sufficient to show that the set ℱ = {x ∈ ℳ;𝜑(x∗x) ≤ 1} is w-closed. Since ℱ ∩ℳ𝜆 is the imageof 𝒢𝜑 ∩ (ℳ𝜆 × (ℋ𝜑)1) by the canonical projection mapping (x, 𝜉) ↦ x, from Lemma 1.10 it followsthat ℱ ∩ℳ𝜆 is w-compact for every 𝜆 > 0. Since ℱ is convex, we infer that ℱ is w-closed.

Consider now the general case. By the earlier argument and by Lemma 1.11, it follows thatℰ∩ℳ0is w-closed in ℳ0. Let x ∈ ℳ+ be w-adherent to ℰ . There exists a net {xi}i∈I ⊂ ℰ such that x1

s→ x.

Also, there exists an increasing net {pk}k∈K ⊂ 𝒫0 with pk ↑ 1. Since ℳ0 is a two-sided ideal in ℳ,for every k ∈ K we have

ℰ ∩ℳ0 ∋ x1∕2i pkx

1∕2i

w→i∈I x

1∕2pkx1∕2 ∈ ℳ0,

hence x1∕2pkx1∕2 ∈ ℰ ∩ ℳ0. Since x1∕2pkx1∕2 ↑ x, using the normality of 𝜑 we infer that 𝜑(x) =supk∈K 𝜑(x1∕2pkx1∕2) ≤ 1, that is, x ∈ ℰ .1.13. We recall that a positive form 𝜑 on the W ∗-algebra ℳ is normal if and only if it is completelyadditive on projections ([L], 5.6, 5.11). This statement cannot be extended to weights, as thefollowing example shows.

Let 𝓁∞(ℕ) be the W ∗-algebra of all bounded complex sequences. The weight 𝜑 defined on 𝓁∞(ℕ)by 𝜑({an}) =

∑n an if the set {n ∈ ℕ; an ≠ 0} is finite, and 𝜑({an}) = +∞ otherwise, is completely

additive on projections, but is not normal.

1.14 Proposition. Let𝜑 be a normal weight on the W ∗-algebraℳ and a, b ∈ 𝔑𝜑. Then the mapping

𝜑(b∗ ⋅ a) ∶ ℳ ∋ x ↦ 𝜑(b∗xa) ∈ ℂ

is a w-continuous linear form on ℳ.

Proof. Since a, b ∈ 𝔑𝜑, for any x ∈ ℳ we have b∗xa ∈ ℛ∗𝜑𝔑𝜑 = 𝔐𝜑, hence 𝜑(b∗ ⋅ a) is well

defined. If xi ↑ x in ℳ+, then a∗xia ↑ a∗xa in ℳ+, and hence 𝜑(a∗xia) ↑ 𝜑(a∗xa), since 𝜑 isnormal. It follows that 𝜑(a∗ ⋅ a) is w-continuous ([L], 5.11) and the general case is obtained using apolarization relation ([L], 3.21).

1.15. Notes. The main result (Thm. 1.3) of this section is due to Haagerup (1975a).

For our exposition we have used Haagerup (1975a) and [L].

10 Normal Weights

2 The Standard Representation

In this section, we prove that every normal semifinite weight is the supremum of an upward directedfamily of normal positive forms; also, we review and complete the results in ([L], Chapter 10)concerning the associated standard representation.

2.1. Let 𝜑 be a normal weight on the W ∗-algebra ℳ.Using ([L], 2.22) and the normality of 𝜑 it is easy to see that

x ∈ ℳ+, 𝜑(x) = 0 ⇒ 𝜑(s(x)) = 0. (1)

If e, f ∈ ℳ are projections and 𝜑(e) = 𝜑( f ) = 0, then 𝜑(e ∨ f ) = 𝜑(s(e + f )) = 0. Thus thefamily ℰ = {e ∈ Proj(ℳ);𝜑(e) = 0} is upward directed. Let e0 = supℰ . By the normality of 𝜑,it follows that 𝜑(e0) = 0, so that e0 is the greatest projection in ℳ annihilated by 𝜑. The projections(𝜑) = 1 − e0 is called the support of 𝜑. Using (1), we obtain

𝜑(x∗x) = 0 ⇔ xs(𝜑) = 0 (x ∈ ℳ). (2)

In particular, 𝜑 is faithful (1.1) if and only if s(𝜑) = 1. Also

𝜑(x) = 𝜑(s(𝜑)xs(𝜑)) (x ∈ ℳ+). (3)

On the other hand, the w-closure 𝔑w

𝜑 of 𝔑𝜑 is a w-closed left ideal of ℳ, hence ℛw

𝜑 = ℳe for

some projection e ∈ ℳ and 𝔐w

𝜑 = eℳe ([L], 3.20, 3.21). The weight 𝜑 is called semifinite ife = 1, that is, if 𝔑𝜑, or equivalently, 𝔐𝜑, is w-dense in ℳ. In this case, there exists an increasingnet {ui}i∈I in 𝔉𝜑 = 𝔐𝜑 ∩ℳ+ such that ui ↑ 1 ([L], 3.20, 3.21).

We abbreviate the words “normal semifinite faithful” to n.s.f. Recall that on every W ∗-algebrathere exists an n.s.f. weight, while the countably decomposable W ∗-algebras are characterized bythe existence of a normal faithful positive form ([L], 10.14, E.5.6).

2.2 Theorem. Let 𝜑 be a normal weight on the W ∗-algebra ℳ. Then the associated GNSrepresentation 𝜋𝜑 ∶ ℳ → ℬ(ℋ𝜑) is normal and nondegenerate. If 𝜑 is semifinite, then

((𝔐𝜑)n)𝜑 is dense in ℋ𝜑 (n ∈ ℕ). (1)

If 𝜑 is an n.s.f. weight, then 𝜋𝜑 is a *-isomorphism of ℳ onto the von Neumann algebra 𝜋𝜑(ℳ) ⊂ℬ(ℋ𝜑).

Proof. Clearly, 𝜋𝜑(1) = 1, hence 𝜋𝜑 is nondegenerate. To show that 𝜋𝜑 is normal, that is,w-continuous, we have to check that 𝜔 ◦𝜋𝜑 ∈ ℳ∗ for every 𝜔 ∈ ℬ(ℋ𝜑)∗. Since the vector formsare total in ℬ(ℋ𝜑)∗ ([L], 1.3) and 𝔑𝜑 is dense in ℋ𝜑, it is sufficient to do this only for 𝜔 = 𝜔a𝜑,b𝜑with a, b ∈ 𝔑𝜑. In this case, we have 𝜔a𝜑,b𝜑 ◦𝜋𝜑 = 𝜑(b

∗ ⋅ a) ∈ ℳ∗, by Proposition 1.14. Since 𝜋𝜑is normal and nondegenerate, 𝜋𝜑(ℳ) ⊂ℬ(ℋ𝜑) is a von Neumann algebra ([L], 3.12).

If 𝜑 is semifinite, then there exists an increasing net {ui}i∈I in 𝔉𝜑 = 𝔐𝜑 ∩ℳ+ with ui ↑ 1. Fora ∈ 𝔑𝜑, we have

‖a𝜑 − (uia)𝜑‖2𝜑 = 𝜑((a − uia)∗(a − uia)) ≤ 2[𝜑(a∗a) − 𝜑(a∗uia)] → 0. (2)

The Standard Representation 11

Since ui ∈ 𝔉𝜑 ⊂ 𝔑∗𝜑 and a ∈ 𝔑𝜑, we have uia ∈ 𝔑∗𝜑𝔑𝜑 = 𝔐𝜑 and from (2) it follows that (𝔐𝜑)𝜑

is dense in 𝔑𝜑, hence also in ℋ𝜑. Statement (1) follows now using (2) repeatedly.Assume that 𝜑 is an n.s.f. weight. If x ∈ ℳ and 𝜋𝜑(x) = 0, then 𝜑((xa)∗(xa)) = ‖𝜋𝜑(x)a𝜑‖2𝜑 = 0

for all a ∈ 𝔑𝜑. Since 𝜑 is faithful it follows that x𝔑𝜑 = 0 and hence x = 0, as 𝜑 is semifinite.Consequently, 𝜋𝜑 is a *-isomorphism.

In the next three sections, we study the majorization relation between weights in terms of theassociated GNS representation.

2.3 Proposition. Let 𝜑,𝜓 be weights on the C∗-algebra 𝒜 such that 𝜓 ≤ 𝜑, that is, 𝜓(x) ≤ 𝜑(x)for all x ∈ 𝒜+. There exists a unique operator T ′ ∈ 𝜋𝜑(𝒜 )′, 0 ≤ T ′ ≤ 1, such that

𝜓(b∗a) = (T ′a𝜑|T ′b𝜑)𝜑 (a, b ∈ 𝔑𝜑). (1)Proof. Since 𝜓 ≤ 𝜑 we have 𝔑𝜑 ⊂ 𝔑𝜓 and, for every a ∈ 𝔑𝜑, ‖a𝜓‖2𝜓 = 𝜓(a∗a) ≤ 𝜑(a∗a) =‖a𝜑‖2𝜑. It follows that there exists a unique linear operator S′ ∶ ℋ𝜑 → ℋ𝜓 , ‖S′‖ = 1, such thatS′a𝜑 = a𝜓 for all a ∈ 𝔑𝜑. Then T ′ = (S′∗S′)1∕2 ∈ ℬ(ℋ𝜑), 0 ≤ T ′ ≤ 1. For every a, b ∈ 𝔑𝜑 andevery x ∈ 𝒜 we have

𝜓(b∗a) = (a𝜓 |b𝜓 )𝜓 = (S′a𝜑|S′b𝜑)𝜓 = (T ′2a𝜑|b𝜑)𝜑 = (T ′a𝜑|T ′b𝜑)𝜑(S′∗S′𝜋𝜑(x)a𝜑|b𝜑)𝜑 = 𝜓(b∗xa) = 𝜓((x∗b)∗a) = (𝜋𝜑(x)S′∗S′a𝜑|b𝜑)𝜑,

hence T ′2 = S′∗S′ ∈ 𝜋𝜑(𝒜 )′. Since 𝜋𝜑(𝒜 )′ is a C∗-algebra, we infer that T ′ = (T ′2)1∕2 ∈ 𝜋𝜑(𝒜 )′.

From (1) it follows that the numbers (T ′2a𝜑|b𝜑)𝜑(a, b ∈ 𝔑𝜑) are uniquely determined by 𝜓 and𝜑, and this implies the uniqueness of T ′.

If 𝜑 and 𝜓 are finite and 𝒜 is unital, then from (1) it follows that

𝜓(x) = (𝜋𝜑(x)T ′1𝜑|T ′1𝜑)𝜑 (x ∈ 𝒜 ). (2)2.4 Corollary. Let 𝜑 be a weight on the C∗-algebra 𝒜 and denote by 𝒯 ′𝜑 the set of all T

′ ∈ 𝜋𝜑(𝒜 )′

such that there exists some 𝜆T ′ > 0 with the property that ‖T ′a𝜑‖𝜑 ≤ 𝜆T ′‖a‖ for all a ∈ 𝔑𝜑. Then𝒯 ′𝜑 is a left ideal of the W

∗-algebra 𝜋𝜑(𝒜 )′ and:(1) for every positive form f on 𝒜 with f ≤ 𝜑 there exist a unique T ′ ∈ 𝒯 ′𝜑 , 0 ≤ T ′ ≤ 1, and a

unique 𝜂 ∈ 𝜋𝜑(𝔑∗𝜑)ℋ𝜑 such that

f (b∗a) = (T ′a𝜑|T ′b𝜑)𝜑 (a, b ∈ 𝔑𝜑);f (x) = (𝜔𝜂 ◦𝜋𝜑)(x) (x ∈ 𝔐𝜑).

(2) for every T ′ ∈ 𝒯 ′𝜑 , 0 ≤ T ′ ≤ 1, there exist a positive form f ≤ 𝜑 on 𝒜 and a unique𝜂 ∈ 𝜋𝜑(𝔑∗𝜑)ℋ𝜑 such that

f (b∗a) = (T ′a𝜑|T ′b𝜑)𝜑 (a, b ∈ 𝔑𝜑);T ′a𝜑 = 𝜋𝜑(a)𝜂 (a ∈ 𝔑𝜑).

12 Normal Weights

Proof. It is clear that 𝒯 ′𝜑 is a left ideal of 𝜋𝜑(𝒜 )′. Also, if f ≤ 𝜑 is a positive form on 𝒜 , we infer

from 2.3.(1) that ‖T ′a𝜑‖𝜑 ≤ ‖f‖1∕2‖a‖(a ∈ 𝔑𝜑), hence T ′ ∈ 𝒯 ′𝜑 .Let {ui}i∈I be a right approximate unit for the left ideal 𝔑𝜑 of 𝒜 ([L], 3.20). For T ′ ∈ 𝒯 ′𝜑 and

a ∈ 𝔑𝜑 it follows that

𝜋𝜑(a)T ′(ui)𝜑 = T ′(aui)𝜑 → T ′a𝜑.

Thus, if ak ∈ 𝔑𝜑, 𝜉k ∈ ℋ𝜑(1 ≤ k ≤ n), and 𝜁 = ∑nk=1 𝜋𝜑(a∗k )𝜉k ∈ 𝜋k(𝔑∗𝜑)ℋ𝜑, then|||||n∑

k=1(𝜉k|T ′(ak)𝜑)𝜑||||| = limi |(𝜁 |T ′(ui)𝜑)𝜑| ≤ 𝜆T ′‖𝜁‖𝜑.

It follows that the mapping 𝜁 ↦∑n

k=1(𝜉k|T ′(ak)𝜑)𝜑 defines a bounded linear form on 𝜋𝜑(𝔑∗𝜑)ℋ𝜑and, consequently, there exists a unique vector 𝜂 ∈ 𝜋𝜑(𝔑∗𝜑)ℋ𝜑 such that

(𝜉|T ′a𝜑)𝜑 = (𝜉|𝜋𝜑(a)𝜂)𝜑 (a ∈ 𝔑𝜑, 𝜉 ∈ ℋ𝜑),that is, T ′a𝜑 = 𝜋𝜑(a)𝜂 for all a ∈ 𝔑𝜑. In particular, f = 𝜔𝜂 ◦𝜋𝜑 is a positive form on 𝒜 andf (b∗a) = (T ′a𝜑|T ′b𝜑)𝜑 for all a, b ∈ 𝔑𝜑.2.5 Corollary. Let 𝜑 be a normal weight on the W ∗-algebra ℳ and f0 a positive form on ℳ. Iff0 ≤ 𝜑, then there exists a normal positive form f on ℳ such that f ≤ 𝜑 and f|𝔐𝜑 = f0|𝔐𝜑.Proof. By Corollary 2.4.(1), there exists a vector 𝜂 ∈ ℋ𝜑 such that f0(x) = (𝜔𝜂 ◦𝜋𝜑)(x) for everyx ∈ 𝔐𝜑. Since 𝜑 is normal, 𝜋𝜑 is normal (2.2), so we can take f = 𝜔𝜂 ◦𝜋𝜑.

2.6. By Haagerup’s theorem (1.3), every normal weight 𝜑 on the W ∗-algebra ℳ is the pointwisesupremum of the family𝔉𝜑 = {f ∈ ℳ+∗ ; f ≤ 𝜑}; thus𝜑 is also the pointwise supremum of the family{f ∈ ℳ+∗ ; there exists 𝜀 > 0 such that (1+ 𝜀)f ≤ 𝜑}. The next result shows, in particular, that everynormal semifinite weight on a W ∗-algebra is the pointwise supremum of an upward directed familyof normal positive forms.

Theorem (F. Combes). Let 𝜑 be a normal semifinite weight on the W ∗-algebra ℳ. Then the family

{f ∈ ℳ+∗ ; there exists 𝜀 > 0 such that (1 + 𝜀)f ≤ 𝜑}is upward directed.

We first prove two general results of independent interest.

2.7 Proposition. Let 𝔑 be a left ideal of the W ∗-algebra ℳ. Then ℱ = (𝔑∗𝔑) ∩ℳ+ is a face ofℳ+,𝔑 = {x ∈ ℳ; x∗x ∈ ℱ } and 𝔑∗𝔑 = lin ℱ . In particular, 𝔑∗𝔑 is a facial subalgebra of ℳ.

Proof. Clearly, 𝔑 ⊂ {xℳ; x∗x ∈ ℱ } and, by the polarization relation ([L], 3.21), 𝔑∗𝔑 = lin ℱ .Let x ∈ ℳ be such that x∗x ≤ b ∈ ℱ . Since b is self-adjoint, using again the polarization relation

we can find xk, yk ∈ 𝔑, (1 ≤ k ≤ n) such thatx∗x ≤ b = n∑

k=1x∗kxk −

n∑k=1

y∗kyk ≤n∑

k=1x∗kxk = a

The Standard Representation 13

By Proposition 1.4, there exist z, zk ∈ ℳ(1 ≤ k ≤ n) such that x = za1∕2, xk = zka1∕2 and∑nk=1 z∗kzk =s(a). It follows that

x = za1∕2 = z

(n∑

k=1z∗kzk

)a1∕2 =

n∑k=1

zz∗kxk ∈ 𝔑.

Hence ℱ is a face of ℳ+ and {x ∈ ℳ; x∗x ∈ ℱ } ⊂ 𝔑.

2.8 Proposition. Let𝒜 be a C∗-algebra and𝔐 a facial subalgebra of𝒜 . Then the set {x ∈ 𝔐∩𝒜+;‖x‖ < 1} is upward directed.Proof. Let x, y ∈ 𝔐 ∩ 𝒜+, ‖x‖ < 1, ‖y‖ < 1, and let u = x(1 − x)−1, v = y(1 − y)−1, z = (u + v)(1 + u + v)−1. Then u, v, z ∈ 𝒜+, ‖z‖ < 1 and x = u(1 + u)−1, y = v(1 + v)−1. Since the function f1defined in Section 1.5 is operator monotone, we have x ≤ z, y ≤ z. Since 𝔐 is a facial subalgebrain 𝒜 ,𝔐 ∩ 𝒜+ is a face of 𝒜+. We have u ≤ ‖(1 − x)−1‖x ∈ 𝔐 ∩ 𝒜+, so that u ∈ 𝔐 ∩ 𝒜+ and,similarly, v ∈ 𝔐 ∩𝒜+ and z ∈ 𝔐 ∩𝒜+.

2.9. Proof of Theorem 2.6. We have to show that for every f1, f2 ∈ 𝔉𝜑 and every 𝜀 > 0 there existsf ∈ 𝔉𝜑 such that (1 − 𝜀)f1 ≤ f and (1 − 𝜀)f2 ≤ f.

Let f1, f2 ∈ 𝔉𝜑 and 𝜀 > 0. By Corollary 2.4, the set 𝒯 ′𝜑 of all the operators T′ ∈ 𝜋𝜑(ℳ)′ such

that there exists some 𝜆T ′ > 0 with the property

‖T ′a𝜑‖𝜑 ≤ 𝜆T ′‖a‖ (a ∈ 𝔑𝜑)is a left ideal of the von Neumann algebra 𝜋𝜑(ℳ)′; there exist T ′1,T

′2 ∈ 𝒯

′𝜑 such that 0 ≤ T ′j ≤ 1

and

fj(b∗a) = (T ′j a𝜑|T ′j b𝜑)𝜑 (a, b ∈ 𝔑𝜑, j = 1, 2). (1)By Proposition 2.7, it follows that (𝒯 ′𝜑 )

∗𝒯 ′𝜑 is a facial subalgebra of 𝜋𝜑(ℳ)′. Using Proposition 2.8,

we obtain an element X′ ∈ (𝒯 ′𝜑 )∗𝒯 ′𝜑 such that (1 − 𝜀)T

′∗j T

′j ≤ X′ ≤ 1, (j = 1, 2). Let T ′ = (X′)1∕2.

Using again Proposition 2.7, we see that

T ′ ∈ 𝒯 ′𝜑 , 0 ≤ T ′ ≤ 1 and (1 − 𝜀)T ′∗j T ′j ≤ T ′∗T ′ (j = 1, 2). (2)By Corollaries 2.4 and 2.5, there exists a normal positive form f ≤ 𝜑, that is, f ∈ 𝔉𝜑, such that

f (b∗a) = (T ′a𝜑|T ′b𝜑)𝜑 (a, b ∈ 𝔑𝜑). (3)From (1), (2), and (3), it follows that

(1 − 𝜀)fj(x) ≤ f (x) (x ∈ 𝔐𝜑, j = 1, 2). (4)Since 𝜑 is semifinite, 𝔐𝜑 is w-dense in ℳ. As f, f1, f2 are w-continuous, it follows that inequalities(4) remain valid for every x ∈ ℳ, that is, (1 − 𝜀)f1 ≤ f, (j = 1, 2).

14 Normal Weights

2.10 Corollary. Let 𝜑 be a normal semifinite weight on the W ∗-algebra ℳ. For each w-continuousfunction ℝ ∋ t ↦ x(t) ∈ ℳ+ such that ∫ ∞−∞ ‖x(t)‖dt < +∞ we have

𝜑(∫

+∞

−∞x(t)dt

)= ∫

∞

−∞𝜑(x(t))dt.

Proof. Since ℳ = (ℳ∗)∗ ([L], 1.10; A.16), the properties of the function t ↦ x(t) show that thereexists a unique element x = ∫ x(t)dt ∈ ℳ+ such that f (x) = ∫ f (x(t))dt for all f ∈ ℳ∗.

By Theorem 2.6, there exists an increasing net {fi}i∈I ⊂ℳ+∗ such that 𝜑 = supi fi. For each i ∈ I,we have fi(x) = ∫ fi(x(t))dt. Since fi(x(t)) ↑i 𝜑(x(t))(t ∈ ℝ), using the classical Beppo-Levi theoremwe obtain 𝜑(x) = supi fi(x) = supi ∫ fi(x(t))dt = ∫ supi fi(x(t))dt = ∫ 𝜑(x(t))dt.2.11. Using the theorems of Haagerup (1.3) and Combes (2.6), we can now extend, without anymodification, the statement and the proof of the standard respresentation theorem ([L], 10.14) forweights that are normal in the sense defined in Section 1.3:

Theorem. Let 𝜑 be an n.s.f. weight on the W ∗-algebra ℳ. Then 𝔑𝜑 ∩ 𝔑∗𝜑, endowed with the*-algebra structure inherited from ℳ and the scalar product of ℋ𝜑, is a left Hilbert algebra𝔄𝜑 ⊂ ℋ𝜑 such that 𝔄𝜑 = 𝔄′′𝜑, 𝜋𝜑(ℳ) = 𝔏(𝔄𝜑) and

𝜑(a) ={‖𝜉‖2 if there exists 𝜉 ∈ 𝔄𝜑 with 𝜋𝜑(a)1∕2 = L𝜉

+∞, otherwise. (a ∈ ℳ+).

Indeed, from Theorems 1.3 and 2.6, it follows that every normal semifinite weight is the supremumof an upward directed family of normal positive forms and it is exactly this definition of normalitywhich is used in the proof of ([L], Thm. 10.14).

Consequently, every n.s.f. weight on a W ∗-algebra is the natural weight associated with a leftHilbert algebra ([L], 10.16). Since these natural weights are sums of normal positive forms ([L],10.18), any n.s.f. weight has the same property. In Section 5.8, we shall give a simpler proof of thisresult.

2.12. We shall use the notation and results of ([L], Chapter 10) for left Hilbert algebras and theobjects associated with them.

Let 𝜑 be an n.s.f. weight on the W ∗-algebra ℳ and 𝔄𝜑 ⊂ ℋ𝜑 the associated left Hilbert algebra(2.11). In this section, we recall some of the notation and results just used, together with some newresults.

Since the weight 𝜑 is faithful, the left ideal 𝔑𝜑 ⊂ℳ will be also considered as a linear subspaceof ℋ𝜑 via the mapping 𝔑𝜑 ∋ x ↦ x𝜑 ∈ ℋ𝜑.

The closed antilinear operator S = S𝜑 in ℋ𝜑 is the closure of the preclosed antilinear operator

S0𝜑 ∶ 𝔄𝜑 ∋ x𝜑 ↦ (x𝜑)∗ = (x∗)𝜑 ∈ ℋ𝜑.

For each 𝜂 ∈ ℋ𝜑, one defines a linear operator R0𝜂 on ℋ𝜑 affiliated to 𝜋𝜑(ℳ)′, with domain

D(R0𝜂) = 𝔄𝜑 = 𝔑𝜑 ∩𝔑∗𝜑 and

R0𝜂x𝜑 = 𝜋𝜑(x)𝜂 (x ∈ 𝔑𝜑 ∩𝔑∗𝜑).

The Standard Representation 15

If 𝜂 ∈ D(S∗), then R0𝜂 is preclosed and its closure R𝜂 = R0𝜂 satisfies R∗𝜂 ⊃ RS∗𝜂

∗). The right Hilbertalgebra 𝔄′𝜑 ⊂ ℋ𝜑 is the set

𝔄′𝜑 = {𝜂 ∈ D(S∗);R𝜂 ∈ ℬ(ℋ𝜑)}

with the scalar product of ℋ𝜑 and with the operations

𝜂b = S∗𝜂, 𝜂1𝜂2 = R𝜂2𝜂1 (𝜂, 𝜂1, 𝜂2 ∈ 𝔄′𝜑).

We have

𝜋𝜑(ℳ)′ = ℜ(𝔄′𝜑) = {R𝜂; 𝜂 ∈ 𝔄′𝜑}so.

If R0𝜂 (𝜂 ∈ ℋ𝜑) is bounded, then

R𝜂x𝜑 = 𝜋𝜑(x)𝜂 (x ∈ 𝔑𝜑). (1)

Indeed, if {yi}i∈I ⊂ 𝔑𝜑 is a norm-bounded net with yis∗→ 1, then for every x ∈ 𝔑𝜑 we have

y∗i x ∈ 𝔑𝜑 ∩𝔑∗𝜑 and, by the definition of R

0𝜂 ,

R𝜂x𝜑 = limi 𝜋𝜑( y∗i )R𝜂x𝜑 = limi R𝜂𝜋𝜑( y

∗i )x𝜑

= limi

R0𝜂( y∗i x)𝜑 = limi 𝜋𝜑( y

∗i x)𝜂 = 𝜋𝜑(x)𝜂.

Similarly, for each 𝜉 ∈ ℋ𝜑 one defines a linear operator L0𝜉 on ℋ𝜑, affiliated to 𝜋𝜑(ℳ), withdomain D(L0𝜉) = 𝔄

′𝜑 and

L0𝜉𝜂 = R𝜂𝜉 (n ∈ 𝔄′𝜑).

If 𝜉 ∈ D(S), then L0𝜉 is preclosed and its closure L𝜉 = L0𝜉 satisfies the relation: L

∗𝜉 ⊃ LS𝜉

∗). ByTheorem 2.11, we have

𝔄𝜑 = 𝔄′′𝜑 = {𝜉 ∈ D(S);L𝜉 ∈ ℬ(ℋ𝜑)}

𝜋𝜑(ℳ) = 𝔏(𝔄𝜑) = {L𝜉 ; 𝜉 ∈ 𝔄𝜑}so.

For 𝜉 ∈ ℋ𝜑, we have

L0𝜉 is bounded ⇔ there exists x ∈ 𝔑𝜑 with 𝜉 = x𝜑;

in this case L𝜉 = 𝜋𝜑(x).(2)

*The equality is not always true (cf. [L], C.10.1 and an example by M. Pimsner).

16 Normal Weights

Indeed, let x ∈ 𝔑𝜑. By (1), for every 𝜂 ∈ 𝔄′𝜑 we have

L0x𝜑𝜂 = R𝜂x𝜑 = 𝜋𝜑(x)𝜂.

Conversely, assume that L0𝜉 is bounded. Then L𝜉 ∈ 𝜋𝜑(ℳ), hence there exists x ∈ ℳ with 𝜋𝜑(x) =L𝜉 . Let x = v|x| be the polar decomposition of x in ℳ. It is easy to check that

L𝜋𝜑(v∗)𝜉 = 𝜋𝜑(v∗)L𝜉 = 𝜋𝜑(v∗x) = 𝜋𝜑(|x|) = |L𝜉| ≥ 0.

Using ([L], 10.8) it follows that 𝜋𝜑(v∗)𝜉 ∈ 𝔄′′𝜑 and then, using Theorem 2.11 we get 𝜑(x∗x) =‖𝜋𝜑(v∗)𝜉‖2𝜑 = ‖𝜉‖2𝜑 < +∞, hence x ∈ 𝔑𝜑. Since Lx𝜑 = 𝜋𝜑(x) = L𝜉 , we conclude that 𝜉 = x𝜑.

Also, note that x ∈ 𝔑𝜑, x𝜑 ∈ D(S𝜑) ⇒ x𝜑 ∈ 𝔄𝜑, that is,

𝔑𝜑 ∩ D(S𝜑) = 𝔑𝜑 ∩𝔑∗𝜑. (3)

Indeed, the inclusion 𝔑𝜑 ∩ 𝔑∗𝜑 = 𝔄𝜑 ⊂ 𝔑𝜑 ∩ D(S𝜑) is obvious. Conversely, if x ∈ 𝔑𝜑 andx𝜑 ∈ D(S𝜑), it follows from (2) that x𝜑 = 𝔄′′𝜑 = 𝔄𝜑 hence x ∈ 𝔑𝜑 ∩𝔑

∗𝜑.

From the polar decomposition S𝜑 = J𝜑Δ1∕2𝜑 of S𝜑, one obtains the modular operator Δ𝜑 = S∗𝜑S𝜑

and the canonical conjugation J𝜑 = J∗𝜑 = J−1𝜑 , associated with 𝔄𝜑 ⊂ ℋ𝜑. Since J𝜑 is antilinear and

J𝜑Δ𝜑 J𝜑 = Δ−1𝜑 , it follows that

J𝜑f (Δ𝜑)J𝜑 = f (Δ−1𝜑 )

for every Borel function f. In particular, J𝜑Δit𝜑 = Δit𝜑 J𝜑(t ∈ ℝ), and S𝜑 = J𝜑Δ

1∕2𝜑 = Δ

−1∕2𝜑 J𝜑, S∗𝜑 =

J𝜑Δ−1∕2𝜑 = Δ

1∕2𝜑 J𝜑.

By Tomita’s fundamental theorem ([L], 10.12), we have Δit𝜑𝔄𝜑 = 𝔄𝜑,Δit𝜑𝔄

′𝜑 = 𝔄

′𝜑(t ∈ ℝ),

J𝜑𝔄𝜑 = 𝔄′𝜑 and

LΔit𝜑𝜉 = Δit𝜑L𝜉Δ

−it𝜑 ,RJ𝜑𝜉 = J𝜑L𝜉J𝜑 (𝜉 ∈ 𝔄𝜑),

RΔit𝜑𝜂 = Δit𝜑R𝜂Δ

−it𝜑 ,LJ𝜑𝜂 = J𝜑R𝜂J𝜑 (𝜂 ∈ 𝔄

′𝜑).

(4)

Using the definition of the operators L0,R0 the validity of (4) can be extended to arbitrary vectors𝜉, 𝜂 ∈ ℋ𝜑, replacing the operators L,R by L0,R0, respectively. If the operators L0𝜉 ,R

0𝜂 are preclosed,

these identities can be extended to their closures, that is, they remain valid in the earlier form.From Tomita’s fundamental theorem it follows that the mapping j𝜑 ∶ x ↦ J𝜑𝜋𝜑(x∗)J𝜑 defines a

*-antiisomorphism j𝜑 of ℳ onto 𝜋𝜑(ℳ)′, which coincides with 𝜋𝜑 on the center of ℳ ([L], 10.13).We note that

J𝜑𝜋𝜑(x)J𝜑y𝜑 = 𝜋𝜑( y)J𝜑x𝜑 (x, y ∈ 𝔑𝜑). (5)

The Standard Representation 17

Indeed, using the extension of (4) as well as (1) and (2); we obtain J𝜑𝜋𝜑(x)J𝜑y𝜑 = J𝜑Lx𝜑J𝜑y𝜑 =RJ𝜑x𝜑y𝜑 = 𝜋𝜑( y)J𝜑x𝜑.

Using the isometric character of J𝜑 it is easy to check that the n.s.f. weight 𝜑′ on 𝜋𝜑(ℳ)′ definedby

𝜑′(j𝜑(x)) = 𝜑(x) (x ∈ ℳ+) (6)

is just the natural weight on 𝜋𝜑(ℳ)′ = ℜ(𝔄′𝜑) associated with the right Hilbert algebra 𝔄′𝜑 ⊂ ℋ𝜑

([L], p. 331), that is,

𝜑′(R∗𝜁R𝜂) = (𝜂|𝜁 )𝜑 (7)for every 𝜂, 𝜁 ∈ ℋ𝜑, such that R𝜂 ,R𝜁 are bounded, in particular for every 𝜂, 𝜁 ∈ 𝔄′𝜑. We note thatthe standard representation of 𝜋𝜑(ℳ)′ associated with the n.s.f. weight 𝜑′ is unitarily equivalent tothe identity representation 𝜋𝜑(ℳ)′ ⊂ℬ(ℋ𝜑) and we have

S𝜑′ = S∗𝜑, Δ𝜑′ = Δ−1𝜑 , J𝜑′ = J𝜑. (8)

On the other hand, it follows from Tomita’s fundamental theorem that the relation

𝜋𝜑(𝜎𝜑t (x)) = Δ

it𝜑𝜋𝜑Δ

−it𝜑 (x ∈ ℳ, t ∈ ℝ)

defines an s∗-continuous group {𝜎𝜑t }t∈ℝ of *-automorphisms of ℳ which act identically on thecenter of ℳ ([L], 10.13). With an argument similar to that used in proving (5), we obtain

R𝜂(𝜎𝜑t (x))𝜑 = 𝜋𝜑(𝜎

𝜑t (x))𝜂 = R𝜂Δ

it𝜑x𝜑

for every x ∈ 𝔑𝜑 and every 𝜂 ∈ 𝔄′𝜑. Letting R𝜂so→ 1, we conclude

(𝜎𝜑t (x))𝜑 = Δit𝜑x𝜑 (x ∈ 𝔑𝜑, t ∈ ℝ). (9)

Since Δ𝜑′ = Δ−1𝜑 for the weight 𝜑′ on 𝜋𝜑(ℳ)′ we have

𝜎𝜑′

t (j𝜑(x)) = j𝜑(𝜎𝜑−t(x)) (x ∈ ℳ, t ∈ ℝ). (10)

The weight 𝜑 is invariant with respect to the modular automorphism group {𝜎𝜑t }t∈ℝ, that is,𝜑 ◦ 𝜎𝜑t = 𝜑(t ∈ ℝ), and satisfies the KMS condition with respect to {𝜎

𝜑t }t∈ℝ in any two elements

x, y ∈ 𝔑𝜑 ∩𝔑∗𝜑, that is, there exists a function f = fx,y defined, continuous and bounded on the strip{𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 1}, analytic in the interior of this strip and such that

f (it) = 𝜑(x𝜎𝜑t ( y)), f (1 + it) = 𝜑(𝜎𝜑t ( y)x) (t ∈ ℝ).

18 Normal Weights

These properties characterize the modular automorphism group associated with 𝜑. More exactly, in([L], 10.17) one actually proves the following uniqueness statement:

if {𝜎t}t∈ℝ is a group of ∗ -automorphisms of ℳ with the properties ∶(a) 𝜑 ◦ 𝜎t = 𝜑 for all t ∈ ℝ;(b) there exists a ∗ -subalgebra 𝒳 ⊂ 𝔄𝜑 such that S𝜑|𝒳 = S𝜑 (11)and 𝜑 satisfies the KMS condition with respect to {𝜎t}t∈ℝin any two elements of 𝒳 ;then 𝜎t = 𝜎

𝜑t for t ∈ ℝ.

If the weight 𝜑 is finite, that is, if 𝜑 ∈ ℳ+∗ , then

xis→ x in ℳ ⇒ ‖(xi)𝜑 − x𝜑‖𝜑 → 0. (12)

Thus, in this case, we can replace 𝒳 in condition (b) of (11) by any w-dense *-subalgebra of ℳ.If 𝜑 is not necessarily finite, we still have the following convergence result

𝔑𝜑 ∋ xiw→ x ∈ ℳ, (xt)𝜑

weakly→ 𝜉 ∈ ℋ𝜑 ⇒ x ∈ 𝔑𝜑, 𝜉 = x𝜑. (13)

Indeed, for 𝜂 ∈ 𝔄′𝜑 and 𝜁 ∈ ℋ𝜑, we have (L0𝜉𝜂|𝜁 )𝜑 = (R𝜂𝜉|𝜁 )𝜑 = limi(R𝜂(xi)𝜑|𝜁 )𝜑 =

limi(𝜋𝜑(xi)𝜂|𝜁 ) = (𝜋𝜑(x)𝜂|𝜁 )𝜑 and using (2) we conclude x ∈ 𝔑𝜑 and 𝜉 = x𝜑.Using (13) and the w-compactness of the closed unit ball of ℳ, we obtain also the following

result:

if {xi} ⊂ 𝔑𝜑 is a norm-bounded net and if {(xi)𝜑} is weaklyconvergent to some 𝜉 ∈ ℋ𝜑, then there exists x ∈ 𝔑𝜑 such that (14)

xis→ x and x𝜑 = 𝜉.

An important technical tool in the standard representation associated with 𝜑 is the Tomita algebra([L], 10.20, 10.21)

𝔗𝜑 ⊂ 𝔄𝜑 ∩𝔄′𝜑 ∩⋂𝛼∈ℂ

D(Δ𝛼𝜑).

Recall that 𝔗𝜑 is a left Hilbert subalgebra of 𝔄𝜑, equivalent to 𝔄𝜑, and J𝜑𝔗𝜑 = 𝔗𝜑,Δ𝛼𝜑𝔗𝜑 =𝔗𝜑 Δ𝛼𝜑|𝔗𝜑 = Δ𝛼𝜑, (𝛼 ∈ ℂ). The identities

Δ𝛼𝜑(𝜉𝜂) = (Δ𝛼𝜑𝜉)(Δ

𝛼𝜑𝜂), J𝜑(𝜉𝜂) = ( J𝜑𝜂)( J𝜑𝜉) (𝜉, 𝜂 ∈ 𝔗𝜑)

are straightforward consequences of (9) and (5).The arguments in ([L], 10.21) prove that for 𝜉 ∈ ℋ𝜑 we have

𝜉 ∈ 𝔗𝜑 ⇔ 𝜉 ∈⋂𝛼∈ℂ

D(Δ𝛼𝜑) and Δn𝜑𝜉 ∈ 𝔑𝜑 ⊂ ℋ𝜑 for all n ∈ ℤ. (15)

The Standard Representation 19

Using this criterion and arguing as in ([L], p. 302), it is easy to obtain the following approximationresult:

for every x ∈ 𝔑𝜑 there exists a sequence {xn} ⊂ 𝔗𝜑 such that‖xn‖ ≤ ‖x‖, xn s→ x and ‖(xn)𝜑 − x𝜑‖𝜑 → 0; (16)in fact we can take

xn =√

n∕𝜋 ∫+∞

−∞e−nt

2𝜎𝜑t (x)dt.

If x ∈ 𝔑𝜑∩𝔑∗𝜑, then the approximation is stronger, namely we have also ([L], Cor. 2/10.21) x∗n

s→ x∗

and ‖(x∗n)𝜑 − (x∗)𝜑‖𝜑 → 0.In the next sections, we define the translation of a weight by certain elements (2.13), consider

analytic elements with respect to a weight (2.14–2.16), give some useful reformulations of theKMS condition (2.17–2.10), study the centralizer of a weight (2.21, 2.22), and use the standardrepresentation in order to introduce a natural topology on the group of all *-automorphisms(2.23–2.26).

2.13 Proposition. Let 𝜑 be an n.s.f. weight on the W ∗-algebra ℳ. If a ∈ 𝔗2𝜑, then the linear form

x ↦ 𝜑(xa) (resp. x ↦ 𝜑(ax))

is defined and w-continuous on the set

{x ∈ ℳ; x𝔗𝜑 ⊂ 𝔄𝜑} (resp. {x ∈ ℳ;𝔗𝜑x ⊂ 𝔄𝜑})

which contains 𝔗𝜑 and can therefore be extended to a w-continuous linear form on ℳ, denoted by𝜑(⋅a) (resp. 𝜑(a⋅)).

The sets {𝜑(⋅a); a ∈ 𝔗2𝜑} and {𝜑(a⋅); a ∈ 𝔗2𝜑} are norm-dense linear subspaces of ℳ.

Proof. Let a = bc∗ with b, c ∈ 𝔗𝜑 and x ∈ ℳ with x𝔗𝜑 ⊂ 𝔄𝜑. Then xb ∈ 𝔄𝜑 ⊂ 𝔑∗𝜑, c∗ ∈ 𝔑𝜑,

hence xa ∈ 𝔐𝜑 and we have

𝜑(xa) = (𝜋𝜑(x)b𝜑|Δ𝜑c𝜑)𝜑. (1)Indeed,

𝜑(xbc∗) = ((c∗)𝜑|(b∗x∗)𝜑)𝜑 = (S𝜑c𝜑|S𝜑(xb)𝜑)𝜑= ( J𝜑Δ1∕2𝜑 c𝜑| J𝜑Δ1∕2𝜑 (xb)𝜑)𝜑 = (Δ1∕2𝜑 (xb)𝜑|Δ1∕2𝜑 c𝜑)𝜑= ((xb)𝜑|Δ𝜑c𝜑)𝜑 = (𝜋𝜑(x)b𝜑|Δ𝜑c𝜑)𝜑.

Similarly,

𝜑(ax) = (𝜋𝜑(x)Δ𝜑b𝜑|c𝜑)𝜑. (2)

20 Normal Weights

This proves the first part of the proposition. Moreover, (1) and (2) give the explicit form of theextensions 𝜑(⋅a) and 𝜑(a⋅).

Assume that the linear subspace {𝜑(⋅a); a ∈ 𝔗2𝜑} is not norm-dense in ℳ∗. Then, by the Hahn–Banach theorem, there exists x ∈ ℳ, x ≠ 0, such that 𝜑(xa) = 0 for all a ∈ 𝔗2𝜑. Using (1) we inferthat (𝜋𝜑(x)𝜉|𝜂)𝜑 = 0 for all 𝜉, 𝜂 ∈ 𝔗𝜑, hence 𝜋𝜑(x) = 0, contradicting x ≠ 0.

Note that, conversely,

if 0 ≤ a ∈ 𝔑𝜑 and 𝔑∗𝜑 ∋ x ↦ 𝜑(xa) is w-continuous, then a ∈ 𝔐𝜑. (3)Indeed, there exists a sequence {en} of spectral projections of a such that aen ≥ n−1en and en ↑ s(a).We have en ≤ n2aena ∈ 𝔐𝜑, so that 𝔑∗𝜑 ∋ en ↑ 1 and hence, by assumption, supn 𝜑(ena) < +∞. Onthe other hand, we have ena = a1∕2ena1∕2 ↑ a, hence 𝜑(a) = supn 𝜑(ena) < +∞, that is a a ∈ 𝔐𝜑.

2.14. Let 𝜑 be an n.s.f. weight on the W ∗-algebra ℳ and a ∈ ℳ.The element a is called analytic in the vertical strip {𝛼 ∈ ℂ; −𝜀1 ≤ Re 𝛼 ≤ 𝜀2}, (0 ≤ 𝜀1, 𝜀2 <

+∞), if there exists an ℳ-valued function F, defined and w-continuous on this strip and analytic inthe interior of the strip, such that

F(it) = 𝜎𝜑t (a) (t ∈ ℝ).

In this case, for each 𝛼 ∈ ℂ,−𝜀1 ≤ Re 𝛼 ≤ 𝜀2, we let𝜎𝜑−i𝛼(a) = F(𝛼).

For 𝛼 ∈ ℂ, we shall write a ∈ D(𝜎𝜑𝛼 ) if the element a is analytic in some vertical strip containingi𝛼. The following statements are easily verified:

a ∈ D(𝜎𝜑𝛼 ) ⇒ a∗ ∈ D(𝜎𝜑�̄� ), 𝜎

𝜑�̄� (a

∗) = 𝜎𝜑𝛼 (a)∗; (1)

a, b ∈ D(𝜎𝜑𝛼 ) ⇒ ab ∈ D(𝜎𝜑𝛼 ), 𝜎

𝜑𝛼 (ab) = 𝜎

𝜑𝛼 (a)𝜎

𝜑𝛼 (b). (2)

Using ([L], 9.21), from the relation 𝜋𝜑(𝜎𝜑t (a))Δit𝜑𝜉 = Δ

it𝜑𝜋𝜑(a)𝜉 we infer that

a ∈ D(𝜎𝜑−i𝛼), 𝜉 ∈ D(Δ𝛼𝜑) ⇒ 𝜋𝜑(a)𝜉 ∈ D(Δ

𝛼𝜑),Δ

𝛼𝜑𝜋𝜑(a)𝜉 = 𝜋𝜑(𝜎

𝜑−i𝛼(a))Δ

𝛼𝜑𝜉 (3)

or, using ([L], 9.24) and replacing 𝛼 by −i𝛼,

a ∈ D(𝜎𝜑𝛼 ) ⇒ 𝜋𝜑(𝜎𝜑𝛼 (a)) = Δi𝛼𝜑 𝜋𝜑(a)Δ−i𝛼𝜑 |D(Δ−i𝛼𝜑 ). (4)

It follows that if the element a is analytic in the strip {𝛼 ∈ ℂ; 0 ≤ Re 𝛼 ≤ 𝜀} then the function𝛼 ↦ 𝜎𝜑−i𝛼(a) is norm-continuous and norm-bounded on this strip.

Also, we have

a ∈ D(𝜎𝜑𝛽 ), 𝜎𝜑𝛽 (a) ∈ D(𝜎

𝜑𝛼 ) ⇒ a ∈ D(𝜎

𝜑𝛼+𝛽), 𝜎

𝜑𝛼+𝛽(a) = 𝜎

𝜑𝛼 (𝜎

𝜑𝛽 (a)); (5)

a ∈ D(𝜎𝜑𝛼 ) ⇒ 𝜎𝜑𝛼 (a) ∈ D(𝜎

𝜑−𝛼), 𝜎

𝜑−𝛼(𝜎

𝜑𝛼 (a)) = a. (6)

The Standard Representation 21

Proposition. Let 𝜑 be an n.s.f. weight on the W ∗-algebra ℳ, a ∈ ℳ and 𝜆 ∈ (0,+∞). Thefollowing statements are equivalent:

(i) 𝜑(ax∗xa∗) ≤ 𝜆2𝜑(x∗x) for every x ∈ ℳ;(ii) x ∈ 𝔑𝜑 ⇒ xa∗ ∈ 𝔑𝜑 and ‖(xa∗)𝜑‖𝜑 ≤ 𝜆‖x𝜑‖𝜑;

(iii) a ∈ D(𝜎𝜑−i∕2) and ‖𝜎𝜑−i∕2(a)‖ ≤ 𝜆.If a ∈ D(𝜎𝜑−i∕2), then

(xa∗)𝜑 = J𝜑𝜋𝜑(𝜎𝜑−i∕2(a))J𝜑x𝜑 (x ∈ 𝔑𝜑) (7)

and if moreover 𝜎𝜑t (aa∗) = aa∗(t ∈ ℝ), then

𝜑(𝜎𝜑−i∕2(a)∗x∗x𝜎𝜑−i∕2(a)) = 𝜑(aa

∗x∗x) (x ∈ 𝔑𝜑). (8)

Proof. It is clear that (i) ⇔ (ii).(ii) ⇒ (iii). From (ii) it follows that there exists T ∈ ℬ(ℋ𝜑), ‖T‖ ≤ 𝜆, such that Tx𝜑 = (xa∗)𝜑

(x ∈ 𝔑𝜑). For x ∈ 𝔄𝜑, we have

Tx𝜑 = (xa∗)𝜑 = S𝜑(ax∗)𝜑 = S𝜑𝜋𝜑(a)S𝜑x𝜑 = J𝜑Δ1∕2𝜑 𝜋𝜑(a)Δ−1∕2𝜑 J𝜑x𝜑

so the operator Δ1∕2𝜑 𝜋𝜑(a)Δ−1∕2𝜑 |𝔄′𝜑 is bounded with norm ≤ 𝜆. Since Δ−1∕2𝜑 |𝔄′𝜑 = Δ−1∕2𝜑 , using

([L], 9.24) we infer that a ∈ D(𝜎𝜑−i∕2) and ‖𝜎𝜑−i∕2(a)‖ ≤ 𝜆.(iii) ⇒ (i). Let x ∈ 𝔄𝜑 = 𝔑𝜑 ∩𝔑∗𝜑. Then (x

∗)𝜑 ∈ D(Δ1∕2𝜑 ) and, using (3) with 𝛼 = 1∕2, we get

Δ1∕2𝜑 𝜋𝜑(a)(x∗)𝜑 = 𝜋𝜑(𝜎

𝜑−i∕2(a))Δ

1∕2𝜑 (x

∗)𝜑.

S𝜑(ax∗)𝜑 = J𝜑𝜋𝜑(𝜎𝜑−i∕2(a))J𝜑S𝜑(x

∗)𝜑,

hence

(xa∗)𝜑 = J𝜑𝜋𝜑(𝜎𝜑−i∕2(a))J𝜑x𝜑 (x ∈ 𝔄𝜑). (9)

It follows that

𝜑(aza∗) ≤ ‖𝜎𝜑−i∕2(a)‖𝜑(z) (z ∈ ℳ+),and this proves (i). Indeed, if 𝜑(z) = +∞ the inequality is obvious and if 𝜑(z) < +∞ then x = z1∕2 ∈𝔄𝜑 and we use (9).

Consider now x ∈ 𝔑𝜑. There is a sequence {xn} ⊂ 𝔄𝜑 such that ‖(xn)𝜑 − x𝜑‖𝜑 → 0. From(ii) it follows that xa∗ ∈ 𝔑𝜑 and ‖(xna∗)𝜑 − (xa∗)𝜑‖𝜑 → 0. Thus, (7) follows from (9) in thelimit.

Finally, we prove (8). Let b = 𝜎𝜑−i∕2(a). Using (1) and (6) it follows that b∗ ∈ D(𝜎𝜑−i∕2) and 𝜎

𝜑−i∕2

(b∗) = a∗. On the other hand, since 𝜎𝜑t (aa∗) = aa∗(t ∈ ℝ), it is obvious that 𝜎𝜑−i∕2(aa

∗) = aa∗. Using

22 Normal Weights

(7) we obtain

𝜑(b∗x∗xb) = ‖(xb)𝜑‖2𝜑 = ‖ J𝜑𝜋𝜑(a∗)J𝜑x𝜑‖2𝜑= (x𝜑| J𝜑𝜋𝜑(aa∗)J𝜑x𝜑)𝜑 = (x𝜑|(xaa∗)𝜑)𝜑 = 𝜑(aa∗x∗x).

2.15. An element a ∈ ℳ such that a ∈ D(𝜎𝛼𝜑) for all 𝛼 ∈ ℂ is called an entire analytic element.We put

ℳ𝜑∞ = {a ∈ ℳ; a is an entire analytic element}.

Using ([L], 10.20, 9.24) we see that

a ∈ 𝔗𝜑 ⇒ a ∈ ℳ𝜑∞ and 𝜎𝜑𝛼 (a) ∈ 𝔗𝜑 for all 𝛼 ∈ ℂ. (1)

From Section 2.14, it follows that ℳ𝜑∞ is an s∗-dense *-subalgebra of ℳ. Moreover, the sets𝔑𝜑,𝔄𝜑,𝔐𝜑 are all invariant under left or right multiplications by elements of ℳ

𝜑∞. Note also that

a ∈ 𝔗𝜑 ⇒ Δi𝛼𝜑 a𝜑 = (𝜎𝜑𝛼 (a))𝜑 and LΔi𝛼𝜑 a𝜑 = 𝜋𝜑(𝜎

𝜑𝛼 (a)). (2)

Indeed, for 𝜉 ∈ 𝔗𝜑 we have

LΔi𝛼𝜑 a𝜑𝜉 = R𝜉Δi𝛼𝜑 a𝜑 = Δ

i𝛼𝜑RΔ−i𝛼𝜑 𝜉a𝜑 = Δ

i𝛼𝜑 La𝜑Δ

−i𝛼𝜑 𝜉

= Δi𝛼𝜑 𝜋𝜑(a)Δ−i𝛼𝜑 𝜉 = 𝜋𝜑(𝜎

𝜑𝛼 (a))𝜉 = L(𝜎𝜑𝛼 (a))𝜑𝜉.

In particular,

RΔi�̄�𝜑 J𝜑a𝜑 = RJ𝜑Δi𝛼𝜑 a𝜑 = J𝜑LΔi𝛼𝜑 a𝜑J𝜑 = J𝜑𝜋𝜑(𝜎𝜑𝛼 (a))J𝜑. (3)

2.16 Proposition. Let 𝜑 be an n.s.f. weight on the W ∗-algebra ℳ, {xk}k∈K ⊂ 𝔄𝜑 a net such that

xks∗→ 1, supk ‖xk‖ ≤ 1 and

ak =√

1∕𝜋 ∫+∞

−∞e−t

2𝜎𝜑t (xk)dt (k ∈ K). (1)

Then {ak}k∈K ⊂ 𝔗𝜑 ⊂ℳ𝜑∞ and for every 𝛼 ∈ ℂ, k ∈ K, we have

𝜎𝜑𝛼 (ak)s∗→ 1, (2)‖𝜎𝜑𝛼 (ak)‖ ≤ exp((Im 𝛼)2). (3)

Proof. Arguing as in ([L], p. 347) we see that ak ∈ 𝔗𝜑 and

𝜎𝜑𝛼 (ak) =√

1∕𝜋 ∫+∞

−∞e−(t−𝛼)

2𝜎𝜑t (xk)dt.

The Standard Representation 23

Let r = Re 𝛼, s = Im 𝛼. Then (t − 𝛼)2 = −s2 + (t − r)2 − 2is(t − r), so

‖𝜎𝜑𝛼 (ak)‖ ≤ es2√1∕𝜋 ∫ +∞−∞ e−(t−r)2‖𝜎t(xk)‖dt ≤ es2√1∕𝜋 ∫+∞

−∞e−t

2dt = es2 .

Let ℳ ⊂ℬ(ℋ ) be realized as a von Neumann algebra. For 𝜉 ∈ ℋ , we have

‖𝜉 − 𝜎𝜑𝛼 (ak)𝜉‖ ≤ es2 ∫ +∞−∞ e−(t−r)2‖𝜎𝜑t (1 − xk)𝜉‖dt= es2 ∫

+∞

−∞e−t

2‖𝜎𝜑t (1 − xk)𝜉‖dt → 0,since limk ‖𝜎𝜑t (1 − xk)𝜉‖ = 0 (t ∈ ℝ), using the Lebesgue dominated convergence theorem.Consequently, 𝜎𝜑𝛼 (ak)

so→ 1 and, similarly, 𝜎𝜑𝛼 (ak)∗

so→ 1.

2.17 Proposition. Let 𝜑 be an n.s.f. weight on the W ∗-algebra ℳ, x, y ∈ ℳ and 𝛼 ∈ ℂ. If x ∈𝔑∗𝜑 ∩ D(𝜎

𝜑𝛼−i), 𝜎

𝜑𝛼−i(x) ∈ 𝔑𝜑 and y ∈ 𝔑𝜑 ∩ D(𝜎

𝜑𝛼 ), 𝜎

𝜑𝛼 ( y) ∈ 𝔑∗𝜑, then

𝜑(xy) = 𝜑(𝜎𝜑𝛼 ( y)𝜎𝜑𝛼−i(x)). (1)

Proof. By Proposition 2.16, there exists a net {ak} ⊂ 𝔗𝜑 such that 𝜎𝜑𝛽 (ak)

s∗→ 1 for all 𝛽 ∈ ℂ.

Using Properties 2.14.(1), 2.14.(4), 2.14.(6) and 2.15.(3), we obtain

𝜋𝜑(ak)y𝜑 = (aky)𝜑 = S𝜑( y∗a∗k )𝜑 = S𝜑𝜋𝜑( y∗)S𝜑(ak)𝜑

= S𝜑𝜋𝜑(𝜎𝜑−�̄�(𝜎

𝜑�̄� ( y

∗)))S𝜑(ak)𝜑 = S𝜑Δ−i�̄�𝜑 𝜋𝜑(𝜎𝜑�̄� ( y

∗))Δi�̄�𝜑 S𝜑(ak)𝜑= J𝜑Δ−i�̄�+(1∕2)𝜑 𝜋𝜑(𝜎

𝜑𝛼 ( y)

∗)Δi�̄�−(1∕2)𝜑 J𝜑(ak)𝜑= J𝜑Δ−i�̄�+(1∕2)𝜑 RΔi�̄�−(1∕2)𝜑 J𝜑(ak)𝜑(𝜎

𝜑𝛼 ( y)

∗)𝜑= J𝜑Δ−i�̄�+(1∕2)𝜑 J𝜑𝜋𝜑(𝜎

𝜑𝛼−(i∕2)(ak))J𝜑(𝜎

𝜑𝛼 ( y)

∗)𝜑

and taking the limit over k it follows that

y𝜑 = J𝜑Δ−i�̄�+(1∕2)𝜑 (𝜎𝜑𝛼 ( y)

∗)𝜑.

Similarly, we obtain

(x∗)𝜑 = J𝜑Δ−i𝛼−(1∕2)𝜑 (𝜎𝜑𝛼−i(x))𝜑.

Consequently,

𝜑(xy) = ( y𝜑|(x∗)𝜑)𝜑= ( J𝜑Δ−i�̄�+(1∕2)𝜑 (𝜎

𝜑𝛼 ( y)

∗)𝜑| J𝜑Δ−i𝛼−(1∕2)𝜑 (𝜎𝜑𝛼−i(x))𝜑)𝜑= (𝜎𝜑𝛼−i(x)𝜑|(𝜎𝜑𝛼 ( y)∗)𝜑)𝜑 = 𝜑(𝜎𝜑𝛼 ( y)𝜎𝜑𝛼−i(x)).

24 Normal Weights

In particular, for 𝛼 = 0, 𝛼 = i, and 𝛼 = i∕2, we have

𝜑(xy) = 𝜑( y𝜎𝜑−i(x)) = 𝜑(𝜎𝜑i ( y)x) = 𝜑(𝜎

𝜑1∕2( y)𝜎

𝜑−1∕2(x)), (2)

whenever x, y ∈ 𝔗𝜑. These identities replace for weights the relation 𝜑(xy) = 𝜑( yx), which is validonly for traces.

2.18. Another similar result, which ( formally) follows from 2.17.(1), is contained in the nextstatement:

a ∈ ℳ𝜑∞, z ∈ 𝔐𝜑 ⇒ 𝜑(z𝜎𝜑𝛼 (a)) = 𝜑(𝜎

𝜑𝛼+i(a)z) for all 𝛼 ∈ ℂ. (1)

We give a direct proof here. Since z ∈ 𝔐𝜑 = 𝔑∗𝜑𝔑𝜑, we may assume z = y∗x with x, y ∈ 𝔑𝜑.

Using Proposition 2.14, we get

𝜑( y∗x𝜎𝜑𝛼 (a)) = ((x𝜎𝜑𝛼 (a))𝜑|y𝜑)𝜑 = ( J𝜑𝜋𝜑(𝜎𝜑−i∕2(𝜎𝜑𝛼 (a)∗)J𝜑x𝜑|y𝜑)𝜑

= ( J𝜑𝜋𝜑(𝜎𝜑𝛼+(i∕2)(a))

∗J𝜑x𝜑|y𝜑)𝜑 = (x𝜑|J𝜑𝜋𝜑(𝜎𝜑𝛼+(i∕2)(a))J𝜑y𝜑)𝜑= (x𝜑|J𝜑𝜋𝜑(𝜎𝜑−i∕2(𝜎𝜑𝛼+i(a)))J𝜑y𝜑)𝜑 = (( y𝜎𝜑𝛼+i(a)∗)𝜑|x𝜑)𝜑= 𝜑(𝜎𝜑𝛼+i(a)y

∗x).

As 𝛼 ↦ (x𝜑| J𝜑𝜋𝜑(𝜎𝜑𝛼+(i∕2)(a))J𝜑y𝜑)𝜑 is an entire analytic function, bounded on horizontal strips,it follows that the functions

𝛼 ↦ 𝜑(z𝜎𝜑𝛼 (a)) and 𝛼 ↦ 𝜑(𝜎𝜑𝛼 (a)z)

are entire analytic and bounded on horizontal strips, for all a ∈ ℳ𝜑∞ and all z ∈ 𝔐𝜑.On the other hand, if a ∈ ℳ𝜑∞, and z ∈ 𝔐𝜑 ∩ℳ+, then for all 𝛼 ∈ ℂ we have

𝜑(𝜎𝜑𝛼 (a)z𝜎𝜑𝛼 (a)

∗) ≤ ‖𝜎𝜑𝛼−(i∕2)(a)‖2𝜑(z),𝜑(𝜎𝜑𝛼 (a)

∗z𝜎𝜑𝛼 (a)) ≤ ‖𝜎𝜑𝛼+(i∕2)(a)‖2𝜑(z). (2)Indeed, z = x∗x with x = z1∕2 ∈ 𝔑𝜑 and using Proposition 2.14, we get

𝜑(𝜎𝜑𝛼 (a)z𝜎𝜑𝛼 (a)

∗) = ‖(x𝜎𝜑𝛼 (a)∗)𝜑‖2𝜑≤ ‖ J𝜑𝜋𝜑(𝜎𝜑−i∕2(𝜎𝜑𝛼 (a)))J𝜑‖2‖x𝜑‖2𝜑 = ‖𝜎𝜑𝛼−(i∕2)(a)‖2𝜑(z)

and the other inequality is verified in a similar way.We note that the right-hand side of (2) depends just on |Im 𝛼|, as the 𝜎𝜑t (t ∈ ℝ) being

*-automorphisms, are isometric.Properties (1) and (2) characterize the entire analytic functions of the type 𝛼 ↦ 𝜎𝜑𝛼 (a) with

a ∈ ℳ𝜑∞ as we shall see in Theorem 2.19.The identity (1) is meaningful also for certain other a and z. Indeed, since 𝔐𝜑 is contained and

w-dense in the set {z ∈ ℳ; z𝔗𝜑 ⊂ 𝔄𝜑,𝔗𝜑z ⊂ 𝔄𝜑}, using Proposition 2.13, we infer from (1),

The Standard Representation 25

taking the limit, the following statement:

if a ∈ 𝔗2𝜑 and z ∈ ℳ; z𝔗𝜑 ⊂ 𝔄𝜑,𝔗𝜑z ⊂ 𝔄𝜑, then

𝜑(z𝜎𝜑𝛼 (a)) = 𝜑(𝜎𝜑𝛼+i(a)z) for all 𝛼 ∈ ℂ.

(3)

Under the same conditions as in statement (3), the functions

𝛼 ↦ 𝜑(z𝜎𝜑𝛼 (a)) and 𝛼 ↦ 𝜑(𝜎𝜑𝛼 (a)z)

are entire analytic and bounded on horizontal strips. Indeed, if a = bc∗ with b, c ∈ 𝔗𝜑, then, using2.13.(1) and 2.13.(2), we get

𝜑(z𝜎𝜑𝛼 (a)) = 𝜑(z𝜎𝜑𝛼 (b)𝜎

𝜑�̄� (c)

∗) = (𝜋𝜑(z)Δi𝛼𝜑 b𝜑|Δi�̄�+1c𝜑)𝜑,𝜑(𝜎𝜑𝛼 (a)z) = 𝜑(𝜎

𝜑𝛼 (b)𝜎

𝜑�̄� (c)

∗z) = (𝜋𝜑(z)Δi𝛼+1b𝜑|Δi�̄�c𝜑)𝜑.2.19 Theorem (A. Connes). Let𝜑 be an n.s.f. weight on the W ∗-algebra ℳ and consider a functionF ∶ ℂ → ℳ such that

(a) for every z ∈ 𝔐𝜑 ∩ ℳ+ we have F(𝛼)z ∈ 𝔐𝜑, zF(𝛼) ∈ 𝔐𝜑 the functions 𝛼 ↦ 𝜑(F(𝛼)z) and𝛼 ↦ 𝜑(zF(𝛼)) are entire analytic and

𝜑(zF(𝛼)) = 𝜑(F(𝛼 + i)z) for all 𝛼 ∈ ℂ;

(b) for every 𝜀 > 0 there exists 𝛿 > 0 such that if 𝛼 ∈ ℂ, |Im 𝛼| ≤ 𝛿, and z ∈ 𝔐𝜑 ∩ ℳ+, then𝜑(F(𝛼)zF(𝛼)∗) ≤ 𝜀𝜑(z) and 𝜑(F(𝛼)∗zF(𝛼)) ∈ 𝜀𝜑(z).

Then F(0) ∈ ℳ𝜑∞ and F(𝛼) = 𝜎𝜑𝛼 (F(0)) for all 𝛼 ∈ ℂ.

Proof. We have to show that F is an entire analytic function and F(t) = 𝜎𝜑𝛼 (F(0)) for all t ∈ ℝ.By assumption, 𝔑𝜑F(𝛼) ⊂ 𝔑𝜑,𝔑𝜑F(𝛼)∗ ⊂ 𝔑𝜑,𝔐𝜑F(𝛼) ⊂𝔐𝜑,F(𝛼)𝔐𝜑 ⊂𝔐𝜑 and, for x ∈ 𝔑𝜑

and 𝛼 ∈ ℂ, |Im 𝛼| ≤ 𝛿, we have‖(xF(𝛼))𝜑‖𝜑 ≤ 𝜀1∕2‖x𝜑‖𝜑, ‖(xF(𝛼)∗)𝜑‖𝜑 ≤ 𝜀1∕2‖x𝜑‖𝜑. (1)

Let a, b ∈ 𝔗𝜑. For every 𝛾 ∈ ℂ, we have 𝜎𝜑𝛾 (ab) = 𝜎

𝜑𝛾 (a)𝜎

𝜑𝛾 (b) ∈ 𝔗𝜑𝔗𝜑 ⊂ 𝔑∗𝜑𝔑𝜑 = 𝔐𝜑. We

define a function G of two complex variables by

G(𝛼, 𝛽) = 𝜑(F(𝛼)𝜎𝜑𝛽 (ab)) = ((𝜎𝜑𝛽 (b))𝜑|(𝜎𝜑𝛽 (a)∗F(𝛼)∗)𝜑)𝜑. (2)

By assumption and by the first equation in (2) it follows that 𝛼 ↦ G(𝛼, 𝛽) is an entire analyticfunction and

G(𝛼 + i, 𝛽) = 𝜑(𝜎𝜑𝛽 (ab)F(𝛼)). (3)

Using (1) and the second equation in (2) it follows that 𝛽 ↦ G(𝛼, 𝛽) is also an entire analyticfunction. Consequently, G is an entire analytic function in both variables, by the Hartogs theorem(Hormander, 1966, p. 2.2.8). On the other hand, using (3) and 2.18.(3), we obtain

G(𝛼 + i, 𝛽 + i) = 𝜑(𝜎𝜑𝛽+i(ab)F(𝛼)) = 𝜑(F(𝛼)𝜎𝜑𝛽 (ab)) = G(𝛼, 𝛽).

26 Normal Weights

Thus, 𝛼 ↦ g(𝛼) = G(𝛼, 𝛼) is an entire analytic function and g(𝛼 + i) = g(𝛼). Using (1) with 𝛿 = 1and (2), for 𝛼 ∈ ℂ with |Im 𝛼| ≤ 1 we get

|g(𝛼)| = |((𝜎𝜑𝛼 (b))𝜑|(𝜎𝜑𝛼 (a)∗F(𝛼)∗)𝜑)𝜑≤ 𝜀1∕2‖(𝜎𝜑𝛼 (b))𝜑‖𝜑‖(𝜎𝜑𝛼 (a)∗)𝜑‖𝜑 = 𝜀1∕2‖Δi𝛼𝜑 b𝜑‖𝜑‖Δi𝛼+(1∕2)𝜑 a𝜑‖𝜑.

Therefore, the entire analytic function g is bounded, and hence constant, by the Liouville theorem.In particular, for t ∈ ℝ we have 𝜑(𝜎𝜑−t(F(t))ab) = 𝜑(F(t)𝜎

𝜑t (ab)) = g(t) = g(0) = 𝜑(F(0)ab). Since

𝔗𝜑 is dense in ℋ𝜑 it follows that F(t) = 𝜎𝜑t (F(0)) for all t ∈ ℝ.

In order to prove that F is an entire analytic function, it is sufficient to show that F is bounded oneach compact subset of ℂ, as the set {𝜑(⋅z); z ∈ 𝔗2𝜑} is norm-dense in ℳ∗ (2.13) and the functions𝛼 ↦ 𝜑(F(𝛼)z)(z ∈ 𝔗2𝜑) are, by assumption, entire analytic (use the Montel theorem and [L],Lemma 9.24).

According to (1), the boundedness of F on compact subsets of ℂ will follow once we establishthe following identity (compare with 2.14.(7)):

J𝜑𝜋𝜑(F(𝛼))J𝜑a𝜑 = (aF(𝛼 + (i∕2))∗)𝜑 (a ∈ 𝔗𝜑, 𝛼 ∈ ℂ). (4)

Since the assumptions are stable under translations 𝛼 ↦ 𝛼 + 𝛼0, it is sufficient to prove (4) only for𝛼 = 0.

To this end, consider a, b ∈ 𝔗𝜑. For 𝛽 ∈ ℂ let

f1(𝛽) = (𝜋𝜑(F(0))Δ−i𝛽𝜑 (a∗)𝜑|Δ−i𝛽𝜑 b𝜑)𝜑,

f2(𝛽) = ( J𝜑Δ−1∕2𝜑 b𝜑|(aF(𝛽)∗)𝜑)𝜑.The function f1 is obviously entire analytic and the function f2 is entire analytic by the assumption(b). For t ∈ ℝ, it is easy to check that f1(t) = f2(t), since F(t) = 𝜎

𝜑t F(0). Hence f1 = f2. In particular,

f1(i∕2) = f2(i∕2), that is,

( J𝜑𝜋𝜑(F(0))J𝜑a𝜑| J𝜑Δ−1∕2𝜑 b𝜑)𝜑 = (aF(i∕2)∗)𝜑| J𝜑Δ−1∕2𝜑 b𝜑)𝜑.Since b ∈ 𝔗𝜑 was arbitrary, we obtain (4) for 𝛼 = 0. □

2.20. The results presented in Section 2.18 involve several variants of the KMS condition.For instance, if a ∈ ℳ𝜑∞ and z ∈ 𝔐𝜑, then from 2.18.(1) it follows that the equation

f (𝛼) = 𝜑(z𝜎𝜑−i𝛼(a)) (𝛼 ∈ ℂ)

defines an entire analytic function f, bounded on vertical strips, such that

f (it) = 𝜑(z𝜎𝜑t (a)), f (1 + it) = 𝜑(𝜎𝜑t (a)z) (t ∈ ℝ).

Also, if a ∈ 𝔗2𝜑 and z ∈ ℳ, z𝔗𝜑 ⊂ 𝔄𝜑,𝔗𝜑z ⊂ 𝔄𝜑, then the same conclusion is obtainedfrom 2.18.(3).

We record here one more variant of the KMS condition, where the similarity to, as well as thecontrast with, the trace property is very striking.

The Standard Representation 27

Proposition. Let𝜑 be an n.s.f. weight on the W ∗-algebra ℳ and let x ∈ 𝔑𝜑. There exists a boundedregular positive Borel measure 𝜇 on (0,+∞) such that

𝜑(x∗𝜎𝜑t (x)) = ∫∞

0𝜆itd𝜇(𝜆), 𝜑(xx∗) = ∫

∞

0𝜆 d𝜇(𝜆) (t ∈ ℝ).

Proof. Let {e𝜆 = 𝜒[0,𝜆)(Δ𝜑)}𝜆>0 be the spectral scale of Δ𝜑 ([L], E.9.10). Since x ∈ 𝔑𝜑, we obtaina bounded regular positive Borel measure 𝜇 on (0,+∞) setting

d𝜇(𝜆) = d(e𝜆x𝜑|x𝜑)𝜑,that is, 𝜇 is “the spectral measure associated with Δ𝜑 and x𝜑 ∈ ℋ𝜑.” According to ([L], E.9.11) weget

𝜑(x∗𝜎𝜑t (x)) = (Δit𝜑x𝜑|x𝜑)𝜑 = ∫ ∞0 𝜆itd𝜇(𝜆)

and if x𝜑 ∈ 𝔑𝜑 ∩𝔑∗𝜑,

𝜑(xx∗) = ‖(x∗)𝜑‖2𝜑 = ‖S𝜑x𝜑‖2𝜑 = ‖ J𝜑Δ1∕2𝜑 x𝜑‖2𝜑 = ‖Δ1∕2𝜑 x𝜑‖2𝜑 = ∫ ∞0 𝜆d𝜇(𝜆).The proof is completed by the remark that x ∈ D(S𝜑) = D(Δ

1∕2𝜑 ) ⇔ ∫ ∞0 𝜆 d𝜇(𝜆) < +∞ and

𝔑𝜑 ∩ D(S𝜑) = 𝔑𝜑 ∩𝔑∗𝜑g (2.12.(3)).

2.21. Let 𝜑 be an n.s.f. weight on the W ∗-algebra ℳ. The centralizer of 𝜑 is the W ∗-subalgebra ofℳ defined by

ℳ𝜑 = {a ∈ ℳ; 𝜎𝜑t (a) = a for all t ∈ ℝ}.

Since 𝜋𝜑(𝜎𝜑t (a)) = Δit𝜑𝜋𝜑(a)Δ

−it𝜑 (t ∈ ℝ), it follows that a ∈ ℳ

𝜑 if and only if 𝜋𝜑(a) commutes withΔ𝜑 ([L], E.9.20, E.9.23).

Clearly,

a ∈ ℳ𝜑 ⇒ a ∈ ℳ𝜑∞ and 𝜎𝜑𝛼 (a) = a for all 𝛼 ∈ ℂ.

Also, it follows from statement (7) of 2.14 that

a ∈ ℳ𝜑, x ∈ 𝔑𝜑 ⇒ xa∗ ∈ 𝔑𝜑 and (xa∗)𝜑 = J𝜑𝜋𝜑(a)J𝜑x𝜑. (1)

The Pedersen–Takesaki theorem ([L], 10.27) shows that if a ∈ ℳ, then

a ∈ ℳ𝜑 ⇔ a𝔐𝜑 ⊂𝔐𝜑,𝔐𝜑a ⊂𝔐𝜑 and 𝜑(ax) = 𝜑(xa) for x ∈ 𝔐𝜑. (2)

The implication (⇒) follows obviously from 2.15 and 2.18.(1). Conversely, we have a𝔗𝜑 ⊂𝔄𝜑,𝔗𝜑a ⊂ 𝔄𝜑, hence (2.20, 2.18(3)) for every x ∈ 𝔗2𝜑 the function fx(𝛼) = 𝜑(a𝜎

𝜑−i𝛼(x)) is an

28 Normal Weights

entire analytic function, bounded on vertical strips and such that

fx(1 + it) = 𝜑(𝜎𝜑t (x)a) = 𝜑(a𝜎

𝜑t (x)) = fx(it) (t ∈ ℝ).

By the Liouville theorem, it follows that fx is constant and hence

𝜑(𝜎𝜑t (a)x) = 𝜑(a𝜎𝜑−t(x)) = fx(−it) = fx(0) = 𝜑(ax) (t ∈ ℝ).

Since x ∈ 𝔗2𝜑 was arbitrary, using Proposition 2.13 we infer that 𝜎𝜑t (a) = a(t ∈ ℝ), so that a ∈ ℳ𝜑.

Let v ∈ ℳ be a partial isometry such that vv∗ ∈ ℳ𝜑. We define a normal semifinite weight 𝜑von ℳ by

𝜑v(x) = 𝜑(vxv∗) (x ∈ ℳ+).

It is easy to check that s(𝜑v) = v∗v.In particular, for every projection e ∈ ℳ𝜑, we have defined a subweight 𝜑e on ℳ with

s(𝜑e) = e.

Proposition. Let 𝜑 be an n.s.f. weight on the W ∗-algebra ℳ and v ∈ ℳ a partial isometry suchthat e = v∗v ∈ ℳ𝜑 and f = vv∗ ∈ ℳ𝜑. Then: 𝜑v = 𝜑e ⇔ v ∈ ℳ𝜑.

Proof. We assume first that v ∈ ℳ𝜑. By (2) we have 𝔑𝜑v ⊂ 𝔑𝜑 and 𝔑𝜑v∗ ⊂ 𝔑𝜑. Since 𝔑𝜑v ={x ∈ ℳ; xv∗ ∈ 𝔑𝜑} and 𝔑𝜑e = {x ∈ ℳ; xe ∈ 𝔑𝜑}, and since e = v

∗v, v∗ = ev∗, it follows that𝔑𝜑v = 𝔑𝜑e and hence 𝔐𝜑v = 𝔐𝜑e . For x ∈ 𝔐𝜑v = 𝔐𝜑e we obtain vxv

∗ ∈ 𝔐𝜑, exe ∈ 𝔐𝜑 and,since v = ve ∈ ℳ𝜑, 𝜑(vxv∗) = 𝜑(vexv∗) = 𝜑(exv∗v) = 𝜑(exe). We conclude that 𝜑v = 𝜑e.

Conversely, assume that 𝜑v = 𝜑e. For every x ∈ ℳ+ we have 𝜑(vxv∗) = 𝜑(exe). Replacingx by v∗xv here, we get 𝜑(v∗xv) = 𝜑( fxf ). If x ∈ 𝔑𝜑, then xf ∈ 𝔑𝜑, as f ∈ ℳ𝜑; consequently,𝜑(v∗x∗xv) = 𝜑( fx∗xf ) < +∞, that is, xv ∈ 𝔑𝜑. Thus, 𝔑𝜑v ⊂ 𝔑𝜑 and, similarly, 𝔑𝜑v∗ ⊂ 𝔑𝜑. Itfollows that v𝔐𝜑 ⊂𝔐𝜑 and 𝔐𝜑v ⊂𝔐𝜑. Then, for x ∈ 𝔐𝜑 we have

𝜑(vx) = 𝜑( fvx) = 𝜑(vxf ) = 𝜑(vxvv∗) = 𝜑(exve) = 𝜑(xve) = 𝜑(xv).

Using (2) we conclude that v ∈ ℳ𝜑.

In particular, for a unitary element u ∈ ℳ we have 𝜑u = 𝜑 if and only if u ∈ ℳ𝜑.

2.22. If 𝜑 is a normal, semifinite, but not necessarily faithful, weight on the W ∗-algebra ℳ, thenwe shall denote by {𝜎𝜑t