Notes Which Was Later Published As Springer Lecture Notes No.128: Simpliﬁ- Cation Was Not an Issue, but the Validity of Tomita’S Claim

STRUCTURE OF VON NEUMANN ALGEBRAS OF TYPE III

Masamichi Takesaki

Abstract. In this lecture, we will show that to every von Neumann algebra M there corresponds unquely a covariant system M, ⌧, R, ✓ on the real line R in such a way that n o f ✓ s M = M , ⌧ ✓s = e ⌧, M0 M = C, \ where C is the center off M. In the case that M fis a factor, we have the following commutative square of groups which describes the relation of several important groups such as thef unitary group U(M) of M, the normalizer U(M) of M in M and the cohomology group of the ﬂow of weights: C, R, ✓ : { } e f 1 1 1

? ? @ ? 1 ? U?(C) ✓ B1( ,?U(C)) 1 ! yT ! y ! ✓ Ry !

? ? @ ? 1 U(?M) U(?M) ✓ Z1( ,?U(C)) 1 ! y ! y ! ✓ Ry ! Ad Ade ? ? ˙ ? @✓ 1 1 Int?(M) Cnft?r(M) H ( ?, U(C)) 1 ! y ! y ! ✓ Ry !

? ? ? ?1 ?1 ?1 y y y

Contents Lecture 0. History of Structure Analyis of von Neumann Algebras of Type III. Lecture 1. Covariant System and Crossed Product. Lecture 2: Duality for the Crossed Product by Abelian Groups. Lecture 3: Second Cohomology of Locally Compact Abelian Group. Lecture 4. Arveson Spectrum of an Action of a Locally Compact Abelian Group G on a von Neumann algebra M. 1 2 VON NEUMANN ALGEBRAS OF TYPE III

Lecture 5: Connes Spectrum. Lecture 6: Examples. Lecture 7: Crossed Product Construction of a Factor. Lecture 8: Structure of a Factor of Type III. Lecture 9: Hilbert Space Bundle. Lecture 10: The Non-Commutative Flow of Weights on a von Neumann algebra M, I. Extended Lecture: The Non-Commutative Flow of Weights on a von Neumann algebra M, II.

Lecture 0. History of Structure Analyis of von Neumann algebras of type III. The foundation of the theory of von Neumann algebras was laid down by John von Neumann together with a collaborator Francis Joseph Murray in a series of the fundamental papers ”Rings of Operators” during the period of 1936 through 1943, although the fundamental bicommutant theorem was proven in 1929. The reason for this 7 year gap was that John von Neumann believed the Wedderbun type theorem for von Neumann algebra: all von Neumann algebras are of type I, unbelievable mistake by the genius. During their active perod, there was no one followed their work, except possibly the work of Gelfand and Neumark in 1943 who succeeded to have given postulates of a C⇤-algebra in a set of simple axioms with an additional condition which is now known to be unnecessary. The decade after the WW II, there was a tidal wave on the theory of operator algebras notably in the US, the Chicago school, and France lead by Jacque Dixmier. In this period, the knowledge of the topologgical properties of an operator algebra was greatly increased, which allowed the people handles the infinite dimensional non-commutative objects successfully. It is safe to say that the misterious subject introduced by the two pioneers is more or less digested by the specialists by then. The fashion after the WWII was somewhat gone by 1955. However, there were quite a few dicult but important problems were left untouched. The specialists lead by R. Kadison continued to attack the field and obtained deep results although the field remained unpopular, notaly Kadison’s transitivity theorem, Glimm’s result on type I C⇤-algebras, Sakai’s characterization of von Neumann algebra as a C⇤-algebra whichi is a dual Banach space. In the 1960’s, there was a dramatic change occured: the invasion of theoretical physicists lead by Rudolf Haag, Daniel Kastler, Nico M. Hugenholtz, Hans Borchers, David Ruel, Huzihiro Araki, Derek Robinson and many others who brought in a set of new ideas from the point of view of physics. It changed the scope the field greatly. New ideas and techniques were brought in. A great deal of progress started to follow. The number of specialists increased M. TAKESAKI 3 dramatically too. Then a big brow came when Araki proved in 1964, [Ark], that the most of von Neumann algebras occuring in the theoretical physics were of type III as there was no general method to treat the case of type III. Generally speaking, during the 60’s the mathematical physicists lead the field: for example in 1967, [RTPW], R.T. Powers proved the existence of continuously many non-isomorphic factors of type III, which are often called the Powers’ factors although they appeared in the work of J. von Neumann in 1938 and 1940 and were studied by L. Pukanszky in 1956, [Pksky]. Af- ter the work of Powers, Araki and Woods,[ArWd], worked on infinite tensor product of factors of type I, abbreviated ITPFI, and classified them with invariant called the asymptotic ratio set and denoted r (M) for such a factor M toward the end of the 60’s. But a von Neumann1 algebra of type III are still hiden deep in mistery. For instance the commutation theorem (M N)0 = M0 N0 was unkown, while the tensor product M N was shown to b⌦e of type II⌦I if either of the components is of type III, prov⌦en by S. Sakai in 1957, [Sk]. So the theory for a von Neumann algebra of type III was badly needed when Tomita proposed his theory at the Baton Rouge Meeting in the spring of 1967. But his preprint was very poorly written and full of poor mistakes: nobady bothers to check the paper. When I wrote to Dixmier in the late spring of 1967 about the validity of Tomita’s work, he responded by saying that he was unable to go beyond the third page and mentioned that it was very improtant to decide the validity of Tomita’s work. At any rate, Tomita’s work was largely ignored by the participants of the Baton Rouge Conference. For the promiss I made to Hugenholtz and Winnink, I started very seri- ously in April, 1967, after returning from the US and was able to resque all the major results: not lemmas and small propositions, many of which are either wrong or nonsense. Then I spent the academic year of 1968 through 1969 at Univ. of Pennsylvania, where R.V. Kadison, S. Sakai, J.M. Fell, E. E↵ros, R.T. Powers, E. Størmer and B. Vowdon were, but non of them believed Tomita’s result. So I checked once more and wrote a very detailed notes which was later published as Springer Lecture Notes No.128: simplifi- cation was not an issue, but the validity of Tomita’s claim. Through writing up the notes, I dicovered that Tomita’s work could go much further than his claim: the modular condition, (called the KMS-condition by physicists), and a lot more. I know that the crossed product of a von Neumann algebra by the modular automorphism group is semi-finite which I didn’t include in the lecture notes because I thought that the semi-finiteness alone was a half cocked claim. When I mentioned firmly the validity of Tomita’s claim, the people at the U. of Pennsylvania decided to run an inspection seminar in which I was allowed to give only the first introductory talk, but not in subsequent seminars, which run the winter of 1969 through the spring and the validity was established at the end. The miracle match of Tomita’s work and HHW work produced the theory of the modular automorphism group. I lectured this at UCLA for the academic year of 1969 through 1970 and 4 VON NEUMANN ALGEBRAS OF TYPE III produced UCLA Lecture Notes which were widely distributed. Among the audience of my lectures were H.A. Dye, R. Herman, M. Walter and Colin Sutherland. The campus disturbance in Japan prevented me from going home in Sendai. I decided to stay in the US and UCLA o↵ered me a professorship. In the academic year of 1970/71, Tulane University hosted a special year project on Ring Theory and invited me to participate. I participated only the Fall quarter, and lectured on the duality on locally compact groups based on Hopf von Neumann algebra techniques, [Tk4], which were later completed as the theory of Kac algebras, [EnSw], by Enock and Scwartz, students of J. Dixmier. Through these lectures, I recognized the need of unbounded functionals, i.e., the theory of weights, which motivated the joint work with Gert K. Pedersen in the succeeding year, [PdTk]. In the spring of 1971, UCLA hosted a special quarter project on operator algebras which invited three major figures: R. Haag, N.M. Hugenholtz and R.V. Kadson along with Gert K. Pedersen who stayed there throughout the spring quoter of 1971. Short term visitors were J. Glimm, J. Ringrose, B. Johnson and many others. The summer of 1971 was a turning point for the theroy of operator algebras as A. Connes participated in the summer school at Battelle Institue Seattle on the mathematical aspect of statistical mechanics, organized by D. Ruelle, O. Landford and J. Lebowitz. It was this Battelle Institute Summer School which motivated Connes to work on operator algebras. At the time of the summer school, he wasn’t sure what he wanted to work on. But the Summer School gave him enough kick to determine his mind to work on operator algeberas despite discouragement from his supervisor J. Dixmier. At any rate, soon after the summer school, Connes started to produce a series of excellent works on the structure analysis of factors of type III introducing his famous invariant S(M) and T (M) and devided factors of type III into those of type III , 0 1. During this period, I was in close contact with Connes. Maybe it is notan exaggeration to say that the period of 1972 through 1977 recorded historically high speed development in the theory of operator algebras. For my part, for the period of 71 through the spring of 72, I was engaged in the advancement of the theory of weights with Gert K. Pedersen as I recognized the need of the theory of weights in the course of the duality theory in which I was also very much interested. Then in the spring of 1972, W. Arveson came down to UCLA from Berkeley to deliver his new discovery of the constructive proof of a theorem of Borchers, [Brch], which asserts that the energy is a limit of local observables: before him the result rest on the derivation theorem of Sakai which is not constructive as it relies on the compactness of a relevant object. Arveson’s theory was so beautiful that I couldn’t resist to try on the structure analysis of a factor of type III. Also his theory equally inspiered Connes to develop his theory of Connes spectrum (↵) of an action ↵ of a locally compact abelian group on a von Neumann algebra. The summer of 1972 was quite dramatic for the structure analysis of a factor of type III M. TAKESAKI 5 on the both sides of Atlantic Oceaan. On the west, Araki was staying at Kingston with E.J. Woods as the host and working on the structure analysis of factors of type III which overlapped heavily with mine and collided at a meeting in Austin, Texas, where the both of us presented almost same results on the structure of a factor of type III. It was E. J. Woods who recognized immediately that the collaboration of myself and Araki would produce a remarkable progress in the structure theory of factors of type III and invited myself to Kingston for the summer of 72. The structure analysis of factors of type III developped in a breath taking pase: on the eastern side Connes was developping his own structure analysis of factor of type III , 0 < 1, while Araki and myself are working on a factor of type III which admists a special kind of states. Observing the rapid progress in Connes’ work, J. Dixmier advised him to postpone his Ph D for another year to complete a comprehensive and complete Ph D thesis and Connes followed his advise: thus his historic thesis which does not lose its glory today at all. The fall of 1972, I was able to prove the duality theorm which takes care of all von Neumann algebras of type III. Then I received a phone call from Daniel Kastler inviting me to spend the academic year of 1973/ 74 in Maraseille along with Connes. At that time Connes agreed to apply for the Hedricks Assistant Professorship at UCLA, but Kastler’s o↵er is more attractive. He invited me along with my students: Trond Digernes and Hiroshi Takai. So I accepted the invitation. Guggenheim Foundation also supported my plan to spend a year in Marseille. Then Connes visited me at UCLA in February of 1973, with which we started to work on the flow of weights: of course we didn’t think about the title. We both just knew that the duality theorem was not the end of the story, instead it was a beginning of something much deeper theory. The collaboration went very successfully: birth of the theory of flow of weights in which we have proved a technically demanding result: the relative commutanat theorem. The year 1973 - 74 has gone very quickly too. It took another year to complete the flow of weights paper. After this year, Connes went to Kingston to fulfil his military duty as a French citizen. There he proved his famous results on injective factors and the classification of a single automorphism of an AFD factor, leaving me far behind. In the succeeding year, I worked on the further completion of the structure theory of a factor of type III, which resulted the canonical construction of the core of a factor of type III and the cocycle conjgacy analysis of amenable discrete countable group actions on an AFD factor, following the work of Vaughan F.R. Jones and Adrian Ocneanu. 6 VON NEUMANN ALGEBRAS OF TYPE III

Lecture 1. Covariant System and Crossed Product. 1.1. Topology on Aut(M). Let M be a von Neumann algebra. An automorhism§ ↵ of M means a bijective linear map ↵ : x M ↵(x) M such that 2 7! 2 ↵(xy) = ↵(x)↵(y), x, y M. ↵(x⇤) = ↵(x)⇤, 2

The set Aut(M) of all automorphisms of M forms a group under the com- position of maps. If there exists a unitary u M such that 2

def 1 Ad(u)(x) = uxu⇤, x M, 2 then the automorphism Ad(u) is called inner. The set Int(M) of all inner automorphims forms a normal subgroup of Aut(M). We denote by

Out(M) def= Aut(M)/Int(M) the quotient group of Aut(M) by Int(M). For each ↵ Aut(M), we consider the following; 2

! ↵ ! < 1, def k i i k U(↵ : !1, , !n) = Aut(M) : 1 1 1 i n . · · · 2 ! ↵ ! < 1,   ( i i ) Then the family U(↵ : !1, , !n) : !1, , !n M deﬁnes a topology which makes Aut({M) a topological· · · group.· · · 2 ⇤}

Remark 1.1. The group Int(M) is not necessarily closed.

Under this neighborhood system, Aut(M) is a complete topological group. In the case that M is separable2, Aut(M) is a Polish group. ⇤ 1.2. Locally Compact Group. Now let G be a locally compact group with§ left invariant Haar measure.3 Let us ﬁx basic notations concerning locally compact group G. We take the left Haar measure ds and consider

def 1The notation = deﬁnes the left by the right of the equality sign. 2We call M separable when the predual M is seprable. A separable von Neumann ⇤ algebra is not separable in norm. In fact, a seperable von Neumann algebra in norm is ﬁnite dimensional. 3For those who are not familier with the general theory of locally compact group, one may take the additive group R of real numbers. M. TAKESAKI 7 the following Banach spaces relative to the left Haar measure:

f(st)dt = f(t)dt, f C (G); 2 c ZG ZG f(st)ds = (t) f(s)ds, f C (G); G 2 c ZG ZG 1 L (G) = f : f 1 = f(s) ds < + ; k k G| | 1 ⇢ Z 1 2 L2(G) = f : f = f(s) 2ds < + ; k k2 | | 1 ( ✓ZG ◆ ) 4 L1(G) = f : g G : f(s) r = 0 for some r R+ ; { |{ 2 | | }| 2 } f = inf r > 0 : s G; f(s) > r = 0 . k k1 { |{ 2 | | }| } The Banach space L1(G) admits an involutive Banach algebra structure:

1 (f g)(t) = f(s)g s t ds; ⇤ 1 ZG f, g L (G). 1 1 2 f ⇤(s) = G(s) f(s ),

We also introduce the notation:

1 f _(s) = f s

On the Hilbert space L2(G) of square integrable functions on G relative to the left Haar measure, we deﬁne unitary representations of G:

1 (G(s)⇠)(t) = ⇠ s t ; 2 1 ⇠ H = L (G), s, t G. (⇢G(t)⇠)(s) = G(t) 2 ⇠(st), 2 2

They are called the left and right regular representations of G respectively. Integrating and ⇢ , we get the -representations of L1(G): G G ⇤

(G(f)⇠)(t) = f(s)(G(s)⇠)(t)ds ZG 1 = f(s)⇠ s t ds = (f ⇠)(t); ⇤ ZG 1 (⇢G(f)⇠)(s) = f(t)(⇢G(t)⇠)(s)dt = f(t)G(t) 2 ⇠(st)dt ZG ZG 1 1 1 1 2 2 _ = f s t s t ⇠(t)dt = ⇠ f (s). G ⇤ G ZG ✓ ◆ ⇣ ⌘ 4The notation E for a Borel subset E G means the measure of E relative to a ﬁxed | | ⇢ Haar measure 8 VON NEUMANN ALGEBRAS OF TYPE III

An action of G on a von Neumann algebra M is a continuous map: g G ↵ Aut(M) such that 2 7! g 2 ↵ = ↵ ↵ , g, h G. gh g h 2 We call the system M, G, ↵ a covariant system on G. In the case that { } G = R, the action ↵ is called a one parameter automorphism group. Indeed, this is most relevant to us for the structure theory of von Neumann algebra of type III. 1.3. Perturbaton by Cocyles and Various Equivalence of Actions. § When two actions ↵ on M and on N of a locally compact group G are considered, we say that ↵ and are conjugate and write ↵ ⇠= if there exists an isomorphism : M N such that 7! 1 ↵ = , g G. g g 2 A continuous map: g G u(g) U(M) which satisﬁes the following cocycle condition: 2 7! 2

u(gh) = u(g)↵ (u(u )), g, h G, g h 2 is called an ↵-cocycle, or more precisely an ↵-one cocycle. We denote the set of all ↵-cocycles by Z↵(G, U(M)). For each ↵-cocycle u Z↵(G, U(M)), we deﬁne the perturbation u↵ of ↵ by the cocycle u as follows:2

↵ = Ad(u(g)) ↵ , g G, u g g 2 which is a new action of G on M. The cocycle identity guarantees that the perturbed u↵ is an action as seen via a simple computation. We say that u↵ the pertubed action of G by the cocycle u. For each w U(M), we set 2

(@wu)(g) = w⇤u(g)↵g(w).

It is easily seen that

1 ↵ = Ad(w) ↵ Ad(w), g G, (@wu) g u g 2 so that ↵ = ↵. More precisely, we write with v = @ u, b Z (G, U(M)), u ⇠ (@w(u)) w 2 ↵ 5 u↵ ⇠= v↵ mod Int(M).

5The notation mod Int(M) indicates the conjugation is actually given by the group Int(M) of inner automorphisms. M. TAKESAKI 9

The set of all cocycle peturbations of an action ↵ of G will be denoted by [↵] and called the cocycle perturbation class of ↵. For each [↵], i.e., 2 g = u↵g, g G, for some ↵-cocycle u, we consider the set Z(G, U(M)) of all -cocycles,2 and

Z (G, U(M)) = Z (G, U(M)) : = ↵ for some u Z (G, U(M)) . [↵] { u 2 ↵ } [ Each u U(M) acts on Z (G, U(M)) by the following: 2 [↵]

@ (v)(g) = u⇤v(g) (u), g G, v Z (G, U(M)), [↵]. u g 2 2 2

We denote the set of all @U(M)-orbits by the following:

H[a](G, U(M)) = Z[↵](G, U(M))/@(U(M)).

Two actions ↵ and of G on M are said to be cocycle conjugate if there exists u Z (G, U(M)) such that 2 ↵

⇠= u↵. 1.4. Crossed Product of a von Neumann algebra M by an Action of§ a Locally Compact Group G. Fix a covariant system M, G, ↵ . We represent M on a Hilbert space H faithfully. We view the{Hilbert }space K = H L2(G) as the Hilbert space L2(H, G) of all H-valued square integrable functions⌦ on G relative to the left Haar measure. Set

1 (⇡ (x)⇠)(g) = ↵ (x)⇠(g), ↵ g 2 1 ⇠ K = L (H, G), x M. (uG(h)⇠)(g) = ⇠ h g , 2 2 It then follows that ⇡↵ is a normal faithful representation of M and u is a unitary representation of G on K which are linked in the following way:

u(g)⇡ (x)u(g)⇤ = ⇡ (↵ (x)), g G, x M, ↵ ↵ g 2 2 the relation called the covariance. The crossed product M o↵ G is by deﬁnition the von Neumann algebra generated by ⇡↵(M) and u(G).

Theorem 1.2. The crossed product M o↵ G does not depend on the Hilbert space on which M acts.

Sketch of Proof. Suppose that M is represented on Hibert spaces H1 and H2 faithfully. Consider the direct sum Hilbert space H = H H and write the 1 2 actions of M on H1 by ⇡1 and on H2 by ⇡2 respectively. Then represent M diagonally on H, i.e.

⇡ (x) 0 x M ⇡(x) = 1 L(H). 2 7! 0 ⇡2(x) 2 ✓ ◆ 10 VON NEUMANN ALGEBRAS OF TYPE III

Let e1 and e2 be the projections to the ﬁrst and second components of H respectively. Then we have

⇡ (M), H = ⇡(M) , ⇡ (M), H = ⇡(M) . { 1 1} ⇠ e1 { 2 2} ⇠ e2 2 Consider the Hilbert space K = L (H, G) on which ⇡(M)o↵ G is represented. 2 Then the Hilbert space K is the direct sum of K1 = L (H1, G) and K2 = 2 L (H2, G). Now the crossed product ⇡(M)o↵G is generated by two kinds of operators: ⇡ (x), x ⇡(M), u (s), s G. ↵ 2 G 2 Let f1 and f2 be the projections of K to K1 and K2 respectively, i.e.,

f = e 1 2 , f = e 1 2 . 1 1 ⌦ L (G) 2 2 ⌦ L (G) Now we observe that

e , e ⇡(M)0; 1 2 2 ve v⇤ : v U(⇡(M)0) = 1 = we w⇤ : w U(⇡(M)0) . { 1 2 } { 1 2 } _ _ As v 1L2(G) U (⇡(M)o↵G)0 for every v U(⇡(M)0). we have ⌦ 2 2 vfiv⇤ : v U (⇡(M)o↵G)0 = 1, i = 1, 2. 2 _ Hence the maps x ⇡(M)o↵G xfi ⇡i(M)o↵G, i = 1, 2, are isomorphisms. Hence we ha2ve 7! 2

⇡1(M)o↵G ⇠= ⇡(M)o↵G ⇠= ⇡2(M)o↵G.

~ Let g G U(g) U(H) be a unitary reprresentation of G on the Hilbert space2 on7!which M2 acts also. If we have

U(g)xU(g)⇤ = ↵ (x), x M, g G, g 2 2 then we say that the action M on H and the unitary representation U of G are covariant and/or that the covariant system M, G, ↵ acts on H covariantly. { } Theorem 1.3. Suppose that an action of a covariant system M, G, ↵ on a Hilbert space H is covariant via a unitary representation U of {G on H, }then we have the following: 2 i) Let ⇡ be the representation of the commutant M0 on K = H L (G) given by the following: ⌦

(⇡(y)⇠)(g) = y⇠(g), y M0, g G, ⇠ K. 2 2 2 M. TAKESAKI 11

Then deﬁne v : g G v (g) U(K) by the following: G 2 7! G 2 1 6 (v (g)⇠)(h) = (g) 2 U(g)⇠(hg), g, h G, G G 2

where G : g G G(g) R⇤ be the modular function of G which 2 7! 2 + makes the above vG a unitary representation G on K. The representation ⇡ of the commutant M0 of M and unitary representation v of G is covariant in the sense that

U(g)⇡(y)U(g)⇤ = ⇡ ↵0 (y) , y M0, g G, g 2 2 where ↵0 is the action of G on the commutant M0 given by the following:

↵0 (y) = U(g)yU(g)⇤, y M0, g G. g 2 2

ii) The commutant of the crossed product M o↵ G represented on K is given by 00 (M o↵ G)0 = ⇡(M0) vG(G) . { [ } iii) The von Neumann algebra (M o↵ G)0 is naturally isomorphic to the

crossed product M0 o↵0 G. iv) If u Z (G, U(M)), then the unitary U deﬁned by the following: 2 ↵ 1 (U⇠)(g) = u g ⇠(g), g G, ⇠ K, 2 2 gives a spatial isomorphism:

U(M o↵ G)U ⇤ = M ou↵ G.

Thus the isomorphism class of the crossed product is stable under a cocycle perturbation.

Corollary 1.4. Let ⇢G be the right regular representation of G on the Hilbert space L2(G), i.e.,

1 2 (⇢ (g)⇠)(h) = (g) 2 ⇠(hg), ⇠ L (G), g, h G. G G 2 2 2 Then the unitary representation ⇢G gives rise to an action ⇢ of G on L L (G) as follows: 2 ⇢ (x) = ⇢ (g)x⇢ (g)⇤, x L L (G) , g G, g G G 2 2 6The unitary representation v is the tensor product U ⇢ of the unitary represen- G ⌦ G tation U and the right regular representation of G. 12 VON NEUMANN ALGEBRAS OF TYPE III which has the following properties: 2 ↵ ⇢ M o↵ G = M L L (G) ⌦ ⇠ ⌦ where N for a covariant system N, G, means the ﬁxed point subalgebra of N under any action . { } Unless G is discrete, there is no way to represent a general element of the crossed product M o↵ G. But fortunately, there are prenty many elements of the crossed product M o↵ G which are written in the form:

x = ⇡↵(x(g))u(g)dg, ZG where g G x(g) M is integrable in the sense that 2 7! 2 x = x(g) dg < + . k k1 k k 1 ZG With this in mind, we deﬁne an involutive Banach algebra structure on 1 L (M, G), the space obtained as the completion of the set Cc(M, G) of all 1 -strongly⇤continuous M-valued functions on G under the L -norm:

1 (x y)(g) = x(h)↵ (y(h g))dh; ⇤ h ZG 1 1 1 x⇤(g) = (g) ↵ x g ⇤; x, y L (M, G). G g 2 x = x(g) dg, k k1 k k ZG For each x L1(M, G), we set 2 ⇡(x) = ⇡ (x(g))u(g)dg L L2(H, G) . ↵ 2 ZG Then ⇡ is a -representation of L1(M, G) and the image ⇡ L1(M, G) is - ⇤ weakly dense in M G. The space C (M, G) of L1(M, G) is a self-adjoint o↵ c subalgebra and dense. The square Cc(M, G) Cc(M, G), i.e., the set of all linear combinations of the products: ⇤ n C (M, G) C (M, G) = x y : x , y C (M, G), i = 1, , n c ⇤ c i ⇤ i i i 2 c · · · (i=1 ) X lies in the domain of the operator valued weight: E(x y) = (x y)(e) M, x, y C (M, G), ⇤ ⇤ 2 2 c where we identify Cc(M, G) with its image

⇡(Cc(M, G)) M o↵ G. ⇢ We denote the domain of E and the positive part of the domain by the following: D(M, G) = C (M, G) C (M, G); c ⇤ c n D(M, G) = x⇤ x : x , , x C (M, G) . + i ⇤ i 1 · · · n 2 c (i=1 ) X M. TAKESAKI 13

1.5 Dual Weight. We fix a covariant system M, G, ↵ . We now inves- tigate§ a semi-finite normal weight on M and the{corresponding} semi-finite normal weight on M o↵ G. Theorem 1.4. The map E : D(M, G) M is extended to a semifinite + 7! normal faithful operator valued weight from M o↵ G to the subalgebra M where M is identified with its image ⇡↵(M) in the crossed product M o↵ G. Definition 1.5. For a semi-finite normal weight ' on M, the semi-finite normal weight ' on M o↵ G defined by the following: ' = ' E b is called the dual weight of '. b Theorem 1.6. Under the above notations and the set up the dual weight ' on M o↵ G of a faithful semi-finite normal weight ' enjoys the following properties: b i) The semi-finite normal weight ' is faithful. ii) The modular automorphism group ' acts on the generators of the

crossed product M o↵ G as followsb : b '(⇡ (x)) = ⇡ ('(x)), x M; t ↵ ↵ t 2 ' b it (u(g)) = G(g) u(g)⇡↵ (D'↵g : D') , t R. t t 2 b iii) If is another faithful semi-ﬁnite normal weight, then we have the Connes cocycle derivative given the following:

D : D' = ⇡↵((D : D')t), t R. t 2 ⇣ ⌘ b b In the theorem, if we take the modular covariant system M, R, ' , then ' { } the modular automophism group : t R is given by Ad(u (t)) : t R , t 2 { R 2 } which gives the following result atn once.b o

Corollary 1.7. The crossed product M = Mo' R of the modular covariant system M, R, ' is semi-finite and its algebraic type is independent of the { } choice of a faithful semi-finite normale weight '. The faithful semi-finite normal trace on the crossed product M is given by the following formula:

1 1 ⌧(x) = ' h 2 xeh 2 , x M , 2 + ⇣ ⌘ where the non-singular positiveb self-adjoint opertatore h aliated with M is the logarithmic generator of the one parameter unitary group u (t) : t R , { R 2 } i.e., e it u (t) = h , t R. R 2 14 VON NEUMANN ALGEBRAS OF TYPE III

Lecture 2: Duality for the Crossed Product by Abelian Groups. 2.1. Fourier Transform and Plancherel Formula. We now consider a§ locally compact abelian group G. We denote the group operation of G additively. Let G be the Kampen-Pontrjagin dual of G, i.e., the group of all continous homomorphism from G to the torus T = C : = 1 . We { 2 | | } write the dualitybof G, G in the following way: n o (g, p) G G g, p T jointly continuous bicharacter; 2 ⇥ 7!bh i 2 g + h, p = g, p h, p , g, p + q = g, p g, q , h bi h ih i h i h ih i g, h G, p, q G 2 2 We choose the Haar measures dg on G and dp on G inb such a way that the Plancherel formula holds, i.e., b (Ff)(p) = g, p f(g)dg, f L1(G), h i 2 ZG Ff (g) = g, p f(p)dp, f L1 G , G h i 2 ⇣ ⌘ Z ⇣ ⌘ b b f Ff b =b Ff f b , b L2(G) L2(G) ⇣ ⌘ ⇣ ⌘ 2 1 2 2 fb bL (G) L (G),bf Lb G L G ; 2 \ 2 \ ⇣ ⌘ ⇣ ⌘ F Ff = f, if f L1(G) and Ff L1 G ; 2 b 2 b b ⇣ ⌘ F Ff = f, if f L1 G and Ff L1(G); b 2 2 b f = Ff , f⇣ ⌘L1(G) L2(G); bkb k2 b k bk2 2b \b b Ff = f , f L1 G L2 G . 2 2 2 \ ⇣ ⌘ ⇣ ⌘ If f L2(G), thenforb beac h compact b subsetb Kb G thereb exists f L2 G 2 b K 2 such that ⇣ ⌘ f (p) = g, p f(g)dg, b b K h i ZK and the net f : K Gb converges to f L2 G in L2-norm as K in- K b 2 creases to G.nWe call thisof the extended Fourier⇣ ⌘ transform of f and b b b write f = Ff. Hence the extended Fourier transform F is indeed an isometry from L2(G) onto L2 G . b b We use the following⇣ ⌘notations and name: b F L1 G = A(G) = L2(G) L2(G) C (G); ⇤ ⇢ 0 ⇣ ⇣ ⌘⌘ F L1(G) = A G = L2 G L2 G C G . b b ⇤ ⇢ 0 ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ b b b b M. TAKESAKI 15

The algebras A(G) and A G are respectively called the Fourier algebra. We now discuss the real⇣ additiv⌘ e group R. In this case, we have b R = R; isp (s, p) R R s, p = e T; 2 ⇥ b7! h i 2 1 isp 1 (Ff)(p) = e f(s)ds, f L (R); p2⇡ 2 ZR 1 isp 1 Ff (s) = e f(p)dp, f L (R). p2⇡ 2 ⇣ ⌘ ZR b b b b Viewing the Fourier transform F as an operator on L2(R), we have

F = F⇤, FF⇤ = 1, F⇤F = 1, unitary F2f(s) = f( s), F2f (p) = f( p), F4 = 1, F4 = 1. b ⇣ ⌘ Namely the Fourier transformbon L2(R) is a unitary operatorb with period 4. ⇡s2 If f(s) = e , then (Ff)(p) = f(p), p R. 2 Deﬁne the convolution and the involution by the following:

1 (f g)(s) = f(t)g(s t)dt, f, g L (R); ⇤ 2 ZR f ⇤(s) = f( s). With this algebraic structure, L1(R) is an involutive Banach algebra. Fur- thermore, the Fourier transform is indeed the Gelfand transform of the Ba- nach algebra L1(R) as seen below:

1 Ff C0(R), f L (R); 2 2 1 F(f g) = (Ff)(Fg), f, g L (R); ⇤ 2 (Ff ⇤) = (Ff); 1 F(fg) = (Ff) (Fg), f, g C0(R) L (R); ⇤ 2 \ 1 Ff = (Ff)⇤, f L (R). 2 2.2. Uniqueness of the Heisenberg Commutation Relation. § Let G be an abelian locally compact group with Kampen - Pontrjagin dual G. A pair U, V of unitary representations of G and G on the same Hilbert space H is called covariant if the following covariance condition holds: b b U(s)V (p) = s, p V (p)U(s), (s, p) G G. h i 2 ⇥ b 16 VON NEUMANN ALGEBRAS OF TYPE III

The map: 2 ((s1, p1); (s2, p2)) G G mG(g1, g2) = s2, p1 T 2 ⇥ 7! h i 2 ⇣ ⌘ where b g = (s , p ) G G, i = 1, 2, i i i 2 ⇥ is a 2-cocycle satisfying the cocycle identity: b m (g , g )m (g g , g ) = m (g , g g )m (g , g ), g , g , g G G. G 1 2 G 1 2 3 G 1 2 3 G 2 3 1 2 3 2 ⇥ 7 We ﬁx the Haar measures on G and G respectively so that the Plancherelb formula holds. Consider the Hilbert space H = L2(G) and deﬁne: b (⇢ (s)⇠)(t) = ⇠(t + s), s, t G; G 2 ( (s)⇠)(t) = ⇠(t s) G (µ (p)⇠)(t) = t, p ⇠(t), p G, ⇠ L2(G). G h i 2 2 Then the following computation: b ( (s)µ (p)⇠)(t) = (µ (p)⇠)(t s) = t s, p ⇠(t s); G G G h i (µ (p) (s)⇠)(t) = t, p ( (s)⇠)(t) = t, p ⇠(t s); G G h i G h i (s)µ (p) = s, p µ (p) (s) G G h i G G shows that the pair G, µG is covariant. We then consider the e↵ect of the Fourier transform on{ them: }

(F (s)F⇤⇠)(p) = t, p ( (s)F⇤⇠)(t)dt = t, p (F⇤⇠)(t s)dt G h i G h i ZG ZG = t + s, p (F⇤⇠)(t)dt = s, p t, p (F⇤⇠)(t)dt h i h i h i ZG ZG 2 1 = s, p (FF⇤⇠)(p) = s, p ⇠(p), ⇠ L G L G , h i h i 2 \ ⇣ ⌘ ⇣ ⌘ (Fµ (p)F⇤⇠)(q) = t, q (µ (p)F⇤⇠)(t)dt = t, q t, p b(F⇤⇠)(t)dtb G h i G h ih i ZG ZG = t, p + q (F⇤⇠)(t)dt = (FF⇤⇠)(p + q) h i ZG = ⇢G(p)⇠ (q) for suciently many ⇠’s bto conclude

F (s)F⇤ = µ (s), s G; G G 2 FµG(p)F⇤ = ⇢ (p) = (p)⇤, p G. G b G 2 b b 7For those of you who are not familier with the theory of locally compact abelian groups, you can assume G = G = R and 1/2⇡ times the Lebesgue measure as the Haar measures on G and G with which Plancherelp formula holds. b b M. TAKESAKI 17

Theorem 2.1. i) The covariant representations G, µG is irreducible in the sense that there is no non-trivial closed subsp{ace M} L2(G) jointly ⇢ invariant under ⇢G and µG. Equivalently, we have

2 (G) µ G 00 = L L (G) . G [ G ⇣ ⇣ ⌘⌘ b ii)The covariant pair G, µG of unitary representations is a unique covariant representation in{the sense} that if U, V is another covariant pair of { } unitary representations of G and G on the same Hilbert space H, then there exists a Hilbert space K such that b 1 , µ 1 , L2(G) K = U, V, H Unitarily Equivalent. G ⌦ K G ⌦ K ⌦ ⇠ { } The algebra L1(G) of all essentially bounded functions on G relative to the Haar measure acts on L2(G) by multiplication:

2 (⇡(f)⇠)(s) = f(s)⇠(s), f L1(G), ⇠ L (G). 2 2

2 2 Then ⇡(L1(G)) L L (G) is a maximal abelian subalgebra of L L (G) . ⇢ For each f L1 G , the multiplication operator by the Fourier transform 2 f = Ff C (G)⇣of ⌘f is given by the following; 2 b 0 b b b b (⇡(f)⇠)(s) = s, p f(p)dp ⇠(s) = f(p) s, p ⇠(s)dp h i h i ✓ZG ◆ ZG b b = b f(p)µG(p)dp ⇠ (s) b ✓✓ZG ◆ ◆ = µG bfb⇠ (s), ⇣ ⇣ ⌘ ⌘ where b µ f = f(p)µ (p)dp L L2(G) . G G 2 ZG ⇣ ⌘ Consequently, b b b

1 ⇡(f) = µ f , f = F⇤f, f L G . G 2 ⇣ ⌘ ⇣ ⌘ b b b b 1 Since A(G) = F L G is norm dense in C0(G), we have ⇣ ⇣ ⌘⌘ b b µG G 00 = ⇡(L1(G)). ⇣ ⌘ b 18 VON NEUMANN ALGEBRAS OF TYPE III

For each f L1 G G , set 2 ⇥ ⇣ ⌘ b ⇡(f) = f(s, p)⇢G(s)µG(p)dsdp;

GZZG ⇥ b ⇡(f)⇡(g) = 0 f(s, p)⇢G(s)µG(p)dsdp10 g(t, q)⇢G(t)µG(q)dtdq1 GZZG GZZG B ⇥ CB ⇥ C @ A@ A = b t, p f(s, p)g(t, q)⇢ (s + t)µ b(p + q)dsdpdtdq h i G G 2 ZGZZGZ ( ⇥ ) = b t, p f(s, p)g(t s, q p)⇢ (t)µ (q)dsdpdtdq h i G G 2 ZGZZGZ ( ⇥ ) = ⇡(f g) b⇤ where (f g)(t, q) = t, p f(s, p)g(t s, q p)dsdp ⇤ h i GZZG ⇥ Then we have b

(⇡(f)⇠)(t) = t, p f(s, p)⇠(s + t)dsdp h i GZZG ⇥ = b t, p f(s t, p)⇠(s)dsdp. h i GZZG ⇥ Let M, G, ↵ be a covariant systemb on a locally compact abelian group G. Assume{ that}the von Neumann algebra M acts on a Hilbert space H. Then the crossed product: N = M o↵ G, is generated by the two kind of operators:

⇡ (x), x M, u(s), s G, ↵ 2 2 and operates on the Hilber space

K = H L2(G) = L2(G, K). ⌦ Now the unitary representation v : p G v(p) U(K) deﬁned by the formula: 2 7! 2 (v(p)⇠)(s) = s, p ⇠(s), ⇠ bK, s G, p G, h i 2 2 2 b M. TAKESAKI 19 behaves in the following way:

v(p)⇡ (x)v(p)⇤ = ⇡ (x), x M, p G; ↵ ↵ 2 2 v(p)u(s)v(p)⇤ = s, p u(s), s G. h i 2 b

Thus the unitary representation v of the dual group G gives rise to an action ↵ of G: ↵p(x) = v(p)xv(p)⇤, x M o↵ G, bp G. 2 2 b b Defition 2.2. The action ↵ of G is said to be dual to the original action ↵ b b or the dual action of ↵. The covariant system M o↵ G, G, ↵ is called the b dual covariant system orb the covariant systemn dual too the original system M, G, ↵ . b b { } Now we are ready to state the duality theorem on the crossed product of a covariant system on a locally compact abelian group. Theorem 2.3. (Duality Theorem). Suppose that M, G, ↵ is a covariant system on a locally compact abelian group G. Then{ the covariant} system

M o↵ G, G, ↵ dual to M, G, ↵ has the following properties: { } n i) The fixeod point algebra (M G)↵ under the dual action ↵ is pre- b b o↵ cisely ⇡↵(M); b ii) A faithful semi-finite normal weight ! on the M o↵ G is dualb to a some faithful semi-finite normal weight ' on M, i.e., ! = ' if and only if ! is invariant under the dual action ↵; iii) Concerning the second crossed product, we have b b 2 (M o↵ G) o↵ G = M L L (G) ⇠ ⌦ under the isomorphism determineb b d uniquely by the following properties:

1 ((⇡ (x))⇠)(s) = ↵ (x)⇠(s), x M, s G; ↵ s 2 2 2 ((⇡↵(u(s)))⇠)(t) = ⇠(t s), s, t G; ⇠ L (H, G) 2 2 u (p) ⇠ (t) = t, p ⇠(t), p G; b G h i 2 b b where uG( ) means the unitary representation of the dual group G corresponding· to the second crossed product; iv) The isomorphismb conjugates the second dual action ↵ to the ac-b tion ↵ ⇢ where ⇢ means the inner automorphism action of G on 2 ⌦ L L (G) given by the right regular representation ⇢G deﬁneb d by the following: (⇢ (s)⇠)(t) = ⇠(s + t), ⇠ L2(G), s, t G; G 2 2 20 VON NEUMANN ALGEBRAS OF TYPE III

v) For each faithful semi-ﬁnite normal weight ', the second dual semi- ﬁnite normal weight 'ˆ is related to the original weight ' in the following way:

ˆ 1 D 'ˆ : D(' Tr) ⇠ (s) = (D('↵s) : D')t⇠(s) ⌦ t ⇣⇣ ⇣ ⌘ ⌘ ⌘ for each ⇠ L2(H, G), s G and t R. 2 2 2 Brief Sketch of the Proof of Theorem 2.3.(iii). The second crossed 2 2 product N = (Mo↵G) o↵ G acts on the Hilbert space H' L (G) L G , ⌦ ⌦ where H' is the semi-cyclic Hilbert space constructed by a faithful semi-⇣ ⌘ b ﬁnite normal weight ' on Mb on which the original von Neumann algebrabM acts. Making use of the inverse Fourier transform 1 1 F, we represent the second crossed product von Neumann algebra N on⌦ K⌦= L2(H , G G,) ' ⇥ which is then generated by the following three types of operators:b

U(s) = 1 (s) (s), s G; ⌦ G ⌦ G 2 V (p) = 1 1 µ (p), p G; ⌦ ⌦ G 2 ⇡↵⇡↵(x) = ⇡↵(x) 1, x M. ⌦ 2 b Consider the following bunitary operator which is called the fundamental unitary of G:

(W ⇠)(r, s) = ⇠(r + s, s), r, s G, ⇠ K. 2 2 Observe

W (1 1 µ (p))W ⇤ = 1 1 µ (p), p G; ⌦ ⌦ G ⌦ ⌦ G 2 W (1 G(s) G(s))W ⇤ = 1 1 G(s), s G; ⌦ ⌦ ⌦ ⌦ 2b W (⇡ ⇡ (x))W ⇤ = ⇡(x), x M, ↵ ↵ 2 where b 1 (⇡(x)⇠)(r, s) = ↵ (x)⇠(r, s), x M, ⇠ K. (r+s) 2 2 We then prove

⇡(M) (C C L1(G)) = ⇡↵(M) L1(G), (*) _ ⌦ ⌦ ⌦ which will implies the following:

00 W NW ⇤ = (C C L1(G)) C C G(G) ⇡(M) ⌦ ⌦ _ ⌦ ⌦ _ = (⇡↵(M) L1(G)) (⇣C C R⇣(G)) ⌘⌘ ⌦ _ ⌦ ⌦ = ⇡ (M) L L2(G) . ↵ ⌦ M. TAKESAKI 21

This completes the proof. Proof of ( ) for a ﬁnite abelian group G. Observe ⇤

⇡(x) = ⇡ (↵ (x))ds ⇡ (M) L1(G), ↵ s 2 ↵ ⌦ ZG so that ⇡(M) ⇡ (M) L1(G). ⇢ ↵ ⌦ Hence we have

W (⇡↵(M) C)W ⇤ = ⇡(M) ⇡↵(M) L1(G). ⌦ ⇢ ⌦

On the other hand, W commutes with C C L1(G), which yields the following: ⌦ ⌦ W (⇡ (M) L1(G))W ⇤ ⇡ (M) L1(G). ↵ ⌦ ⇢ ↵ ⌦ Simlarly, we have

W ⇤(⇡ (M) L1(G))W ⇡ (M) L1(G). ↵ ⌦ ⇢ ↵ ⌦ Hence, we conclude that

W (⇡ (M) L1(G))W ⇤ = ⇡ (M) L1(G). ↵ ⌦ ↵ ⌦ This completes the proof of ( ). ⇤ ~ 22 VON NEUMANN ALGEBRAS OF TYPE III

Lecture 3: Second Cohomology of Locally Compact Abelian Group. 3.1. Two cocycles of a separable locally compact abelian group. § Fix a separable locally compact abelian group G, where we write the group operation multiplicatively to shorten formulae. A Borel function m : G G T is called a 2-cocycle if the following cocycle identity holds: ⇥ 7!

m(g , g )m(g g , g ) = m(g , g g )m(g , g ), g , g , g G. 1 2 1 2 3 1 2 3 2 3 1 2 3 2 If there exists a Borel function b : G T such that 7! m(g , g ) = b(g )b(g )b(g g ), g , g G, 1 2 1 2 1 2 1 2 2 then m is called a coboundary and written

m = @b.

We also say that the function b cobounds m. The set of all cocycles is 2 written Z (G, T) and becomes an abelian group under the pointwise multiplication, i.e.,

2 (mn)(s1, s2) = m(s1, s2)n(s1, s2), s1, s2 G, m, n Z (G, T). 2 2 2 2 The set B (G, T) of all coboundaries is a subgroup of Z (G, T). The quotient group: 2 2 2 H (G, T) = Z (G, T)/B (G, T) is called the second cohomology group of G. 2 Fixing a cocycle m Z (G, T), we consider 2

Gm = T G, ⇥ and deﬁne a map:

(( , s ), ( , s )) G G ( m(s , s ), s s ) G . 1 1 2 2 2 m ⇥ m 7! 1 2 1 2 1 2 2 m

Observe that Gm is a group. We consider the Borel structure in Gm generated by product sets E F : E T, F G of Borel subsets E T and F G and consider the{integration:⇥ ⇢ ⇢ } ⇢ ⇢

f(, s)dds, for Borel function f 0, ZGm where d means the normalized Haar measure on the torus T. This integration gives an invariant standard measure on Gm, which makes the group Gm 2 a locally compact group. Hence each m Z (G, T) gives rise to an exact 2 M. TAKESAKI 23 sequence of locally compact groups equipped with associated cross-section, (i.e., a right inverse of the map ⇡m):

i ⇡m 1 T Gm G 1 ! ! ! ! um such that i(T) is contained in the center of Gm and

u (r)u (s) = i(m(r, s))u (rs), r, s G. m m m 2 If we have another cocycle n Z2(G, T ), then we have another exact sequence: 2

i ⇡n 1 T Gn G 1. ! ! ! ! um

These exact sequences are in the following commutative diagram:

i ⇡m 1 T Gm G 1 ! ! ! !

i ? ⇡n 1 T G?n G 1 ! ! y ! ! if and only 2 [m] = [n] H (G, T). 2 In particular, the above exact sequence split in the sense that the quotient 2 map ⇡m admits a left inverse u if and only if m B (G, T). 2 For every closed subgroup H G and the subgroup of the dual group G: ⇢ b H? = p G : s, p = 1, s H , 2 h i 2 n o the Kampen-Pontrjagin Duality Theoremb for locally compact abelian groups asserts that [ H ⇠= G/H?, G/H ⇠= H?, under the natural correspbondence.b This means that the above exact sequence splits in the sense that the quotiend map ⇡m admits a left inverse if and only if Gm is commutative. 2 For each m Z (G, T) consider the following asymmetrization: 2

(ASm)(s , s ) def= m(s , s )m(s , s ), s , s G. 1 2 1 2 2 1 1 2 2 24 VON NEUMANN ALGEBRAS OF TYPE III

Theorem 3.1. i) If m is a 2-cocycle on a locally compact abelian group G, then the assymetrization ASm is a skew symmetric bicharacter in the following sense:

(ASm)(s1s2, r) = (ASm)(s1, r)(ASm)(s2, r); r, s1, s2, s G, (ASm)(r, s) = (ASm)(s, r), 2

Furthermore, the cocycle m is a coboundary if and only ASm = 1, in the sense that ASm(r, s) = 1, r, s G. Consequently, the second cohomology 2 2 group H (G, T) is isomorlphic to the group X2(G, T) of all continuous T- valued squew symmetric bicharacters on G. ii) If G is closed under the squar root, i.e. if every element of G is square of another element of G, then every 2-cocycle is cohomologous to a skew symmetric bicharacter. We are now returning to the additive notation on the group G. A skew symmetric bicharacter X2(G, T) is said to be symplectic if for every non-zero s G there exists2 some t G such that 2 2 (s, t) = 1. 6 Example 3.2. i) Let E be an even dimensional real vector space. Let be a symplectic bilinear form on E, i.e.,

(r, s) = (s, r); (r, s) = 0 for all s E r = 0, 2 ) Then the bicharacter

(r, s) = exp(i(r, s)), r, s E. 2 is a symplectic bicharacter of E. ii) Let E = Rn Rn, n N and write element r E by ⇥ 2 2

r = (x1, , xn; p1, , pn); · · · · · · xi, yi, pj, qj R, 1 i, j n. s = (y , , y ; q , , q ), 2   1 · · · n 1 · · · n Then set n (r, s) = (x q p y ) i i i i i=1 X Then is a symplectic bilinear form on E. iii) Every symplectic bilinear form on an even dimensional real vector space E is of the above form after choosing an appropriate coordinate system. M. TAKESAKI 25

Theorem 3.3. Suppose that E is an even dimensional real vector space equipped with a symplectic bilinear form and set

(s, t) = exp(i((s, t))), s, t E. 2 i) The Hilbert space L2(G) of square integrable functions on E relative to the ﬁxed Haar measure is a unimodular left Hilbert algebra under the twisted convolution and the involution deﬁned by the following:

(x y)(s) = (s, t)x(s)y(t s)ds; 2 ⇤ G x, y L (G), s G. Z 2 2 x⇤(s) = x( s),

ii) Choose a basis e1, , en; f1, , fn of E so that for a pair of vectors: { · · · · · · } n n u = xiei + pjfj; i=1 j=1 X X n n v = yiei + qjfj, i=1 j=1 X X the symplectic form takes the following form:

n (u, v) = (x q y p ). i i i i i=1 X 2 iii) Set F = e1, , en E and consider the Hilbett space L (F ) of square inteh grable· · · functionsi ⇢ on F relative to the Lebesgue measure. Then the twisted convolution:

f ⇠(x) = exp( i p, y )f(x, p)⇠(y x)dxdp. ⇤ h i ZZE for every f L2(E) and ⇠ L2(F ) converges and gives a bounded operator: 2 2

⇠ L2(F ) ⇡(f)⇠ = f ⇠ L2(F ). 2 7! ⇤ 2 The map ⇡ : L2(E) ⇡(f) L L2(F ) gives rise to an isomorphism 7! 2 between L2(E) and the algebra L2 L2(F ) of Hilbert-Schmidt class operators on L2(F ) and 2 Tr(⇡(g)⇤⇡(f)) = (f g), f, g L (F ). | 2 26 VON NEUMANN ALGEBRAS OF TYPE III

Lecture 4. Arveson Spectral of an Action of a Locally Compact Abelian Group G on a von Neumann algebra M. We discuss further the Arveson spctrum of actions of a separable locally compact abelian group G on a von Neumann algebra M. [Arv]. Let G be a locally compact abelian group and X the dual Banach space of another Banach space X on which G acts in such a way that the map s G ⇤ 2 7! ↵s(x), ' C is continuous for every ' X . An action G on X means h i 2 2 ⇤ that a homomorphism ↵ : s G GLw(X), where GLw(X) means the set of all (X, X )-continous inv2ertible7! operators on X, such that ⇤

lim '↵s ' = 0, ' X s 0 ⇤ ! k k 2 sup ↵ : s G < + . {k sk 2 } 1 Let ↵ be an action of G on X which will be ﬁxed for a while. Recall that the Fourier algebra A G is the Fourier transform of the convolution algebra L1(G), equivalently⇣the⌘set of all convolution of two square integrable functions on G relative to theb Haar measue. For each f L1(G), we write and set 2 b f(p) = Ff (p) = s, p f(s)ds, p G; Gh i 2 ⇣ ⌘ Z b b b ↵f (x) = f(s)↵s(x)ds, f A G , x X. G 2 2 Z ⇣ ⌘ b Then ↵ L(X) and map ↵ : f A G ↵ b L(X) is a representation f 2 2 7! f 2 of a Banach algebra A G on the Banac⇣ ⌘h space X, i.e., b b b b ⇣ ⌘ b ↵ = ↵ ↵ , f, g A G . fg f g 2 ⇣ ⌘ b b b For a ﬁxed element x Xb, the set b b b 2 I (x) = f A G : ↵ (x) = 0 ↵ 2 f n ⇣ ⌘ o b b b is a closed8 ideal of A G . Hence we get a closed subset ⇣ ⌘ b Sp (x) = p G : f(p) = 0, f I (x) G. ↵ 2 2 ↵ ⇢ n o and call it the ↵-spectrum of bx bX. b b With 2 I(↵) = f A G : ↵ = 0 = I (x), 2 f ↵ x X n ⇣ ⌘ o \2 b b b 8The norm topology in A G is inherited from L1(G). ⇣ ⌘ b M. TAKESAKI 27 the zero point set of the ideal I(↵) of A G is called the Arveson spectrum of ↵ and written Sp(↵). ⇣ ⌘ The ↵-spectrum Sp (x) of x X bis the oscillation mode of the map: ↵ 2 s G ↵s(x) X as seen below. Then the following properties of the ↵-sp2 ectrum7! of x 2 M is easily shown: 2 Sp (↵ (x)) supp(f) Sp (x). ↵ f ⇢ \ ↵ b Definition 4.1. For a subset E G, we bset ⇢ X↵(E) = x bX : Sp (x) E . { 2 ↵ ⇢ } Example 4.2. Suppose that a map x( ) is a X-valued continuous function · on G such that

↵s(x(p)) = s, p x(p), p G. b h i 2 Set b x = x(p)dp X. 2 ZG For each f L1(G), let b 2

f(p) = Ff (p) = s, p f(s)ds, Gh i ⇣ ⌘ Z b b and compute

↵f (x) = f(s)↵s x(p)dp ds ZG ✓ZG ◆ b = f(s)↵ (xb(p))dsdp = f(s) s, p (x(p))dsdp s h i GZZG GZZG ⇥ ⇥ = fb(p)x(p)dp. b ZG b Then conclude that b

f I (x) supp(f) supp(x( )) = . 2 ↵ , \ · ; Thus we get in thisb case

Sp (x) = supp(x( )). ↵ · 28 VON NEUMANN ALGEBRAS OF TYPE III

Theorem 4.3. (Arveson’s Theorem). Suppose that be another action of G on X such that GL (X) and s 2 w

lim 's ' = 0, ' X . s 0 ⇤ ! k k 2 Then the following statements are equivalent: i) ↵ = , s G; s s 2 ii) X↵(E) X(E) for all compact subset E G; ⇢ ⇢ iii) X(E) X↵(E) for all compact subset E G; ⇢ ⇢ iv) X(E) = X↵(E) for all compact subset E Gb. ⇢ b The proof is quite technical. So we omite the proof.b The following result indicates the property of the Arveson spectrum. Proposition 4.4. If M, G, ↵ is a covariant system on locally compact abelian group G, then {the Arveson} spectrum of each x M enjoys the following properties: 2 ↵ ↵ i) Sp↵(x⇤) = Sp↵(x), x M, i0) M (E)⇤ = M ( E), E G, 2 ↵ ↵ ↵ ⇢ ii) Sp↵(xy) Sp↵(x) + Sp↵(y), ii0) M (E)M (F ) M E + F . ⇢ ⇢ b We continue to work on the covariant system M, G, ↵ withG a locally compact abelian group, and set { }

p↵(E) = sup s (x) : x M↵(E) ; { ` 2 } q↵(E) = sup s (x) : x M↵(E) , { r 2 } for each closed subset E G. where s`(x) and sr(x), x M mean respectively the left and right supp⇢ ort of x, i.e., 2 b 1 1 n n s`(x) = -strong⇤ lim (xx⇤) , sr(x) = -strong⇤ lim (x⇤x) , x M. n n !1 !1 2 Theorem 4.5. Under the above notation, both p↵(E) and q↵(E) for a closed subset E G are projections in the center C↵ of the ﬁxed point subalgebra M↵. ⇢ b Making use of the above discusion, the following reﬁned theorem of Borchers can be shown. Theorem 4.6 (Borchers-Arveson). For a one parameter automorphism group M, R, ↵ of a von Neumann algebra M, the following two conditions are equivalent{ : } i) There exists a one parameter unitary group u(t) : t R in the unitary group U(M) such that { 2 }

↵t(x) = u(t)xu(t)⇤, x M, t R; 2 2 M. TAKESAKI 29

and the one parameter unitary group u(t) : t R has a positive spectrum in the sense that the spectral inte{ gration2: } 1 u(t) = eitde() Z0 gives the spectral decompostion of u(t) over the positive real line [0, ); { } ii) 1 inf q↵([t, + )) : t R = 0. { 1 2 } If the condition (ii) holds, then the spectral measure e( ) of u(t) is given explicitly in the following form: · { } e([t, + )) = inf q↵([s, + )) : s < t . 1 { 1 } Returning to the original setting X, X , ↵ on locally compact abelian group G, we have the following { ⇤ } Theorem 4.7. i) Let A be the Banach subalgebra of L(X) generated by ↵ , f A G . Then the spectrum Sp(A) of the commutative Banach algebra f 2 A is natur⇣ally⌘ identiﬁed with the Arveson spectrum Sp(↵) of the action ↵. b ii) The folb lowing two conditions on X, X , ↵ are equivalent: { ⇤ } a) The map: s G ↵s L(X) is contiuous in norm; b) The Arveson2spe7!ctrum2Sp(↵) is compact.

Definition 4.8. Let A be a C⇤-algebra and : x A (x) is called a derivation if is a linear map such that 2 7! (xy) = (x)y + x(y), x, y A. 2 When we have (x⇤) = (x)⇤, x A, 2 it is called a -derivation. Setting ⇤ ⇤(x) = (x⇤)⇤, x A, 2 we obtain another derivation ⇤ and we decompose the derivation into the linear combination of -derivations: ⇤ 1 1 = ( + ⇤) + i ( ⇤). 2 2i Theoem 4.9. (Sakai’s Theorem). A derivation on a C⇤-algebra A is always bounded and generates a norm continuous one parameter automorphism group of A in the sense n t n ↵t(x) = (x), t R. n! 2 n X2Z+ Each automorphism ↵ is a -automorphism if and only if is a -derivation. t ⇤ ⇤ Theorem 4.10. (Sakai - Kadison). A derivation on a von Neumann algebra M is inner in the sense that there exists a M such that 2 (x) = [a, x] = ax xa, x M. 2 30 VON NEUMANN ALGEBRAS OF TYPE III

Lecture 5: Connes Spectrum. We return to the study of a covariant system M, G, ↵ over a locally compact abelian group G. We denote the ﬁxed poin{t subalgera} of M under the action ↵ by M↵, i.e., M↵ = x M : ↵ (x) = x, s G , { 2 s 2 } or in the previous lecture’s notation M↵ = M↵(0). Definition 5.1. In the above setting, if e Proj(M↵), then the action 2 ↵ leaves the reduced algebra Me globally invariant, so that the restriction e ↵s, s G, of ↵s to the reduced algebra Me = eMe makes sense. We call this 2 e e restriction Me, G, ↵ a reduced covariant system and ↵ the reduced action of ↵{by the pro}jection e M↵. The intersection: 2 (↵) = Sp(↵e) : e Proj(M↵), e = 0 { 2 6 } is called the Connes spectrum\ of ↵. Let us state the main theorem concerning the Connes spectrum of a covariant system M, G, ↵ with G a locally compact abelian group. { } Theorem 5.1 (Connes - Takesaki). Suppose that M, G, ↵ with G is a covariant system over a locally compact abelian group{G. Then} the Connes spectrum (↵) has the following properties: i) The Connes spectrum (↵) is a closed subgroup of G; ii) (↵) + Sp(↵) = Sp(↵); iii) If is another action of G on M which is cocyclebconjugate to ↵, then (↵) = (); iv) The Connes spectrum is precisely the kernel of the restriction

CN, G, ↵ n o of the dual system ↵ to the center C of the crossed product N = b b N M o↵ G. Corollary 5.2. If M is a factorb of type III, then (') is a closed subgroup of the real additive group R and independent of the choice of a faithful semi- ﬁnite normal weight ' on M , hence it is an algebraic invariant of the factor M.

Definition 5.3. A factors of type III is called of type III with 0 < < 1 if

' ( ) = (log )Z for any faithful semi-ﬁnite normal weight '; of type III1 if ' ( ) = R for any faithful semi-ﬁnite normal weight '; and of type III0 in the remaining case. M. TAKESAKI 31

Lecture 6: Examples. We discuss examples. Fix 0 < < 1 and let

Mn = M(2, C), n N. 2 For each n N, deﬁne a state ! on Mn by the following: 2 n 1 1 x11 x12 2 2 x11 x12 !n = 1 1 x11 + 1 1 x22, Mn. x21 x22 2 2 2 2 x21 x22 2 ✓ ◆ + + ✓ ◆ Consider the algebraic inﬁnite tensor product:

M0 = ⌦Mn, n Y2N which is the set of linear combinations of the elements of the from:

x1 xn 1 1 , xi Mi, i N. ⌦ · · · ⌦ ⌦ ⌦ · · · 2 2

Then M0 is naturally an involutive algebra on C, which admits a unique C⇤-norm such that

n x x 1 1 = x . k 1 ⌦ · · · n ⌦ ⌦ ⌦ · · ·k k ik i=1 Y

The completion M of M0 under this norm is called the inﬁnite C⇤-tensor product. We will write this C⇤-algebra as follows:

1 M = ⌦Mn. n=1 Y b On this C⇤-algebra M we deﬁne a state ! in the following way:

1 !(x x 1 1 ) = ! (x ) 1 ⌦ · · · n ⌦ ⌦ ⌦ · · · i i i=1 Y where xn+1 = 1, xn+2 = 1, , which gives rise to a representation ⇡, H of M via the GNS construction· · · and obtain the von Neumann algebra{equipped} with a faithful normal state !:

1 R, ! = ⌦ Mn, !n . n=1 Y To deﬁne an inﬁnite tensor product of von Neumann algebras we need to specify state on each component von Neumann algebra. It is easy to see that 32 VON NEUMANN ALGEBRAS OF TYPE III

each !n is a faithful state on Mn and its modular automorphism group n is given by the following:

it 2 0 t,n = Ad it . 0 2 ✓ ◆ From the modular conditon, it is easy to see that the modular automorhism group of ! is the tensor product automorphism group:

! 1 = ⌦ , t R. t t,n 2 n=1 Y ! We write t for this heavy notation t instead. With 2⇡ T = > 0, log we have = id, n N, T,n 2 so that each component modular automorphism group n has common period T and consequently T = id. Let S be the finite permutation group of N, i.e. each element S permute 2 only finitely many members of N, which then acts on M0 in the following way: (x x 1 1 ) = x x x 1 ⌦ · · · n ⌦ ⌦ ⌦ · · · (1) ⌦ (2) ⌦ · · · ⌦ (n) ⌦ · · · if (n + k) = n + k, k N, which is then extended by linearity to M0 and leaves the state ! inv2ariant. Furthermore, the automorphism acts only on the first n-components and leaves the rest invariant and the first n-tensor product is isomorphic to the matrix algebra M(2n, C) of order 2n. Hence it is given by a unitary, say

n U() ⌦M . 2 k kY=1 As leaves ! invariant, U() belongs to the centerlizer:

n ! ⌦Mk . ! kY=1 Now for each n N, consider the permutation n which permutes the ﬁrst n terms with the n2 terms starting from 2n + 1 and leaves the rest unchanged. Then it is easy to see that

lim ! (n(x)y) = ! (x)! (y), x, y M0, n !1 2 M. TAKESAKI 33 which is called the mixing property of !. This property entails the following -weak convergence:

lim n(x) = ! (x) -weakly for every x R. n !1 2

Now each is given by Ad(U( )) and U( ) (R ) . Consequently n n n 2 !

(R)0 R = C. ! \

! In particular, the centralizer R,! of ! is a factor. Hence = (log)Z R. Thus the factor R is of type III. We now come to ⇣the result⌘ of Powers ⇢in 1967: Theorem 6.1 (R.T. Powers). If 0 < = µ < 1, then 6 R = R . 6⇠ µ Now we ﬁx 0 < = µ < 1 such that 6 log Q, log µ 62 which guarantees the following:

it it µ = 1 except for t = 0. 6 To each n N, we asign the 3 3 matrix algebra: 2 ⇥

Mn = M(3, C), and consider the state given by the following:

x x x 11 12 13 1 µ ! x x x = x + x + x n 0 21 22 23 1 1 + + µ 11 1 + + µ 22 1 + + µ 33 x31 x31 x33 @ A n The modular automorphism group t of !n is give by the following one parameter unitary group { }

1 0 0 it un(t) = 0 0 , t R. 0 0 0 µit 1 2 @ A The choice of and µ entails that

un(t) T for all non-zero t R. 62 2 34 VON NEUMANN ALGEBRAS OF TYPE III

Consequently, the modular automorphism group n is not periodic. Let { t } M be the inﬁnite C⇤-tensor product:

1 M = ⌦Mn, n=1 Y b and set 1 R, ! = ⌦ M , ! , { } { n n} n=1 Y i.e., 1 00 ! = ⌦!n, R = ⇡!(M) . n=1 Y By the similar arguments as in the previous case, we can conclude that

R0 R = C. ! \ Thus we get ! ( ) = R.

Thus the factor R is of type III1. This factor is not isomorphic to any of R, 0 < < 1, and does not admit a faithful semi-ﬁnite normal weight with periodic modular automorphism group. M. TAKESAKI 35

Lecture 7: Crossed Product Construction of a Factor. 7.1. Transformation Groups of a Standard Measure Space. Let §X, µ be a standard measure space with -ﬁnite measure µ which gives rise { } to an abelian von Neumann algebra A = L1(X, µ) of all essentially bounded complex valued measurable functions on X which acts on the Hilbert space 2 H0 = L (X, µ) of square integrable functions. Let G be a separable locally compact group. By an action G on X, µ we mean a Borel map T : (g, x) G X T x X such that { } 2 ⇥ 7! g 2 i) for each s G the map: x X Tsx X is a non-singular transformation2 of X, i.e., 2 7! 2

µ(T (N)) = 0 µ(N) = 0, N X; s , ⇢ ii) T T = T , s, t G; s t st 2 iii) T x = x, x X, for the identity e G. e 2 2 We naturally assume that the action is e↵ective in the sense described below: T x = x for almost every x X s = e. s 2 ) We call the system G, X, µ, T a G-measure space. This is equivalent to { } an action ↵ of G on the abelian von Neumann algebra A = L1(X, µ) and the relation between the action ↵ and the transformation T is described in the following: Let ↵ be an action of G on the abelian von Neumann algebra A which entails a non-singular transformation group Tg : g G of the measure space X, µ : { 2 } { } 1 ↵ (a)(x) = a T x , a A = L1(X, µ), s G, x X. s s 2 2 2 In this setting, we have the Radon-Nikodym derivative:

dµ T ⇢(s, x) = s (x), s G, x X. dµ 2 2

The ⇢-function satisﬁes the following cocycle identity:

⇢(st, x) = ⇢(s, T x)⇢(t, x), s, t G, x X, t 2 2 and 1 1 f T x dµ(x) = f(x)⇢(s, x)dµ(x), f L (X, µ). s 2 ZX ZX 36 VON NEUMANN ALGEBRAS OF TYPE III

This tells us that the transformation rules on Lp(X, µ) and Lq(X, µ), 1 p = q < + , are di↵erent:  6 1 1 1 q 1 p ↵ (f)(x) = ⇢ s , x f T x , f L (X, µ), s G; s s 2 2 1 1 q 1 q ↵ (f)(x) = ⇢ s , x f T x , f L (X, µ), s G; s s 2 2 1 1 p p 1 p 1 ↵s(f) p = ⇢ s , x f Ts x dµ(x) k k X ✓Z ⇣ ⌘ ◆ 1 p 1 1 p = ⇢ s , x f Ts x dµ(x) ✓ZX ◆ 1 1 p p 1 p p = f T T x dµ(x) = f(x) dµ(x) s s | | ✓ZX ◆ ✓ZX ◆ = f . k kp For p = 2, we write U(s)⇠, ⇠ L2(X, µ), and obtain a unitary representation U( ) of G on L2(X, µ) which2implements the action ↵ in the following sense: ·

↵ (a) = U(s)aU(s)⇤, s G, a A = L1(X, µ), s 2 2 where we regards A as a von Neumann algebra acting on L2(X, µ) by multiplication. The measurability of the map:

s G ! (U(s)) = (U(s)⇠ ⌘), ⇠, ⌘ L2(X, µ) 2 7! ⇠,⌘ | 2 entails the strong continunity of s G U(s) L L2(X, µ) , and hence it is a unitary representation of G and 2A, G,7!↵ is a2covariant system. Associated with this is the crossed product v{on Neumann} algebra

R(X, µ, G, ↵) = A o↵ G.

The measure space theory always involves almost everywhere arguments, which is quite sticky in the case that G is a continuous group as it is not countable. For this reason, if we can replace X, µ, G, T by a topological { } transformation group, it is much easier to handle. So we consider the C⇤- covariant system.

7.2. C⇤-Covariant System for a Commutative Covariant Sys- tem.§ We begin by defenition: Definition 7.1. Let A be a C⇤-algebra and G a locally compact group . An action ↵ of G on A is a map ↵ : s G ↵s Aut(A) is a homomorphism such that 2 7! 2 lim ↵s(x) ↵t(x) = 0, x A. s t ! k k 2 M. TAKESAKI 37

Proposition 7.2. If M, G, ↵ is a covariant system, then the set { } ↵ Ac = x M : lim ↵s(x) x = 0 2 s ek k n ! o is a unital -weakly dense C⇤-subalgebra of M. If M is separable, then we ⇤ can choose a separable -weakly dense unital C⇤-subalgebra A A which ⇢ c is globally invariant under ↵ and the restriction A, G, ↵ is a C⇤-covariant system. { }

7.3. Continuity as a Result of Measurability. As lim s(f) f = § s ek k1 0, f L1(G), we can consider the subset of M: ! 2 ↵ (x) : x M, f L1(G) f 2 2 which is a subset of Ac and -weakly dense in M, because

↵ (↵ (x)) = ↵ (x) and ↵ (x) f x , x M, f L1(G). s f s(f) k f k  k k1k k 2 2 So if X, µ, G, T is an action of separable locally compact group G on a standard{measure space,} then we obtain a covariant system A, G, ↵ as before { } and here we take a seprable C⇤-covairant system A, ↵ to obain a compact space Y = Sp(A) on which G acts as a topological{ transformation} group leaving a probability Radon measure ⌫ on Y quasi-invariant corresponding to the original measure µ on X after replacing µ by an equivalent probability measure. So we will consider only the case that X, µ is a compact space and µ is a probability Radon measure and X, G,{ T }is a toplogical transformation group of X which leaves the measure{ µ quasi-in} variant. 7.4. Free and Ergodic Covariant System. With X, G, µ, T as above, w§ e then consider the crossed product { }

R(X, G, µ, T ) = A o↵ G, A = L1(X, µ).

We often omit the letter T and write R(X, G, µ). Definition 7.3. If the ﬁxed point reduces to the scalar, A↵ = C, then we say that the system X, G, µ, T is ergodic. The action T is called free if for any compact subset{ K G suc} h that e K and for any Borel subset E X with µ(E) > 0 then there⇢ exists a Borel62 subset F E such that ⇢ ⇢ T (F ) F = , s K, and µ(F ) > 0. s \ ; 2 In terms of von Neumann algebra lungage, the freeness is phrased as in the following: for any compact subset K G such that e K and any non-zero projection e Proj(A) there exists f⇢ Proj(A) such 62that 2 2 f ↵ (f), s K, and f = 0. ? s 2 6 38 VON NEUMANN ALGEBRAS OF TYPE III

Theorem 7.4. If A, G, ↵ is a covariant system with A abelian, then the following two conditions{ on}the covariant system are equivalent: i) The action T of G on X is free; ii) The original abelian von Neumann algebra A in the crossed product Ao↵G is maximal abelian. For the proof on a general locally compact group G, it involves the duality theory of unitary representations of G. For an abelian group G, we can prove as an application of the Arveson - Connes spectral theory for actions. Corollary 7.5. Under the assumption that A, G, ↵ is free, the crossed { } product Ao↵G is a factor if and only if the system is ergodic. Theorem 7.6. Let A, G, ↵ is an ergodic and free covariant system with A abelian, then we have{ the fol} lowing properties on the crossed product R = Ao↵G: i) R is of type I if and only if the measue is concentrated on a single orbit, equivalently the action is transitive. ii) R is of type II1 if and only if G is discrete and there exists a ﬁnite invariant measure ⌫ equivalent to the original measure µ. iii) R is of type II if and only if a) The action1 is not transitive; b) There exists a Borel measure ⌫ equivalent to µ such that

d⌫ T (s) s (x) = 1 for every s G and x X. G d⌫ 2 2

where G( ) is the modular function of G, and c) if G is discr· ete, then the measure ⌫ is inﬁnite. iv) R is of type III if and only if there is no nontrivial Borel measure equivalent to µ which satisﬁes the condition (iii-b).

In the sequel, we consider only ergodic case as the genral case is decom- posed into ergodic systems. S7.5. Concrete Example of Each Type. Now we are going to discuss examples which realizes the above criteria. We continue to use the above notations unless we say contrary. Example 7.7. If we take X, µ to be the group G itself with left Haar measure µ, then the left translation{ } action of G on X is obviously transitive. i) If G is inﬁnite, then R is a factor of type I . 1 ii) If G is of oder n, then R is a factor of type In.

Example 5.8. Let X = T = z C : z = 1 be the one dimensional torus. Fix an irrational number ✓ and{ set2 | | }

T x = exp(2⇡i✓)x, x X. 2 M. TAKESAKI 39

The Lebesgue measure µ is invariant under T . Viewing T as an action of the additive integer group G = Z, then the action is free and ergodic with ﬁnite invariant measure. As each orbit is countable, the action can not be transitive. Hence the resulted factor R is of type II1. Example 7.9. Take G = Q to be the additive group of rational numbers and let it act on X = R by translation:

Tsx = x + s, x X = R, s Q. 2 2 Take the Lebesgue measure dx on X. Then the action is free and ergodic. The measure is invariant under the translation action T of G = Q and inﬁnite. Hence the resulted factor R is a factor of type II . 1 Example 7.10. Let G be the ax + b group, i.e., it is given by the following:

a b G = g = g(a, b) = : a Q⇤ , b Q . 0 1 2 + 2 ⇢ ✓ ◆ Let G act on X = R in the following way:

Tg(a,b)x = ax + b

The subgroup 1 b H = : b Q 0 1 2 ⇢✓ ◆ acts on X = R ergodically by translation. So any other invariant measure on X equivalent to the Lebesgue measure is propotional to the Lebesgue measure. But each element g(a, 0), a Q+⇤ , a = 1, transforms the Lebesgue measure µ to aµ. Hence it does not2leave µ6 invariant. Consequently, there is no invariant measure equivalent to the Lebesgue measure. So the resulted factor is of type III. Example 7.11. Let X = R and

n b G = : b Z() 0 1 2 ⇢✓ ◆ for a ﬁxed 0 < < 1, where Z() means the additive subgroup of R generated by the powers n, n Z, of : 2 n a n a 1 n Z() = + + + a0 + a1 + an : ai Z, n N , n · · · · · · 2 2 (k= n ) X then the resulted factor R is of type III. 40 VON NEUMANN ALGEBRAS OF TYPE III

Example 7.12. Fix an irrational positive number ✓. On L2(R) we deﬁne two unitaries:

2 (U⇠)(x) = ⇠(x + 1) (V ⇠)(x) = exp(2⇡i✓x)⇠(x), ⇠ L (R). 2

The von Neumann algebra R generated by U, V is of type II1 and the relative 2 2 { } dimension dimR L (R) of L (R) over R is ✓. 7.6 Flow of Weigh ts for Ergodic Free Transformation Group. First w§ e begin by deﬁnition: Definition 7.13 For a factor M, the restriction C, R, ✓ of the trace { } scaling covariant system N, R, ⌧ with { }

M = N o✓ R to the center C = CN of N is called the ﬂow of weights on M. We now dicuss the ﬂow of weights on the factor R = R(X, µ, G) constructed from a free erogodic system X, G, µ in the case that G is discrete. The continuous case can be handled similarly{ .}So suppose that the covariant system A, G, ↵ is free and ergodic with the ⇢-function ⇢( , ) on G X. Recall the{ cocycle} identity: · · ⇥

dµ g ⇢(g, s) = (x), x X, g G; dµ 2 2 ⇢(gh, x) = ⇢(g, hx)⇢(h, x), g, h G, x X. 2 2

The cocycle identity allows us to make the product space X = R+⇤ X an R G-space as follows: ⇥ ⇥ e s Ts,g(, x) = e ⇢(g, x) , gx), (s, g) R G, (, x) X; 2 ⇥ 2 e f(, x)dµ˜(, x) = f(, x)ddµ(x), f 0. e X X Z ZZR+⇤ ⇥ Since e 1 1 1 f(a)d = f(), d, a > 0, a Z0 Z0 we have for a positive measurbale function f on X,

1 s e f Ts,g(, x)dµ˜(, x) = f e ⇢(g, x), gx ddµ(x) ZZX ZX Z0 s 1 1 e =e e ⇢(g, x) f(, gx)d dµ(x) ZX ✓Z0 ◆ = es f(, x)ddµ(x) = es f(, x)dµ˜(, x). X X ZZR+⇤ ⇥ ZZ e M. TAKESAKI 41

Theorem 7.14 (Connes - Takesaki). The ﬂow of weights for the factor R = R(X, µ, G) associated with the ergodic free G-measure space X, µ, G , { } the core N, R, ⌧, ✓ is given by the following: { } i) The core von Neumann algebra N is the group measure space von Neumann algebra N = R(X, µ˜, G) of the system:

e N = L1 X, µ˜ o↵ G; ⇣ ⌘ 1 e (↵˜g(a))(, x) = a T0,g (,ex) ⇣ ⌘ 1 1 = a ⇢e(g , x), g x , a L1 X, µ˜ ; 2 ⇣ ⌘ ii) The one parameter automorphism group ✓ which scalese down the trace is the lifting of the ﬂow:

s T : (, x) X e , x X s,e 2 7! 2 e e e to the crossed product N = L1 X, µ˜ o↵ G. The flow C, , ✓ of weights on R is then⇣ given⌘ by the flow on the fixed point R e e subalgebra{: } G C = L1 X, µ˜ . ⇣ ⌘ The flow is the restriction of the flow eTs,e : s R to the fixed point subal- G { 2 } gebra C = L1 X, µ˜ . ⇣ ⌘ e 42 VON NEUMANN ALGEBRAS OF TYPE III

Lecture 8: Structure of a Factor of Type III. Trace Scaling One Parameter Automorphism Group 8.1. Trace Scaling One Parameter Automorphism Group. Suppose § that N, R, ✓, ⌧ is a covariant system over the real line R which scales a faithful{ semiﬁnite} norma trace ⌧ down, i.e., s ⌧ ✓s = e ⌧, s R. 2 Set M = N✓. Fix T > 0. Then for any non-zero projection e M, there exists a projection e e such that 2 0  e ✓ (e ). 0 ? T 0 Such a projection is called a wondering projection of ✓T . Set s (e ) = ✓n (e ) N✓T . T 0 T 0 2 n X2Z Choose a maximal family of wondering projections ei : i I for ✓T such that { 2 } ✓ (e ) e , i I; T i ? i 2 s (e ) s (e ) for i = j, i, j I. T i ? T j 6 2 The maximality of the family e : i I implies that { i 2 } ✓ (e) e, ✓n (e) = 1, with e = e . T ? T i n i I X2Z X2 Set k fn = ✓ (e), n N. T 2 k n |X| Then we have f 1 n %

I (f ) = ✓ ✓k (e) ds = ✓k ✓ (e)ds ✓ n s0 T 1 T s ZR k n k n ✓ZR ◆ |X| |X| @ (mA+1)T k = ✓T ✓s(e)ds mT k n m Z Z ! |X| X2 T k m = ✓T ✓T ✓s(e)ds 0 k n m Z Z !! |X| X2 T k m = ✓T ✓T (e) ds = (2n + 1)T < + . 0 0 1 1 Z k n m Z ! |X| X2 @ A mI✓ fnNfn, n N. 2 Hence we conclude the following: M. TAKESAKI 43

Lemma 8.1. In the above setting, the action ✓ is integrable in the sense that -weak closure mI✓ = N. Now choose a faithful semi-ﬁnite normal weight ' W (M) and set 2 0 ' = ' I W (N). ✓ 2 0 Then the faithful semi-ﬁnite bnormal weight ' is ✓-invariant and therefore we get 1 1 s ✓s((D' : D⌧)t) = D'✓s : D⌧ ✓sb t = (D' : D(e ⌧))t ist = e (D' : D⌧)t b b b it for every s, t R. Hence we obtain a one parameter unitary group h' : t R such that 2 b 2 it ist it ✓s h' = e h' such that ' it = Ad h M, t R; t ' | 2 ' N = M o' R, ✓ = . Theorem 8.2.(Connes-Takesaki). If N, , ⌧, ✓ is a trace scaling covari- R c ant system, then it is necessarily the dual{system such} that with M = N✓ the covariant system is decribed in the following way:

' it = Ad h M, t R; t ' | 2 ' N = M o' R, ✓ = for any faithful semi-ﬁnite normal weight ' W (M). 2 c0 Theorem 8.3. (Connes - Takesaki). If N, R, ⌧, ✓ is a trace scaling { } covariant system, then the relative commutant M0 N of the ﬁxed point M = N✓ is the center C of N. \ Definition 8.4 The normalizer

U(M) = u U(N) : uMu⇤ = M { 2 } of M in N is called thee extended unitary group of M. Proposition 8.5. For u U(N) to be a member of the extended unitary 2 group U(M) it is necessary and sucient that

e (@✓u)t = u⇤✓t(u) U(C), t R. 2 2

The map c : t R c(t) = (@✓u)t is necessarily a cocycle and hence a 1 2 7! member of Z✓(R, U(C)). Recall that the one parameter automorphism group ✓ is stable in the sense that every ✓-cocycle is a coboundary. Hence the 44 VON NEUMANN ALGEBRAS OF TYPE III

1 coboundary map @✓ : u U(M) @✓(u) Z✓(R, U(C)) is a surjection. So we have the following exact2 sequence:7! 2 e

@✓ 1 1 U(M) U(M) Z (R, U(C)) 1 ! ! ! ✓ ! e Restricting the above exact sequence to the U(C), we get the following exact sequences:

1 1 1

? ? @✓ 1 ? 1 T? U?(C) B (R?, U(C)) 1 ! y ! y ! ✓ y !

? ? @✓ 1 ? 1 U(?M) U(?M) Z (R,?U(C)) 1 ! y ! y ! ✓ y ! e Adding the last low, we obtain the following characteristic square of Aut(M) ⇥ R-equivariant exact sequences:

1 1 1

? i ? @✓ 1 ? 1 T? U?(C) B (R?, U(C)) 1 ! y ! y ! ✓ y ! i i i

? i ? @✓ 1 ? 1 U(?M) U(?M) Z (R,?U(C)) 1 ( ) ! y ! y ! ✓ y ! ⇤ Ad Ade ˙ ? i f ? @✓ 1 ? 1 Int?(M) Cnt?r(M) H (R?, U(C)) 1 ! y ! y ! ✓ y !

? ? ? ?1 ?1 ?1 y y y Summerizing, we get the following:

Theorem 8.6. (Katayama-Sutherland-Takesaki). Associated with a factor M is the above Aut(M) R equivariant characteristic square ( ). ⇥ ⇤ 8.3. Characteristic Invariant. Looking at the middle vertical column of§ the characteristic square, we recognize the following equivariant exact M. TAKESAKI 45 sequence:9

N = Cntr(M), H = U(M), A = U(C), G = Aut(M) R; ⇥ j E : 1 A i H N 1 ! e! ! ! s 1 s(m)s(n)s(mn) = µ(m, n) A, m, n N; 2 2 1 1 ↵ s g mg s(m) = (m, g) A. g 2 If we change the cross-section s to another one s0 : m N s0(m) H, then the di↵erence is in A, i.e., 2 7! 2

1 s0(m)s(m) = f(m) A, m N. 2 2 We then have

s0(m)s0(n) = f(m)s(m)f(n)s(n) = f(m)f(n)s(m)s(n) = f(m)f(n)µ(m, n)s(mn)

1 = f(m)f(n)f(mn) µ(m, n)s0(m, n)

= µ0(m, n)s0(mn); 1 µ0(m, n) = f(m)f(n)f(mn) = (@1f)(m, n)µ(m, n);

µ0 = (@1f)µ; 1 1 1 ↵g s0 g mg = 0(m, g)s0(m) = ↵g f g mg s g mg 1 1 = ↵g f g mg ↵g s g mg 1 = (m, g)↵g f g mg s(m) 1 1 = (m, g)↵gfg mgf(m) s0(m); 1 1 0(m, g) = (m, g)↵g f g mg f(m)

= (m, g)(@2f)(m, g)

N G 2 The group Z(G, N, A) of pairs (, µ) A ⇥ Z (N, A) satisfying the natural cocycle identity is called{ the }characteristic2 ⇥ cocycle group and each element of the group

B(G, N, A) = @f = (@ f, @ f) : f AN 1 2 2 is called a coboundary. Thenthe quotient group:

⇤(G, N, A) = Z(G, N, A)/B(G, N, A)

9The subgroup i(A) / H is a subgroup of the center of H and G acts on H through ↵, but N does not act on A. 46 VON NEUMANN ALGEBRAS OF TYPE III is then called the characteristic cohomology group. If = [, µ] comes from the above equivariant short exact sequence E, then it is called the characteristic invariant of E and writtend (E) = [, µ] ⇤(G, N, A). 2 Coming back to the Aut(M) R-equivariant exact sequence: ⇥ E : 1 U(C) U(M) Cnt (M) 1, ! ! ! r ! we obtain the characteristic invarianet:

M = (E) ⇤mod ✓(Aut(M) R, Cntr(M), U(C)), 2 ⇥ ⇥ which is called the intrinsic invariant of M. Now we state the major result on the cocycle conjugacy classiﬁcation of amenable group actions on an AFD factor due to A. Connes, V.F.R. Jones, A. Ocneanu, C. Sutherland, Y. Kawahigashi, K. Katayama and M. Takesaki. Theorem 8.7. i) If ↵ is an action of a locally compact group G on a factor M, then the pull back of the intrinsic invariant:

↵ = ↵⇤(M) ⇤↵ ✓(G R, N, U(C)) 2 ⇥ ⇥ 1 is a cocycle conjugacy invariant where N = ↵ (Cntr(M)) / G, need not be a closed subgroup but a Borel subgroup. ii) If M is AFD and G is an amenable countable discrete group, then the combination:

1 mod, N = ↵ (Cntr(M)), ↵

Hom(G, Aut✓(C)) N ⇤↵ ✓(G R, N, U(C)) 2 ⇥ ⇥ ⇥ ⇥ is a complete invariant of the cocycle conjugacy class of the action ↵, where N is the set of all normal subgroups of G. Also every combination of invariants with natural restriction occurs as the invariant of an action of G. 8.3. Modular Fiber Space and Cross-Section Algebra. Let M be a § factor of type III. We have seen that the crossed product N = M o' R is semi-ﬁnite for each faithful semi-ﬁnite normal weight ' on M and that

M o' R ⇠= M o R for any other faithful semi-ﬁnite normal weight on M. Furthermore, if ✓ is the action on N dual to ', then we have

N o✓ R ⇠= M. We want to find a construction of the semi-finite von Neumann algebra N without fixing a faithful semi-finite normal weight ' on M. So let W0 = M. TAKESAKI 47

W0(M) be the set of all faithful semi-ﬁnite normal weights on M. Fix t R and consider a relation on the set M W by the following: 2 ⇠t ⇥ 0 (x, ') (y, ) x(D' : D ) = y ⇠t , t The chain rule of the Connes derivatives yields that the relation is an ⇠t equivalence relation on M W0. We denote the equivalence class [x, '] by x'it and set ⇥

M(t) = (M W )/ = x'it : x M, ' W . ⇥ 0 ⇠t 2 2 0 It is easily seen that the map: x M x'it M(t) is a bijection so that we can deﬁne the vector space structure2 7! on M2(t), transplanting that of M and also set x'it = x , x M, ' W . k k 2 2 0 it Then the bijection x M x' M(t) gives the dual Banach space structure on M(t). We2then deﬁne7! a binary2 operation from M(s) M(t) to M(s + t) as follows: ⇥

it is ' i(s+t) x' y' = x (y)' , s, t R, x, y M. t 2 2 Then set ˙ F = M(t) : t R : disjoint union. { 2 } [ We call this fibre space F the modular fibre space of M and write F(M) when we need to indicate F coming from M. We then define the involution on F: it ⇤ ' it it x' = t(x⇤)' , x' M(t). 2 Then we get

(x + µy)⇤ = ¯x⇤ + µ¯y⇤; (xy)⇤ = y⇤x⇤; x, y F, , µ C, 2 2 x⇤⇤ = x, ', W0, t R, 2 2 it it ' = (D' : D )t, whenever the sum and the product are possible. Now for each ' W0 deﬁne a map ' : M R F as follows: 2 ⇥ 7!

it '(a, t) = a' F, (a, t) M R. 2 2 ⇥ 48 VON NEUMANN ALGEBRAS OF TYPE III

Theorem 8.8. The maps ' : ' W0 have the following properties: { 2 } 1 i) On a bounded subset B M, the map is a homeomorphism ⇢ ' on B R relative to any operator topology on B except the norm ⇥ topology and the usual topology on R; 1 ii) On the entire space M R the map ' is a homeomorphism ⇥ relative to the Arens-Mackey topology ⌧(M, M )-topology on M and ⇤ the usual topology on R; iii) The Connes derivative takes the following form: 1 (D' : D ) = '(1, t), t R. t 2 In fact, we have, for each x M and t R, 2 2 1 '(x, t) = (x(D' : D ) , t) M R. t 2 ⇥ This theorem allows us to introduce a topology on the fiber space F by transplanting the product topology of ⌧(M, M )-topology on M and the usual ⇤ real line R through the map ' which does not depend on the choice of faithful semi-finite normal weight '. Remark 8.9. The norm topology on M cannot be tranplanted to F independent of the choice of ' W , because the Connes derivative 2 0 (D' : D ) : t R { t 2 } is not norm continuous. Definition 8.10. The ⌧-topology on F means the topology introduced by the last theorem based on the Arens-Mackey topology ⌧(M, M )-topology on M. ⇤ The above theorem also allows us to introduce a Borel structure on the fiber space F from the topology introduced above which is indepent of the choice of ' W . Consequently, we can consider a measurable cross-section 2 0 as well as an integrable cross-sections of F, i.e., a Borel map s R x(s) F such that 2 7! 2 x(s) M(s), s R; 2 2 x = x(s) ds, k k1 k k ZR recalling that each M(s) is a Banach space. Let 1(F) denote the set of all integrable cross-sections. It then follows that 1(F) is an involutive Banach algebra under the following structure: (x + µy)(t) = x(t) + µy(t);

(xy)(t) = x(s)y(t s)ds; ZR x⇤(t) = x( t)⇤; x = x(s) ds. k k1 k k ZR M. TAKESAKI 49

We call 1(F) the modular cross-section algebra for M and write 1(F(M)) when it is neccessary to indicate where it is from. To have a von Neumann algebra structure based on the bundle algebra, we need to construct a HIlbert space on which the cross-section algebra acts naturally. So we consider a von Neumann algebra M represented on a Hilbert space H. Let N be the opposit algebra (M0) of the commutant which makes the Hilbert space H an M N bimodule. We are going to construct a bundle of Hilbert spaces on whichF(M) acts from the left and F(N) acts from the right, where F(M) is the ﬁber space constructed above based on M and F(N) is the one based on N. 50 VON NEUMANN ALGEBRAS OF TYPE III

Lecture 9: Hilbert Space Bundle. 9.1. Spatial Derivative Recall, [Cnn8, Tk2, Chapter IX, 3]. Let §M, H be a von Neumann algebra M represented on a Hilbert space §H. Con- { } sider the opposit von Neumann algebra N = (M0) of the commutant M0. Let 0 be a faithful semi-ﬁnite normal weight on M0 and be the correspond- 2 ing semi-ﬁnite normal weight = ( 0) on N. Let L (N) be the standard 2 Hilbert space of N. We then consider the N-right module HN = L (N) HN. def We then consider the von Neumann algebra R = L HN , i.e., the von Neu- e mann algebra of all bounded operators commuting⇣with⌘the right action of N. Each elelment x of R has a 2 2 matrix representation:e ⇥ x x x = 11 12 x x ✓ 21 22 ◆ with components from the following spaces:

x N, x L L2(N) , H 10, 11 2 12 2 N N x L H , L2(N) , x L(H ) = M. 21 2 N N 22 2 N On R we deﬁne the balanced faithful semi-ﬁnite normal weight ⇢ out of ' W (M) of W (N) which is given by the following: 2 0 2 0 x x ⇢ 11 12 = (x ) + '(x ). x x 11 22 ✓ 21 22 ◆

def def 2 Set e = e11 and f = e22, i.e., the projection of H to L (N) and to H respectively which are of course projections in R such that e 2 eRe = N, eRf = L L (N)N, HN , fRf = M. We then set

2 n (H) = fn e = x L H , L (N) : (x⇤x) < + ; ⇢ 2 N N 1 def 1 D(H, ) = = ⇠ H : ⇠x C 0(xx⇤) 2 , x⇤ n for some C 0 . 2 k k  ⇠ 2 ⇠ n o For each x n (H), we have x⇤x N and with polar decomposition x = 2 1 2 uh, h = (x⇤x) 2 N, we set 2 ⌘ (x) = u⌘ (h) H. 2 Similarly, for each ' W (M), we set 2 0 2 n (H) = x L L (N) , H : '(x⇤x) < + . ' 2 N N 1 10 2 The notation L L (N)N, HN means the set of all bounded operators from H to L2(N) which commutes with the right action of N. M. TAKESAKI 51

Then set S ⌘ (x) = ⌘ (x⇤), x n (H) n (H)⇤; ', ' 2 \ ' 1 2 d' ', = S'⇤, S', = , d 0 which is a non-singular self-adjoin t positive operator on H such that

it it d' d' x = '(x), x M; d d t 2 ✓ 0 ◆ ✓ 0 ◆ it it d' d' 0 y = (y), y M0, d d t 2 ✓ 0 ◆ ✓ 0 ◆ d' where 0 W0(M0). We call the operator the spatial derivative of ' 2 d 0 by 0. The spatial derivative enjoys the following properties:

d' it d' it 1 = (D' : D' ) 2 ; d 1 2 t d ✓ 0 ◆ ✓ 0 ◆ 1 d' d = 0 . d d' ✓ 0 ◆ ✓ ◆ 9.2. Hilbert Space Bundle. We ﬁx a von Neumann algebra M, H and § { } set N = (M0) , the opposit von Neumann algebra of the commutant M0. Set

X = R W0(M) H W0(N). ⇥ ⇥ ⇥ For each t R we set an equivalence relation t by the following relation: 2 ⇠ (r , ' , ⇠ , ) (r , ' , ⇠ , ) 1 1 1 1 ⇠t 2 2 2 2 whenever ir1 ir2 d'1 d'2 ⇠1 = ⇠2(D 2 : D 1)t d d ✓ 1 ◆ ✓ 2 ◆ This is an equivalence relation in X. We then set

H(t) def= X/ ⇠t and denote the equivalence class [r, ', ⇠, ]t H(t) under this equivalence relation by the following: 2

ir i(t r) def ' ⇠ = [r, ', ⇠, ] H(t). t 2 Thus in H(t) we have

it it d' it ' ⇠ = ⇠ , ' W0(M), ⇠ H, W0(N); d 2 2 2 ✓ ◆ it it it d' ' ⇠ = ⇠. d ✓ ◆ 52 VON NEUMANN ALGEBRAS OF TYPE III

Observe the equivalence relation on X is generated by the subrelations: ⇠t it it ' ' (D' : D' ) , ' , ' W (M); 1 2 ⇠ 1 2 t 1 2 2 0 it it (D : D ) , , W (N); 1 2 ⇠ 1 2 t 1 2 2 0 it it it d' ' ⇠ ⇠. ⇠ d ✓ ◆ In the set H(t), we deﬁne the following vector space structure and inner product:

ir i(t r) ir i(t r) ir i(t r) ' ⇠ + µ ' ⌘ = ' (⇠ + µ⌘) ;

⇣ i⌘r i(t⇣ r) it i(t⌘r) ' ⇠ ' ⌘ = (⇠ ⌘), | ⇣ ⌘ which makes H(t) a Hilbert space and does not depend on the choice of ' W0(M) and W0(N) nor on r R. Each pair ' W0(M) and 2 W (N) give rise 2to unitaries from H on2 to H(t): 2 2 0 U (t)⇠ def= 'it⇠ H(t), ⇠ H; ' 2 2 V (t)⌘ def= ⌘ it H(t), ⌘ H. 2 2 Then we have d' it V (t)⇤U'(t) = . d ✓ ◆ We then deﬁne a multilinear map: M(r) H(s) N(t) H(r + s + t) as follows: ⇥ ⇥ 7!

(x'ir, 'is⇠, y it) M(r) H(s) N(t) x'ir'is⇠y it 2 ⇥ ⇥ 7! ir is it i(r+s) ' it x' ' ⇠y = ' (r+s)(x)⇠y i(r+s) d' i(r+s+t) = x ⇠r+s(y) . d ✓ ◆ This trilinear map does not depend on the choice of faithful semi-ﬁnite normal weight ' W0(M) and W0(N) and is associative whenever the consequtive pro2duct is possible. 2 Let G be the disjoint union of H(t):

˙ G = H(t) : t R { 2 } [ Now with the above preparation, we will construct a Hilbert space on which 1(F(M)) and 1(F(N)) act from the left and the right. Making use of the maps U'(t) : t R we can transplant the product topology of H R { 2 } ⇥ M. TAKESAKI 53 to G. Let H = 2(G) be the Hilbert space of square integrable cross-sections. To make the Hilbert space H a left and right bimodule over 1(F(M)) and 1(F(N)).e We need to discuss the opposit algebra of 1(F(N)). Recall N is deﬁned to be the oppositealgebra (M0) . We set the correspondence as follows: y N y M0; 2 ! 2 W (N) W (M0); 2 0 ! 2 0 (y) = (y), y N; 2 t (y ) = t (y) ; y = y, y N, = . 2

The -operation on F(N) and F(M0) is then given by the following:

it it it y = ( ) y = t (y)( ) , y N, W0(N). 2 2 So we set 1 1 y(t) = y( t), y (F(N)) (F(M0)), 2 [ and we get 1 1 (F(N)) = (F(M0)), the -operation is an anti-isomorphism of the modular cross-section algebra. Now we let 1(F(M)) and 1(F(N)) act from the left and right as follows:

(x⇠)(t) = x(s)⇠(t s)ds, x 1(F(M)); 2 ZR (⇠y)(t) = ⇠(s)y(t s)ds, y 1(F(N)). 2 ZR 1 Also (F(M0)) acts on H from the left:

1 ye⇠ = ⇠y, y (F(M0)), 2 which means that the action is given by the following:

(y⇠)(t) = (⇠y)(t) = ⇠(s)y(t s)ds ZR = ⇠(s)(y(s t))ds = ⇠(s + t)y(s)ds ZR ZR 54 VON NEUMANN ALGEBRAS OF TYPE III

Lecture 10: The Non-Commutative Flow of Weights on a von Neumann algebra M. Theorem 10.1. The von Neumann algebra M, H generated by the left 1 1 action of (F(M)) on H has the commutant Mn 0 genero ated by (F(M0)). e e Definition 10.2. The von Neumann algebra M on H generated by e e the left action of the modular cross-section algebra 1(F(M)) is called the canonical core of M, or simply the core of M. eOn thee Hilbert space 2(F(M)), we define a one parameter unitary group: ist 2 (U(s)⇠)(t) = e ⇠(t), ⇠ (F(M)), s, t R. 2 2 Theorem 10.3. For a von Neumann algebra M, H the core represnted on { } H through the above procedure has the following properties: i) The one parameter unitary group U(s) : s R defined above gives e { 2 } rise to a one parameter automorphism ✓s : s R of M such that the original von Neumann algebra is precisely{ the2 fixe} d point subalgebra M✓ of the one parameter automorphism group ✓. e ii) For each ' W0(M), there is a natural isomorphism ⇧' from M to 2 ' thee crossed-product M o' R, which conjugate the dual action on M o' R and ✓. e iii) The integral I✓ along the one parameter automorphism group ✓c:

I (x) = ✓ (x)ds, x M , ✓ s 2 + ZR is a faithful normal semi-ﬁnite M-valued eoperator valued weight and the compostion:

' = 'I✓ ' it corresponds to the dual weight ' on M o' R and = Ad ' , t t 2 R. e e iv) Viewing the logarithmic generator of 'it : t R , i.e., b 2 1 d ' = exp 'it , i dt ✓ t=0◆ the weight ⌧ deﬁned by the following: 1 1 1 ⌧(x) = ' ' x = lim' (' + ") 2 x(' + ") 2 , x M+, " 0 2 & ⇣ ⌘ is a faithfule semi-ﬁnite enormal trace on M. e v) The trace ⌧ does not depend on the choice of ' W (M) and scaled 2 0 by ✓: e s ⌧ ✓s = e ⌧, s R. 2 vi) 2 M o✓ R = M L L (R) . ⇠ ⌦ e M. TAKESAKI 55

Theorem 10.4. i) The correspondence: M M, ⌧, R, ✓ is a functor ! from the category of von Neumann algebra with isomorphismsn oas morphisms to the category of semi-ﬁnite von Neumann algebrase equipped with a trace scaling one parameter automorphism group ✓ with conjugation as morphism which transforms the trace one to another. ii) The correspondence M, H M, ⌧, R, ✓, H is a functor from the { } ! category of spatial von Neumann algebrnas with spatialoisomorphisms as morphisms to the category of semi-ﬁnite vone Neumanne algebras equipped with trace scaling one parameter automorphis group ✓ represented on a Hilbert space with spatial isomorphisms intertwining ✓ and ⌧.

Definition 10.5. i) The covariant system M, R, ✓, ⌧ for M is called the noncommutative ﬂow of weights on M.n o e ii) The group U(M) = u U(M) : uMu⇤ = M , the normalizer of M , 2 is called the extended unitaryn group of M. For oeach u U(M), we write e e 2 Ad(u) = Ad(u) Aut(M) e |M 2 and set f Cnt (M) = Ad(u) : u U(M) / Aut(M). r 2 n o Theorem 10.6. (Connes f- Takesakie Relative Commutant Theo- rem). Let M, R, ✓, ⌧ be the core of a von Neumann algebra M. Then the relative commutantn of oM in M is the center of the core M: e

M0 Me= C = The Center of M. e \ e e Theorem 10.7 (The Stability of Non-Commutative Flow). Suppose that M, R, ✓, ⌧ is a semi-finite von Neumann algebra equipped with a trace ⌧ scalingn one paroameter automorphism group ✓. Then the core of the fixed point Me = M✓ is conjugate to the original system. Every ✓-cocycle is a coboundary. e Definition 10.8. Each ↵ Aut(M) is extended to an automorphism 2 ↵ Aut(M) such that 2 e e ↵✓s = ✓s↵, s R, ⌧ ↵ = ⌧. 2 The restriction of ↵ toethe centereC of the core Me is called the module of ↵ and denoted by mod (↵) Aut(C) which commutes with the flow of weights 2 ✓ . e |C e 56 VON NEUMANN ALGEBRAS OF TYPE III

Theorem 10.9. The automorphims group Aut(M) is identiﬁed with the following subgroup Aut⌧,✓(M) of Aut(M):

Aut(M) = Aut (M) = Aut(M) : ✓ = ✓ , ⌧ = ⌧ . ⌧,✓ 2 s s n o In other words, each ↵, ↵ Aut(M), belongse to Aut⌧,✓(M) and if an element Aut (M) leaves each2element of M invariant, then = id. 2 ⌧,✓ Theorem 10.10 (Characteristice Square). To each factor M there corresponds a commutative square of equivariant exact sequences relative to the action of Aut(M) R: ⇥ 1 1 1

? ? @ 1 ? 1 T? U?(C) B (R?, U(C) 1 ! y ! y ! ✓ y !

? ? @✓ 1 ? 1 U(?M) U(?M) Z (R,?U(C)) 1 ! y ! y ! ✓ y ! Ade ˙ ? f ? @✓ 1 ? 1 Int?(M) Cnt?r(M) H (R?, U(C)) 1 ! y ! y ! ✓ y !

? ? ? ?1 ?1 ?1 y y y where C is the center of the non-coummutative ﬂow of weights M, R, ✓, ⌧ of M and U(M) is the extended unitary group of M. n o e e M. TAKESAKI 57

Extended Lecture I: The Non-Commutative Flow of Weights on a von Neumann algebra . Continued. E.1. Grading and Lp(M), 1 < p + and Duality. First we need a  1 definition:§  1 Definition E.1. Let N, ⌧, H be a semi-finite von Neumann algebra equipped with a faithful semi-finite{ } normal trace ⌧ represented on a Hilbert space H. A densely defined closed operator T is said to be aliaed to N if

uT u⇤ = T, u U(N0). 2 It is said to be ⌧ -tamed if

1 2 lim ⌧ [,+ ) (T ⇤T ) = 0, + 1 ! 1 ⇣ ⇣ ⌘⌘ where [,+ ) means the characteristic function of the half line [, + ) R. We will write1 1 ⇢

E( T ) = [,+ )( T ), | | 1 | | Note that (1 E ( T )H D(T ). | | ⇢ [>0 Lemma E.2. Suppose en; n N and fn : n N are decreasing sequences of projections in N such{ that 2 } { 2 }

lim ⌧(en) = lim ⌧(fn) = 0. n n !1 !1 Then

lim ⌧(en fn) = 0. n !1 _ Let M(N, ⌧) be the set of all ⌧-tamed operators aliated to N on H. The following facts makes the significance of the ⌧-tame property: i) A ⌧-tame operator T has no proper extension as a closed operator: a property called hypermaximality. ii) The sum S + T and the product ST are both preclosed and their closures are also aliated to N and ⌧-tamed. iii) The adjoint T ⇤ of a ⌧-tamed operator is also ⌧-tamed. iv) Consequently the set M(N, ⌧) of ⌧-tamed operators forms an involutive algebra over C if we replace the sum and the product of ⌧-tamed operators by their closurs. v) If the trace ⌧ is finite, then every densely defined closed operator aliated to N is ⌧-tamed. 58 VON NEUMANN ALGEBRAS OF TYPE III

Example E.3. Let N = L1(R) and ⌧ be the trace obtained by the integration relative to the Lebesgue measure on R. We then represent N on L2(R) by multiplication. Each densely deﬁned closed operator T aliated to N is easily identiﬁedd with the multiplication operator by a measurable function f on R relative to the Lebesgue measure. Now T is ⌧-tamed if and only if

ess. lim sup f() < and ess. lim sup f() < , | | 1 + | | 1 !1 ! 1 So the mutiplication operator given by the identity function: R R 1 2 ! 2 is not ⌧-tamed while the one by R R, = 0, is ⌧-tamed, where the singular point 0 of the function2does!not matter2 as6 it has measure zero.

Let M, R, ✓, ⌧ be the canonical core of M = M✓. Each faithful semi- ﬁnite normaln weighot ' W0(M) gives rise to a one parameter unitary group it e 2 e ' : t R in U(M) U(M) as seen already. Then we view ' as a densely 2 ⇢ deﬁned self-adjoin t positive operator aliated to M. The trace ⌧-scaling automorphism groupe ✓ transformse 'it in the following way: e it ist it ✓s ' = e ' , s, t R; 2 s ✓s(') = e '

We denote by A' by the von Neumman subalgera of M generated by the one parameter unitary group 'it : t R . The uniqueness of the Heisenberg covariant system implies the existence2 of a unique isomorphisme ⇡' from the covariant system L1(R), R, ⇢ to the covariant system A', R, ✓ conjugat- ing ⇢ and ✓, where{ } { }

(⇢sf)(x) = f(x + s), f L1(R); 2 ' 1 x (⇡ ) (') (x) = e , s, x R, R+. 2 2 ⇣ ' ⌘ E(')(= ⇡ ( , log] , 1 Proposition E.4. The restriction of the trace ⌧ to A' either purely inﬁnite or semi-ﬁnite which corresponds to the integration on R shown below:

' x ⌧(⇡ (f)) = f(x)e dx, f L1(R). 2 ZR Furthermore, ' is ⌧-tamed if and only if ' is bounded, i.e., '(1) = ' < + . k k 1 Remark E.5. So far we discussed only faithful semi-ﬁnite normal weights. But we extend the above theory to non-faithful semi-ﬁnite normal weights with a little bit of extra work by considering the reduced von Neumann algebra M. TAKESAKI 59

it Ms('). So we then identify the set ' : t R of one parameter set of partial isometries such that 2 is it i(s+t) ' ' = ' , s, t R; 2 is is is is ' ' = ' ' = s('), s R. 2 With this remark, we may discuss the ⌧-tamed operator ' aliated with M which corresponds to a normal positive functional on M, i.e., an element of M+. ⇤ e Defnition E.6. For each ↵ C and a densely deﬁned closed operator T 2 aliated to M, we say that the operator T has grade ↵ if

↵s e ✓s(T ) = e T, s R, 2 and write ↵ = grad(T ). We then set

M(↵) = T ⌘ M : grad(T ) = ↵ and T M(M, ⌧) . 2 n o e e The algebra M M, ⌧ is “graded”. Now we can ⇣state ⌘the signicance of the canonical core: e Theorem E.7. The grading on M(M, ⌧) has the following propeties: i) If ↵ < 0, then M(↵) = 0 and M(0) = M. < { } e 1 ii) If p = ↵ > 0, then for every T M(↵) the 1/p-th power T p of the absolute< value of T is given by2' M+. | | 2 ⇤ iii) For any a M with I (a) = 1, 2 + ✓ 1 1 1 e a 2 M(1)a 2 L M, ⌧ , ⇢ ⇣ ⌘ 1 1 e the value ⌧ a 2 T a 2 is independent of the choice of such a. We will write this value⇣ T⌘for T M(1). 2 + iv) The positive part M(1)+ of M(1) is precisely the positive part M of R ⇤ the predual M when M+ is embedded in M1 M, ⌧ . ⇤ ⇤ If ' M(1), ' M+, then ⇣ ⌘ 2 2 ⇤ e ' = '(1) = '. k k Z The duality between M and M is given by the integral: ⇤

xT = x, T , x M, T M(1). h i 2 2 Z 60 VON NEUMANN ALGEBRAS OF TYPE III

Thus we conclude that M = M(1). ⇤

Discussion of the Theorem. Suppose h M(1)+ and e = s(h). Setting hit(1 e) = 0, we consider, with ' W (2M) ﬁxed, 2 0 it it u(t) = h ' M. 2 Then we have

' u(s + t) = u(s)s (u(t)), s, t R; ' 2 u(s)⇤u(s) = s (e) and u(s)u(s)⇤ = e.

So there exists ! W(M) such that 2 it it (D! : D') = u(t) = h ' , t R. t 2 We then have

1 1 !(x) = !(I (x)) = ⌧ h 2 xh 2 , x M . ✓ 2 + ⇣ ⌘ e Consider the vonbNeumann subalgebra A of Me generated by h. Then it is identiﬁed with L1(R) with the action ✓s given by the following: e (✓ (f))(p) = f(p + s), f A. s 2 The ⌧-tamed property of h and the trace scaling property of ✓ yield that ⌧ is semi-ﬁnite on A and ⌧ is given by the integral:

⌧(f) = C f(p)epdp, for some constant C > 0; R Z p h(p) = e ;

1 1 !(f) = ⌧ h 2 fh 2 = C f(p)dp. ⇣ ⌘ ZR The semi-ﬁniteness ofb ⌧ on A guarantees the existence of the conditional expectation E from Me to A. Futhermore, the scaling property of ⌧ under ✓ implies that E commutes with the ﬂow ✓, i.e., for every f A we have 2 e 1 ⌧(E(✓s(x))f) = ⌧(✓s(x)f) = ⌧ ✓s x✓s (f) s 1 s 1 = e ⌧ x✓s (f) =e ⌧ E(s)✓s f s 1 = e ⌧✓s (✓s(E(x))f) = ⌧(✓s(E(x))f);

E(✓s(x)) = ✓s(E(x)), x Me. ) 2 e M. TAKESAKI 61

Let e0 = [0,1] L1(R) = A. Then we have 2

I✓(e0)(p) = [0,1](p + s)ds = 1; ZR !(e ) = C e (p)dp = C < + ; 0 0 1 ZR !(e ) = !(I (e )) = !(e) = !(1) < + . 0b ✓ 0 1 Suppose a M and I (a) = 1. Then we have 2 + b ✓

e1 = E(I✓(a)) = E ✓s(a)ds = E(✓s(a))ds ✓ZR ◆ ZR

= ✓s(E(a))ds = (E(a))(p + s)ds. ZR ZR 1 1 1 Hence we have E(a) L (R). Now we evaluate the trace of a 2 ha 2 in the following: 2

1 1 1 1 ⌧ a 2 ha 2 = ⌧ h 2 ah 2 = !(a) = !(I (a)) = !(1) < + . ✓ 1 ⇣ ⌘ ⇣ ⌘ 1 1 1 1 1 Thus we conclude that a 2 M(1)a 2 bL M, ⌧ and the value ⌧ a 2 ha 2 is ⇢ independent of the choice of a M with⇣ I (a⌘) = 1. ⇣ ⌘ 2 + e ✓ ~ Theorem E.8. For ↵ C with p = ↵ 1, the quontity: 2 e < p 1 T = T p , T M(↵), k k | | 2 ✓Z ◆ makes M(↵) a Banach space. Unlike the usual integration, the intersection of di↵erent M(↵)’s is 0 . So the norm T is deﬁned only for one value of ↵ for a homogenous T { M} relative to thek gradingk given by ✓. 2 Theorem E.9. If T , , T M M, ⌧ and { 1 · · · n} ⇢ ⇣ ⌘ grad(T ) + + grad(T ) = 1, 1 · · · e n then T T M(1) and 1 · · · n 2 T T T T ; k 1 · · · nk  k 1k · · · k nk

T1 Tn = T2 Tn 1T1. · · · · · · Z Z Theorem E.10. If ↵ C satisﬁes the inequality 1 ↵ 1, then Banach spaces M(↵) and M(1 2 ↵) are in duality under thebiliner< form: S, T = ST, S M(↵), T M(1 ↵). h i 2 2 Z 62 VON NEUMANN ALGEBRAS OF TYPE III

E.2 Local Characteristic Square. For each non-zero semi-ﬁnite normal § ' weight ' W, let ⇡ be the equivariant isomorphism of L1(R), R, ⇢ into 2 { } the non commutative ﬂow of weights M, R, ✓, ⌧ such that ' corresponds x to the function x R e R+ andnset o 2 7! 2 e ' ' ' ' A = ⇡ (L1(R)), D = A C. _ Proposition E11. We have the following:

' (A )0 M = M ; \ ' D' M C = The center of M ; \ ⇢ ' ' ' ' ' D , R, ✓ = (M D ) A , R, id ✓ . { } { \ ⌦ ⌦ }

1 C = L1(X, µ), (✓s(a))(x) = a T x , x X, s R. s 2 2 1 c(s + t; x) = c(s; x)c t; Ts x ' p A = L1(R), '(p) = e (✓s(f))(p) = f(p + s), f L1(R); 2 (p+s) s (✓s('))(p) = e = e '(p).

Lemma EII.1. Suppose that A1 and A2 are two commutative von Neumann algebras and that ⇡1 and ⇡2 are faithful normal representations of A1 and A2 such that ⇡ (A ) ⇡ (A )0. 1 1 ⇢ 2 2 Then there exists a measure µ on X = X X such that 1 ⇥ 2 i) L1(X X , µ) = ⇡ (A ) ⇡ (A ); 1 ⇥ 2 ⇠ 1 1 _ 2 2 ii) The projections pr1 and pr2 from X to the respective cooridnates carry the measure µ into µ1 and µ2 of X1 and X2 respectively.

1 For each c Z (R, U(C)) and setting 2 ✓ (b (c))(p, x) = c(p, T x), b D', ' p 2 we compute

1 (@✓b'(c))t(p, x) = b(p, x)b p + t, Tt x 1 = c(p, Tpx)c p + t, Tp+tTt x

= c(p, Tpx)c(p + t, Tpx) 1 = c(p, Tpx)c(p, Tpx)c t, Tp Tpx = c(t, x). M. TAKESAKI 63

Hence (@ b) U(C), so that b (c) U(M) and ✓ t 2 ' 2 c(t, ) = (@ b) . · e ✓ t 1 ' Observe that the map b' : c Z✓(R, U(C)) b'(c) U(D ) is a group homomorphism. 2 7! 2 We now observe the following cobounary operation:

it it it ist @✓ ' (s) = ' ✓s ' = e , s, t R 2 Setting ist c (s; x) = e , x X, t 2 1 we have a cocycle ct Z (R, U(C)). We then check b'(ct) in the following: 2 ✓ ipt it (b'(ct))(p, x) = ct(p, Tpx) = e = ' (p), so that (b (c )) = 'it A'. ' t 2 Theorem E.12. To each faithful semi-ﬁnite normal weight ' W0(M) 1 ' 2 there corresponds a right inverse: c Z✓(R, U(C)) b'(c) D U(M) of ' 2 7! 2 \ the coboundary map @ : u D U(M) @(u)s = u⇤✓s(u), s R such that 2 1 \ 7! 2 e i) @✓(b'(c)) = c, c Z✓(R, U(C)). 1 2 e ' ii) b' : c Z✓(R, U(C)) b'(c) U(M)) D is a continuous homomorphism.2 7! 2 \ iii) e it b'(ct) = ' . iv) For every ↵ Aut(M), we have the covariance: 2 1 b 1 = ↵ b ↵ '↵ ' v) If ' W (M) is dominant, then 2 0 e e ' Ad(b'(c)) = c the extended modular automorphism. vi) For a pair ', W (M) of dominant weight, we have f 2 0 1 b'(c)b (c)⇤ = (D' : D ) U(M), c Z (R, U(C)), c 2 2 ✓

and for a general pair ', W0(M), the left side of the above ex- pression belongs to U(M) and2 therefore we write

def 1 (D' : D ) = b'(c)b (c)⇤, c Z (R, U(C)). c 2 ✓ 1 vii) For a pair c1, c2 Z (R, U(C)), we have 2 ✓ (D' : D ) = (D' : D ) (D' : D ) . c1c2 c1 c1 c2 64 VON NEUMANN ALGEBRAS OF TYPE III

1 Definition E.13. For each ' W0(M) and c Z✓(R, U(C)), the automorphism 2 2 ' = Ad(b (c)) Aut(M) c ' 2 is called the extended modular automorphism corresponding to a f cocycle 1 c Z✓(R, U(C)). 2 ✓ Definition E.14. i) The group deﬁned by the folloing:

Mod'(M) = Ad(u) : u D' U(M) 2 \ n o is called the extended modularfgroup of ' We0(M). ii) The abelian von Neumann subalgebra D2 = D' M is called the ' \ strong center of the centralizer M', ' W0(M). ' 2 iii) We set Mod0 (M) = Ad(u) : u U(D') . With these notations, w{e get the follo2 wing lo}cal characteristic square: Theorem E.15 (Falcone-Takesaki). To each faithful semi-ﬁnite normal weight ' W , there corresponds a commutative square of exact sequences: 2 0 1 1 1

? ? @ 1 ? 1 T? U?(C) B (R?, U(C)) 1 ! y ! y ! ✓ y !

? ' ? @✓ 1 ? 1 U(D? ') U(D )? U(M) Z (R,?U(C)) 1 ! y ! y\ ! ✓ y ! e ˙ ?' ?' @✓ 1 ? 1 Mod?(M) Mod?(M) H (R?, U(C)) 1 ! y0 ! y ! ✓ y !

? ? ? ?1 ?1 ?1 y y y and

' 1 U(D ) U(M) = U(D') Z (R, U(C)) by the isomorphism b'. \ ⇠ ⇥ ✓ DefinitioneE.16. We call the last square the the local characteristic square. Theorem E.17 (Connes-Kawahigashi-Sutherland-Takesaki). If R is an AFD factor, then the auotomorphism group Aut(R) has the following normal subgroups: i) Ker(mod) = Int(R). M. TAKESAKI 65

ii) Let C be the set of all bounded sequences xn `1(R, N) such that { } 2

lim !xn xn! = 0, ! M . n ⇤ !1k k 2

Then group Cntr(R) is precisely the group of all automorphisms ↵ Aut(M) such that 2

-strong⇤ lim (↵(xn) xn) = 0 for every xn C. n !1 { } 2 iii) The exact sequence:

1 Ker(mod) i Aut(R) mod Aut (C) 1 ! ! ! ✓ ! splits. Hence we have

Aut(R) ⇠= Ker(mod) o Aut✓(C).

Note The last assertion (iii) was proven by R. Wong in his thesis but never published. So the proof was presented in [ST3]. E.3. A Factor of Type III : The Mother of Factors of Other § 1 Types.. Suppose that M is a separable factor of type III1. Then its core M is a factor of type II equipped with a trace scaling one parameter automorphism group ✓. 1 e Theorem E.18. In the above setting, to each closed subgroup T Z, T > 0, of ✓T T the addtivie group R a factor M of type III with = e corresponds. If T1Z T2Z, i.e., if T1/T2 Z, then we have ⇢ 2 e ✓ ✓ ✓ M T1 M T2 M = M . · · · Let A, R, ↵ is an ergodic ﬂow. We represent the covariant system { } e e e C, R, ↵ by an ergodic ﬂow of non-singular transformations: { } 1 C = L1(X, µ), (↵s(f))(x) = f T x , s R, x X, f C; s 2 2 2 µ(N) = 0 µ(T (N)) = 0, N X; , s ⇢ dµTs ⇢(s, x) = (x), s R, x X. dµ 2 2

Now consider the tensor product: N = C M and view it as the von Neu- ⌦ mann algebra L1 M, X, µ of all essentially bounded M-valued measurable e functions on X. W⇣e are going⌘ to build a one parameter automorphism group ˜ e e ✓s : s R which scales a semiﬁnite normal trace on N down and the re- 2 strictionn ofo✓˜ to C is exactly the given ergodic ﬂow ↵. Set

˜ 1 ✓s(a) (x) = ✓s log⇢(s 1,x) a Ts x , a N = L1 M, X, µ . 2 ⇣ ⌘ ⇣ ⌘ e 66 VON NEUMANN ALGEBRAS OF TYPE III

We then compute for a N : 2 +

1 ˜ 1 (⌧ µ) ✓s(a) = ⌧ ✓s log⇢(s ,x) a Ts x dµ(x) ⌦ X ⇣ ⌘ Z s 1 1 = e ⇢ s , x ⌧ a Ts x dµ(x) ZX s 1 1 = e ⇢ s , x ⌧ a Ts x dµ(x) ZX s = e ⌧(a(x))dµ(x) X sZ = e (⌧ µ)(a). ⌦ Therefore, the one parameter automorphism group ✓˜ transforms the trace ⌧˜ = ⌧ µ exactly in the way we want. Hence the covariant system N, R, ✓˜, ⌧˜ ⌦ ✓˜ is the core of the factor M0 = N . The flow of weights on M0 isnexactly theo one we started, i.e., C, R, ↵ . If the flow C, R, ↵ is properly ergodic, then { } { } the factor M0 is of type III0. Theorem E.19. Every erogic flow C, R, ↵ appears as the flow of weights { } on a factor of type III0 . If the original factor M is AFD, then the factor M0 constructed above is also AFD. The flow of weights on an AFD factor of type III 0 is a complete algebraic invariant. Hence e all AFD factors are isomorphic to the ones obtained through the above construction.

Fix an ergodic ﬂow C, R, ✓ . Let Aut✓(C) be the group of automorphisms { } of C commuting with ✓. Each ↵ Aut✓(C) gives rise to a non-singular transformation T of the measure space2 X, µ : ↵ { } 1 (↵(a))(x) = a T x , a C, ↵ Aut (C). ↵ 2 2 ✓ We have also the ⇢-function:

dµ T ⇢(↵, x) = ↵ (x), ↵ Aut (C). dµ 2 ✓

We then set 1 (↵(a))(x) = ✓ log ⇢(↵ 1x) a T↵ x . ( , ) Then the map: ↵ Aut (C) ↵ Aut (N )is an injectiv e homorphism e ✓ ⌧,✓ and gives rise to an2automorphism7! 2↵¯ Aut(M ) such that mod(↵¯) = ↵. 2 0 e Theorem E.20. The group Aut(M0) is splits as the semi-direct product:

Aut(M0) ⇠= Ker(mod) o Aut✓(C). M. TAKESAKI 67

Note on the Paper of R. Wong. For an AFD factor R of type III0, the above result was proven by R. Wong in his thesis, although his method was much more based on the ergodic theory. But his papaer was never published. So the proof was given in [ST3]. The reason why his paper was never published was that when his paper was ready to go printing, the Transaction of American Mathematical Society needed his agreement on the copy right, which was never sent back to the AMS oce: he had left the Univ. of New South Wales before the copy right signature request was sent to him. His supervioser was Prof. Colin E. Sutherland. Even he was unable to trace whereabout of Dr. R. Wong. Probably he had left the UNSW in a despair on the academic job market at that time. 68 VON NEUMANN ALGEBRAS OF TYPE III

Concluding Remark Now we come to the end of the lecture series. As seen through the lectures, the theory of von Neumann algebras is very much like a number theory in analysis. There are many mathematical problems which associate operators on a Hilbert space. The theory of operator algebras looks at the enviroment of the operators in quesion and give a guide how to attack the problem. Namely, the theory of operator algebras will give us the framework for how to attack the problem. For instance, the unitary operators on L2(R) of the translation operator by one and the mutiplication operator by trigonometric funcions discussed Example 7.12 is not exotic at all and has a close relation to di↵erence equations. I believe that the theory of operator algebras has been sucently developped that one can now apply them in many areas of mathematics and mathematical physics. One should try to rephrase the existing mathematical problems in terms of operator algebras. The noncommutative geometry of Alain Connes is exactly one of these attempts.

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