1 OD eT a © Copyright by INCREST, 1982
T-THEOREM FOR L’-SPACES ASSOCIATED WITH A VON NEUMANN ALGEBRA
HIDEKI KOSAKI
0 INTRODUCTION
Following the development of the Tomita-Takesaki theory [13], Araki [1], introduced a family of positive cones P?.0 < « < 5 , associated with a von Neumann algebra @ admitting a cyclic and separating vector (See § 1.) In [9], [10], we observed that the cones are closely related to (the positive parts of) associated non-commutative L’-spaces, | < p < co, [2], [8], [10], [11] In fact, the surjectivity of the map: €e€ P*++@,¢.%; is exactly the validity of a “7-theorem” for the L?-space with œ = > — ; - which will be made precise in § 2 Also, for finite von
p
Neumann algebras, this T7-theorem holds always The main purpose of the paper, however, is to prove the converse Namely, we shall prove that the validity of this “T-theorem”’ for the L’-space, 2 < p < oo, implies the finiteness of the algebra.” in question (For p = oo, this is known as the 7-theorem due to Dye, Murray-von Neumann {5].) For a factor of type either I,, or Iil,, 0 < 2 < 1, we shall Prove a slightly stronger result
We shall freely use the basic facts and notations of (relative) modular theory [3], [13], and non-commutative 7-spaces [2], [8], [10], [11]
The author is indebted to Dr Christian Skau for some ideas in the paper Also, the present research was done while he was at the University of California, Los Angeles, and he thanks Professor Masamichi Takesaki for constant encou- ragement
1 PRELIMINARIES
in this section, we collect some basic definitions and properties partially to
Trang 2268 HIDEKI KOSAKE
Let C4, #, J, 2) be a standard form ƒ6] For each e.Z‡ , there correspondS 3a unique implementing vector in PK , which will be denoted by ¢,, that is,
W(x) = @; (x) = (XếuÌlếu), xe.# For a faithful @ and an arbitrary y, the
1
admits the polar decomposition JA;, Here the positive self-adjoint operator đụ, ¡s known as the relative modular operator In case @ = Ứ, 4„„== 4„ is cxactly the usual modular operator determined by the pair (CW, @) in the sense of [I3] The partial isometry (Dy: De), = 7 »tER, in @ 1s known as the Radon-Nikodym cocycle (of y with respect to @), [3]
, 1 ;
Derrnirion 1.1 (Araki [I] For 0 < «< 5? and a faithful pe.@3 , PZ is the closure of the positive cone 4%.4@.¢€, in #
1
in the literature, P? and P2 are denoted by P} and PP respectively We also
1
note that P; is exactly the natural cone z* (for any ø@) In [9], we showed the following Radon-Nikodym type theorem:
v Tá aas i
THEOREM 1.2 The map: [€ PZ+>a;¢.W@,; is bijective for O<a< 40 This
1
is also the case for O<4%< > provided that @ is finite
In [10] we also observed that the cones are closely related to Connes-Hil- sum’s L?-spaces, |<p<oo, [2], [8], which are isomorphic to other L’-spaces, [7], [11] In our set up, their L?-spaces are described as follows:
DEFINITION 1.3 For a faithful gp ¢.@, , we set g’(x’) = (x'€,'¢,), WE.’
so that o’ belongs to #@* For each lI<p < co, Connes-Hilsum’s L?-space L?(4; 0’) consists of all densely-defined closed operators T on # , with the polar decompositions T = u!T', satisfying the following three conditions :
(i) w belongs to 4;
(ii) (homogeneity) T x’ = o@',,(x')|\T\"; te R, x’ €.@’, where o®’ is the modular automorphism group on @’ associated with 9’ ;
P (iii) (integrability) €, belongs to Z(T;”)
(Also, we set L°(@; gp’) =.@.) Uf the above operator T satisfies (i) and (ii), T is
1
Trang 3T-THEOREM FOR L”-SPACES 269 Notice that 7' being affiliated with Ƒ®(⁄; p’) in the above sense is equivalent to
T being affiliated with.@ in the classical sense We also notice that the positive part (as
2
operators) of L°C4, ’) is exactly the set of all Af,, wed
2 T-THEOREM FOR L?-SPACES
Setting up our “‘7-theorem’’, we state our main results, which are the converses of the 7T-theorem
We recall the next result in [10]
THEOREM 2.1 (Theorem 4.1, [10]) Let @ be a fixed faithful functional in
Ms and2<p<oo For each west, the following four conditions are all equivalent :
1 1
(i) the (densely-defined) operator AjgA2 1 * is closable ,
(ii) there exists an operator T affiliated with LP(⁄; @') such that TE, = €,
(with €, € B(T)),
11 1_ 1
(iii) (resp (v)) there exists a vector in Pp?» (resp PAR” +)) salisfying
Úý(x) = (xé | É), x e./
Before going further, we remark that in the above theorem the condition (i) can also be mentioned differently Namely, (i) can be replaced by (i)’: there exists
1 aL
an operator A affiliated with L?(.@; 0’) such that €,¢ D(A) and AAZ ” is a
1
densely-defined closable operator whose closure is đậy In fact, G)’ immediately
2 3424
implies (ii) Also, if one assumes (i), 4 =(Ayo42 7) is affiliated with L°.7; 9’)
1 1
and €,¢ Q(A) Furthermore, it is easy to prove that Ad; ? is densely-defined
1 1
and closable (See the proof of Theorem 4.1, [10].) Noticing that (44? 7)” is
von
(- 4.}-nomogencous (2D and that (44Z£ 7)” &, == €y, ome concludes that
Ad +
(A42 7)“ = Age
COROLLARY 2.2 For @ and p as ín the above theorem, the following four con- ditions are equivalent:
1 1 1
Trang 4270 HIDEKI KOSAKI
(i) for any ¢ €#, there exists an operator T affiliated with L?(.@;0') such
that €=Té,,
A002 221
(iii) (resp (iv)) the map: Ee Pp” (EE DAP *)) +a, €.E is surjective
Proof If one had Pf" instead of # in (ii), the corollary would be a trivial con- sequence of the theorem We thus have to prove (ii) by assuming that (ii) with 2”
instead of # is valid
Let ¢ be an arbitrary vector in # with the “polar decomposition” € == u ;¢' in the sense of [1] and [6] so that w is a partial isometry in with w*u = LWEl] and lé' e2”, The vector ¡Š; belongs to P" so that there exists an operator T affiliated with 7; »’) such that Té, =: '¢| (Notice that we do not know if uT is closable because 7 may or may not belong to L(Z; @.)
Let “, be the set of “smooth” elements +’ €.#’ with respect to a’, that is,
zt» of? (x’) is an “’-valued entire function We claim that the restriction of uT to Ad 9 is closable To prove it, let us assume that a sequence {x,é,} in @9¢, tends to
Tuy ~ 1
0 and that {w7x,¢,}, is Cauchy Since T is (- ~~ | romogencous (with respect P
to 9’),
Txyế, = 0% (ea) Ey = 0% OIE,
Dp P
belongs to the initial space of u so that ty} 1s also a Cauchy sequence Thus
the closedness of 7 impHes that lim 7 Xzế„ == 0 so that lim UT X;ŠG = 0, H
Let T be the closure of the restriction of u7 to ⁄g6„ By construction, ế„c€ A(T) and Té, = ¢ Furthermore, Lemma 2.1 in [10], guarantees that T is
affiliated with L°(Z; 0’) Q.E.D
In the above corollary, we assumed that 2 < p < co so that the exponent 1 ae of A, appearing in (i) is strictly negative On the other hand, we know 1 ?
Lorin 1 1.1
that Az AP
1
¬ > > 0, that is, p<2 (See [8].) Thus, combining Corollary 2.2 and the
P
second half of Theorem 1.2, we have:
* is closable and (43,4? 7) belongs to #(Z; @') whenever
THEOREM 2.3 Let @ be as in the theorem;
Trang 5T-THEOREM FOR L?-SPACES 271 (ii) d<p< 2, 4 is arbitrary) For any €€H there exists a (unique) T in LM; 0') satisfying §=TE,
We remark that, for the special value of p = oo, the statement (i) of Theorem 2.3 is exactly the celebrated 7-theorem of Murray-von Neumann
DEFINITION 2.4 For a pair (4%, @) consisting of a von Neumann algebra #
and a faithful functional in Wf, we say that the T-theorem for L*(4; 0’) holds,
if one (hence all) of the four conditions in Corollary 2.2 is satisfied
Although we fixed a standard form at the beginning of § 1, everything there- after does not depend on the choice of our standard form so that the above defi- nition is legitimate Thus, the above Theorem 2.3, (i), can be rephrased as follows:
THEOREM 2.3’ (T-theorem for L?-spaces, 2 < p< 00) If M is finite, the T-theorem for L°(M; 0’) holds for each faithful functional in ME and 2 < p<co
As stated in the introduction, the main result of the paper is the converse of ‘the above “T-theorem’’ For the special value of p = co, the converse was proved by Dye [5] (See also [12].) We thus assume 2 < p < 0 in the rest of the paper, unless the contrary is stated In the rest, we will concentrate on the proofs of the following two theorems, which are the main results of the paper:
THEOREM 2.5 If, for any faithful pe “= (and a single p, 2 < p<oo), the T-theorem for L?(M; `) holds, then M is finite
THEOREM 2.6 Assume that M is a factor of type either \,, or I11,,0 <4 < 1, and that p e ]2, oo] Then the T-theorem for L°(.4@; @) holds for no @
3 TECHNICAL LEMMAS
To prove the two main results, we prepare some technical lemmas We prove
the stability of our “T-theorem” under certain perturbations and a normal pro-
jection of norm 1
LEMMA 3.1 For an automorphism « of M, the T-theorem for L°(M; 9’) holds if and only if the same is true for L?(M; (po «)’)
Proof Let « = Adv* be the canonical implementation, [1], [6] As a conse-
quence of the uniqueness of the polar decomposition, it is shown that
A Lilt + ALL
Ayo = VU" Ayea, goat, 0, WEME One thus obtains Aj942 ° =ø*Aÿ4s,s.e A?.v 20
so that the result follows from Corollary 2.2 (i) Q.E.D LEMMA 3.2 Assume that faithful ọ, W in “jy satisfy o<lw and W<i,@ with
some 1,, 1,20 For any — 5 <a <<: , (Az) is exactly 2(4ÿ)
Trang 6272 HIDEKI KOSAKE
Before going to its proof, we recall that z ~ (De: DW), €.@ is bounded and
1 ¬
(z-weakly) continuous on — > <Imz<0 and analytic in the interior if and only
if @ < lp for some 1/20 (See U Haagerup, Operator valued weights in von Neumann
algebras I, J Functional Analysis, 32(1979), 175 — 206; Lemma 3.3.) Thus, when
o<lw and <i;o, both of (Dọ: Dự), and (Dự: Do), (- + <Im:<0] make
sense as elements in./# and satisfy
(Dg: Dy), (Dy: Dọ), =: (Dự: De), (De: Dy), = 1
because of the uniqueness of analytic continuation (For z= te R, this equality is well-known.) In other words, (Dg :Dw),¢.@ is invertible and its inverse is
(Dy : Do), (Furthermore, due to (Do : Dy), = (De : DW);?)* = (Dw : De),
1
the above mentioned facts remain valid for |Imz|< >?
Proof of Lemma 3.2 At first we remark (i) J457 = 4¿* (and J4ÿJ = 4ÿ°),
(ii) J4š,Ƒ= AzZ, 1 1
(1) ⁄⁄Zš„ is a common core for 4¿ and Ayo
In fact, (i) is well-known, [13], and (iii) is obvious from the construction Also,
(ii) follows from (i) and the well-known 2 x 2-matrix argument Due to (i), we may and do assume that O<a< >in what follows We now consider the following two functions (for each x €.4):
Zo Ay Xo z — (Dy : Dg), AExé,
i 1
Because of x€,€ G(Aye) D(dg) and the remark before the proof, these two
1
functions are bounded and continuous on — > <JImz<0O and analytic in the interior For z= fe¢R, one obviously has
(Dy : Dg), AU xé, = AY AG MAUKE, = Avpx€, -
Trang 7T-THEOREM FOR L?-SPACES 273 It is well-known that “€, isa common core for Ay, and 42 (01) and Lemma 4, [1)) Furthermore, (Dw : De)_, is invertible (so that Mé, is acore for (Dy : Dg)_j,4§ Therefore, the above equality actually means
= (Dự : Dọ)_;„4$ =
(1)
= A (Dự : D@)* ie
(due to the self-adjointness of 47„, 42 and the boundedness of (Dứ : Dø)_¡„) Taking the inverse of this non-singular positive self-adjoint operator, we have
Ag = (Dy : De)%;,)* 45% =(De@ : D)* „4
Then (i) and (ii) yield:
= J(Dø : DỰ)*¡„JJA;*J =
=/(Dọ : DỤ)*¡„J4 By changing the roles of ø and , we now have
(2) At, = J(Dự : Dø)*¡„J4%
Einally, (1) and (2) yield:
Ae = (Dự : Dø)~Í,Aÿ„ =
—
= (Dy: De)=}.J(Dy : Dọ)* „J4ÿ ›
so that the invertibility of (Dy : De)zi, J(Dự :Dò)*¡„J gives the result Q.E.D By Corollary 2.2 (iv) we have:
COROLLARY 3.3 For o and in the above lemma, the T-theorem for LP(; @')
holds if and only if the same is true for L?(4; ý)
Lemma 3.4 Let ¢ be a normal projection of norm 1 from onto a von Neumann subalgebra VW, and ~ be a faithful functional in Wf If the T-theorem
for L’(.@; (po8)’) holds, then the same is true for LN; @)
Proof We may and do assume that goe = Oz, with a cyclic and separating vector €) and that & is also a cyclic and separating vector in the subspace K ==(NE,] for the restricted von Neumann algebra | # In this set up, the modular operator associated with (4%, @ oc «) is diagonalized as follows:
¬ "|
Trang 8274 HIDEKI KOSAKI
where A, is the modular operator on X determined by (1, ¢@) (See [14].) In
particular, we have
2(4z.,)n # = 9(45),
so that the lemma follows from Corollary 2.2 (iv) Q.E.D
4 A SPECIAL CASE (A FACTOR OF TYPE I,,)
In this section, we prove (a stronger result than) our main result for a factor of type I
Let # be an infinite dimensional separable Hilbert space and @ =: B(#) We realize a standard form for @ by the quadruple (4, #, J = *, #,) Here # denotes the Hilbert space of all Hilbert-Schmidt class operators on % and @==B(X) acts on & as left multiplications We denote the usual trace on @ by Tr For a
faithful @ (resp an arbitrary ) in @; , there exists a unique non-singular positive
trace class operator h, on % (resp a unique positive trace class operator /,, on
1 +
2ý) satisfying œ = Tr(h„-) (resp =: Tr(z„-)) In other words, hg (resp hj) is a unique implementing vector in # , for @ (resp Ủ)
The next result is essentially due to Dixmier:
Lemma 4.1 For faithful ọ, c./‡, there always exists a unitary operator u
1 1 1
on # such that the (densely-defined) operator hệ he —® an 2Ý is not closable, where X= ubu™ and h, = uhyu*
1 1
Proof Since h, is a non-singular compact operator on 4 and -— „<0
i 1
i P + 2
he is not bounded Similarly, hy 2 3 is not either
Then, by Lemma 8.3, [4], there always exists a unitary u on © satisfying
1
ALL _—Ă+L ALL a2
Bho *)nuD(hy *) == DAZ ”)n 20, ”) = {0}
Therefore, one computes
1 1 1
Bho “hệ) =h„ + 9 {20 *)n Dh, *)} = {0} 1
1 1 1 1 1 ko
*isexactlyhe 7h7,
Sr
Trang 9T-THEOREM FOR L?-SPACES 275
For vectors €, € in #, the rank-one operator € @C€on & is defined by (E@O(y) = (1 | E,ne H#, as usual Clearly, € @ € belongs to #, on which @ acts The following results can be checked by straightforward calculation:
LEMMA 4.2 Let @, (resp @2) be a faithful (resp an arbitrary) element in My and €, € be vectors in X#
1
ai 1
(i) If Ce Dhy,*), then € @ Ce HAe 0)? O<a< ¬ and A % ((@Ð= (4,9) @ (age)
aL 1
(ii) If fe Dhy,*), then ÿ @{eØ(A7* ), 0< #< 5” and
4z» ( @ 0 =0250 @ (hg 0)
LEMMA 4.3 For o and x in Lemma 4.1, the (densely-defined) operator
42,42 * on ¥ is not closable
11 _ Pali i
Proof By the previous lemma, (4742 Bye @O =hjh; "E@h, Cl,
1 1 1 1
whenever ¢ € Dh? he —?) and £e 20w Py, At first, we fix a non-zero C) in
1
Why ?), Then, it follows from Lemma 4.1 that there exists a sequence {ế,} in
11 1 1 1 1
2(h2h? *) such that {&,} converges to 0 and {hyhd ~ eR converges to a non- zero € (in #)
1 1 +
Checking a sequence {&, © C,}, in WA? A? *), we easily conclude that
1 1 1
42,42 ? is not closable Q.E.D
Summing up the arguments in this section, one obtains:
Proposition 4.4 Let @ be a factor of type I,,, and os ý be faithful functionals
1
in “@} There always exists a unitary u in M@ such that A sư ? 2? ¡s not closable Here x is given by y = uỤuU" In particular, the T-theorem for L’(; 9’) never holds
5, PROOFS OF MAIN RESULTS
We still keep our assumption: 2 < p <oo
Proof of Theorem 2.5 To show the theorem by contradiction, we assume that
Trang 10276 HIDEKL KOSAKI
Cutting @ by the largest finite central projection, we may and do assume that /
is actually properly infinite Thus @ can be written as the tensor product of a von
Neumann subalgebra and a factor of type I Then we choose a normal projection e of norm 1 from @ onto A and a faithful functional g in By the assumption, the T-theorem for L’(@; ( © £)’) holds so that the same is true for L241; @’) by
Lemma 3.4, which contradicts Proposition 4.4 Q.E.D
Proof of Theorem 2.6 We may assume that #@ is a factor of type III, due
to Proposition 4.4 To show the theorem by contradiction, we also assume that
the 7-theorem for 7?(⁄Z; 0’) holds for some faithful @ ¢.@7 Due to Theorem 2.5, it suffices to show that the 7-theorem for L°(#; w’) holds for a generic faithful
ý c.Zz However, since # isa factor of type III,, 0 < 4 < 1, there always exists
an inner automorphism « and positive numbers /,, /,20 satisfying @<Jw oa,
yw oa<i,g (See [3], Ch I, Corollary 4.2.) Thus, Lemma 3.1 and Corollary 3.3
yield that the 7-theorem for L?(.Z; @') would hold as well Q.E.D Acknowledgement We would like to thank the referee for useful suggestions Among them, the referee informed us that’a lemma closely related to Dixmier’s lemma
used in the proof of Lemma 4.1 was obtained by von Neumann (Zur theorie der
unbeschrankten Matrizen, J fiir Mathematik, 161(1929), 208—234; Satz 18)
REFERENCES
1 ARAKI, H., Some properties of modular conjugation operators of a von Neumann algebra and a non-commutative Radon-Nikodym theorem with a chain rule, Pacific J Math.,
50(1974), 309—354
2 CONNES, Ấ., On the spatial theory of von Neumann algebras, J Fictional Analysis, 35(1980),
153—164
3 Connes, A.; Takesaxt, M., The flow of weights on factors of type II!, Tohoku Math J., (2) 29(1977), 473 —575
4 DIXMIER, J., Etude sur les variétés et les opérateurs de Julia avec quelques applications, Bull Soc Math France, 7701949), 11—101
5 Dre, H., The Radon-Nikodym theorem for finite rings of operators, Trans Amer Math Soc., 72(1952), 243—280
6 Haacerup, U., The standard form of von Neumann algebras, Math Scand., 37(1975),
271 —283
HAAGERUP, U., L?-spaces associated with an arbitrary von Neumann algebra, preprint Hitsum, M., Les espaces L? d’une algébre de von Neumann, preprint
, KOSAKI, H., Positive cones associated with a von Neumann algebra, Math Scand., to appear 10 Kosak1, H., Positive cones and L?-spaces associated with a von Neumann algebra, J Operator
Theory, 6(1981), 13—23
11 Kosaxs, H., Application of the complex interpolation method to a von Neumann algebra (Non-commutative L?-spaces), preprint
\Ð
6
Trang 11T-THEOREM FOR L?-SPACES 277 12 Skau, C., Positive self-adjoint extensions of operators affiliated with a von Neumann algebra,
Math, Scand., 44(1979), 171—195
13 TAKESAKI, M., Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Math., No 128(1970), Springer
14 Takesaki, M., Conditional expectations in von Neumann algebras, J Functional Analysis, 9(1972), 306—321 AIDEKI KOSAKI Department of Mathematics, University of Kansas, Lawrence, Kansas 66045, U.S.A,
Received March 19, 1981; revised August 19, 1981
Added in proof Recently, the author showed that the mapping in Theorem 1.2 is