Notes on the bicategory of W*-bimodules

Transcription

1 Submitted to Journal of the Mathematial Soiety of Japan Notes on the biategory of W*-bimodules By Yusuke Sawada and Shigeru Yamagami (Reeived July 6, 2017) (Revised Jan. 17, 2018) Abstrat. Categories of W*-bimodules are shown in an expliit and algebrai way to onstitute an involutive W*-biategory. Introdution As in the purely algebrai ase, it is fairly obvious to extrat a biategory from operator-algebrai bimodules provided that the relevant monoidal struture is based on ordinary module tensor produts. Although it needs an operator-algebrai modifiation to have natural tensor produts (see [3], [7], [9]), we know that W*-bimodules (i.e., Hilbert spaes with von Neumann algebras ating ontinuously) still supply a biategory of W*-algebrai nature, alled a W*-biategory (see [11], [13]). In the present notes, we shall show that the whole onstrution of the biategory in question as well as the aompanied involution is possible in an expliit and algebrai manner without detailed knowledge of modular theory. The organization is as follows: Related with W*-bimodules, we introdue two W*- biategories M, M and show that these are monoidally equivalent based on the unit objet property of standard W*-bimodules. We then notie the fat that the operation of taking dual bimodules gives an anti-multipliative equivalene between M and M, whih is utilized to get involutions on M and M respetively so that the monoidal equivalene between these preserves involutions as well. In this way, we have a single W*-biategory with involution, whih reovers the one dealt with in [11]. 1. Preliminaries 1.1. Biategories In this paper, a linear biategory is simply referred to as a biategory (see [6] for ategorial bakgrounds). Thus our biategory is a kind of ategorifiation of linear algebra and onsists of a family of linear ategories A L B indexed by a pair (A, B) of labels with the following information: A speial objet I A (alled a unit objet) in A L A is assigned to eah label A. A bivariant funtor ( ) ( ) : A L B B L C A L C is assigned to eah triplet (A, B, C) of labels. Isomorphisms (alled unit isomosphisms) l X : I A X X and r X : X I B X are assigned to eah objet X in A L B Mathematis Subjet Classifiation. Primary 46L10; Seondary 18D10. Key Words and Phrases. W*-bimodule, biategory, involution.

2 2 Y. Sawada and S. Yamagami An isomorphism (alled assoiativity isomorphism) a X,Y,Z : (X Y ) Z X (Y Z) is assigned to eah triplet (X, Y, Z) whih is admissible in the the sense that X A L B, Y B L C and Z C L D for some labels A, B, C, D. These are then required to satisfy the following onditions: (i) l X and r X are natural in X and satisfy the triangle identity in the sense that they make the following triangular diagrams ommutative. (ii) a X,Y,Z is natural in X, Y, Z and satisfies the pentagon identity in the sense that it makes the following pentagonal diagrams ommutative. (X I) Y X (I Y ) (W X) (Y Z) X Y ((W X) Y ) Z W (X (Y Z)) (W (X Y )) Z W ((X Y ) Z) If a biategory onsists of C*-ategories (or W*-ategories) and all the relevant morphisms are unitary, it is alled a C*-biategory (or a W*-biategory). See [4] (f. also [13]) for more information on operator ategories. By an involution on a biategory L, we shall mean a family of ontravariant funtors AL B B L A, whih we denote by X X, Hom(X, Y ) f t f Hom(Y, X ) for objets X, Y in A L B, together with natural families of isomorphisms { X,Y : Y X (X Y ) } (anti-multipliativity) and {d X : X (X ) } (duality) making the following diagrams ommutative (X Y ) Z 1 (Y X) Z a X (Y Z ) 1 X (Z Y ) (Z (Y X)) t a, ((Z Y ) X) X Y d d d (X Y ) t X Y, (Y X ) and fulfilling t d X = d 1 X : X X. (The naturality means t (f g) t g t f and f d t ( t f).) We remark here that the operation (X X, f t f) together with satisfying anti-multipliativity is an anti-monoidal funtor and we see that f t ( t f)

3 biategory of bimodules 3 gives a monoidal funtor with the multipliativity t 1 : X Y (Y X ) (X Y ). In literature, our involution is named in various ways; it is referred to as, for example, having duals in [2] with extra onditions assumed in onnetion with unit objets, whih turns out to be redundant. For C*-biategories (espeially for W*-biategories), it is natural to assume the ompatibility with the *-operation on morphisms as studied in [12]; all the relevant strutural isomorphisms are assumed to be unitary and the operation f t f satisfies t (f ) = ( t f). To avoid the onfusion in this situation, we have used the different symbols X and t f to denote a single funtor. (Other remedy is to use t X for objets, whih looks however apparently awkward.) 1.2. Bimodules We here review relevant fats from [11] (f. [1] also). Let A and B be W*-algebras. By an A W* B bimodule X, we shall mean a Hilbert spae X on whih W*-algebras A and B are normally (i.e., weak-ontinuously) represented in an A-B bimodule fashion. We often write A X B to indiate the ating algebras. Given A W* B bimodules X and Y, the Banah spae of bounded A-B linear maps of A X B into A Y B is denoted by Hom(X, Y ). With these as hom-sets, we have the W*-ategory of A W* B bimodules, whih is denoted by A M B in what follows. We regard a left W*-A module (resp. a right W*-B module) as an A W* B bimodule for B = C (resp. for A = C). The so-alled standard representation (spae) of a W*-algebra A ([5]) is nothing but the regular representation of A and denoted by L 2 (A) in this paper. Reall that L 2 (A) is an A W* A bimodule, whih is linearly spanned by symbols ϕ 1/2 (ϕ A + ) so that the redued left GNS spae [ϕ]aϕ 1/2 is identified with the redued right GNS spae ϕ 1/2 A[ϕ] by J ϕ (aϕ 1/2 ) = (aϕ 1/2 ) = ϕ 1/2 a for a [ϕ]a[ϕ]. Here [ϕ] denotes the support projetion of ϕ, J ϕ stands for the modular onjugation assoiated to the GNS vetor ϕ 1/2 of the redued algebra [ϕ]a[ϕ] and the anonial *-operation, whih is designated by here, is well-defined on the whole L 2 (A). Note that ϕ is faithful when restrited to [ϕ]a[ϕ]. See [10] and [14] for further information. For a W*-bimodule A X B, we write A( 1/2)X = Hom( A L 2 (A), A X), XB( 1/2) = Hom(L 2 (B) B, X B ) with the obvious operations of these on X by right and left multipliations respetively, whih are A-B bimodules by α(af b) = ((αa)f)b and (agb)β = a(g(bβ)) for f A( 1/2)X, g XB( 1/2), a A, b B, α L 2 (A) and β L 2 (B). (The irle for opposite algebra is plaed in the definition of A( 1/2)X to indiates that it ats on X from the right.) Moreover, with this onvention, we introdue an A-valued inner produt A [, ] on A( 1/2)X and a B-valued inner produt [, ] B on XB( 1/2) by α( A [f, f]) = (αf )f, ([g, g ] B )β = g (g β). Here f, f A( 1/2)X and g, g XB( 1/2). Note that L 2 (B)B( 1/2) = End(L 2 (B) B ) is equal to B, whereas A( 1/2)L 2 (A) = End( A L 2 (A)) is indetified with A by the right ation of A. Notie that A [a f, af] = a A[f, f]a and [gb, gb ] B = b [g, g ] B b

4 4 Y. Sawada and S. Yamagami for a, a A and b, b B. Remark 1. The notation suh as XB( 1/2) an be explained as follows: When B admits a faithful normal state ϕ, there is a one-to-one orrespondene between x Hom(L 2 (B) B, X B ) and a ϕ-bounded element ξ X B by the relation x(ϕ 1/2 ) = ξ, whih suggests the formal expression x = ξϕ 1/2. Given index sets I, J, we introdue a matrix extension of a W*-algebra N by M I,J (N) = Hom(l 2 (J) L 2 (N) N, l 2 (I) L 2 (N) N ), whih is identified with a subspae of bounded N-valued matriial sequenes in N I J and eah (x i,j ) M I,J (N) is approximated by its finitely supported uts in any weaker operator topology. Note that M I (N) = M I,I (N) is a von Neumann algebra on l 2 (I) L 2 (N) and weaker operator topologies are well-defined on M I,J (N) as a orner subspae of M I J (N). The L 2 -version of matrix extension is introdued analogously: A matrix extension (a Hilbert-Shmidt extension) of a W*-bimodule A X B is defined by I X J = {(ξ i,j ); ξ i,j X, ξ i,j 2 < }, i I,j J whih is an MI (A)W* MJ (B) bimodule in an obvious way. Reall that unilateral W*- modules are projetive in the sense that we an find an index set I and a projetion e M I (A) so that A X = A L 2 (A) I e or X A = e I L 2 (A) A. This is nothing but a paraphrase of the Dixmier s struture theorem on normal *-homomorphisms between von Neumann algebras. We refer to this as a projetive module realization of X in what follows. When X is an A-B bimodule, the isomorphism End( A L 2 (A) I e) = em I (A)e gives rise to a normal *-homomorphism B em I (A)e. Similarly for a B-A bimodule X and a realization X A = e I L 2 (A) A. 2. W*-biategories of W*-bimodules As observed in [11] from the view point of modular algebras, W*-bimodules onstitute an involutive W*-biategory, whih we shall reonstrut here based on operator-valued inner produts. Aording to two possibilities of them, there are two ways of forming tensor produt bimodules, whih are disriminatingly denoted by X Y = XB( 1/2) B Y, X Y = X B B( 1/2)Y. Here algebrai module tensor produts XB( 1/2) B Y and X B B( 1/2)Y are pre- Hilbert spaes with their inner produts defined by (x B η x B η ) = (η [x, x ] B η ), (ξ B y ξ B y ) = (ξ ξ B[y, y]) respetively and the bar denotes the Hilbert spae ompletion. Given morphisms f : A X B A X B and g : BY C B Y C, bounded A-C linear maps f g : X Y X Y and f g : X Y X Y are well-defined by (f g)(x B η) = (fx) B (gη), (f g)(ξ B y) = (fξ) B (gy). Here gy B( 1/2)Y is defined by β(gy) = g(βy) (β L 2 (B)). Unit isomorphisms are

5 biategory of bimodules 5 defined by l A,B : L 2 (A) X = A A X a A ξ aξ X, r A,B : X L 2 (B) = XB( 1/2) B L 2 (B) x B β xβ X and l A,B : L 2 (A) X = L 2 (A) A A( 1/2)X α A x αx X, r A,B : X L 2 (B) = X B B ξ B b ξb X. Note that, in view of projetive module realizations of X, orrespondenes for r A,B and l A,B are redued to B B L 2 (B) b β bβ L 2 (B) and L 2 (A) A A α a αa L 2 (A) respetively, whih reveals the unitarity of these. To introdue assoiativity isomorphisms, we remark that the algebrai module tensor produt XB( 1/2) B Y C( 1/2) is anonially embedded into (X Y )C( 1/2) in suh a way that (XB( 1/2) B Y C( 1/2)) C Z is dense in (X Y )C( 1/2) C Z. Likewise, we have a anonial embedding B( 1/2)Y C C( 1/2)Z B( 1/2)(Y Z) so that X B B( 1/2)Y C C( 1/2)Z is dense in X (Y Z). Assoiativity isomorphisms are now defined by a X,Y,Z : (X Y ) Z (x B y) C ζ x B (y C ζ) X (Y Z) for x XB( 1/2), y Y C( 1/2) and ζ Z. a X,Y,Z : (X Y ) Z (ξ B y) C z ξ B (y C z) X (Y Z) for ξ X, y B( 1/2)Y and z C( 1/2)Z. The pentagon identity on a quadruple produt W X Y Z then follows from that on W A( 1/2) A XB( 1/2) B Y C( 1/2) C Z. Similarly for the produt. The triangle identities for unit isomorphisms are also witnessed on dense subspaes. For X L 2 (B) Y, this is redued to the ommutativity of the diagram whih is traed by XB( 1/2) B B B Y X (L 2 (B) Y ) (X L 2 (B)) Y X Y x B b B η x B (b B η) (x B b) B η xb B η = x B bη for x XB( 1/2), b B and η Y. In summary, we have two W*-biategories of W*-bimodules, whih are denoted by M and M from here on. Remark 2. In [8], the assoiativity isomorphism is aptured as (X Y ) Z =,

6 6 Y. Sawada and S. Yamagami X (Y Z) in our notation. Although X Y = X Y in a anonial way (see below, f. [3], [7] also), the existene of these isomorphisms does not automatially mean the oherene for quadruple tensor produts. 3. Canonial Equivalene Two W*-biategories of W*-bimodules are now shown to be anonially equivalent. This is reognized in [11] through natural identifiations in modular tensor produts. Here we shall establish this by onstruting an expliit funtor of equivalene. We first observe how tensor produts behave under matrix extensions. Consider W*- bimodules A X B, B Y C and their olumn and row extensions I X B and B Y J by index sets I, J. Then ( I X) (Y J ) and ( I X) (Y J ) are naturally identified with Hilbert-Shmidt extensions I (X Y ) J and I (X Y ) J of X Y and X Y respetively. Note that algebrai sums I XB( 1/2) and B( 1/2)Y J are weakly dense in ( I X)B( 1/2) and B( 1/2)(Y J ) respetively. When this observation is applied to the standard bimodule L 2 (B), the unit isomorphisms l = r : L 2 (B) L 2 (B) L 2 (B) and l = r : L 2 (B) L 2 (B) L 2 (B) are enhaned to M I (B)-M J (B) linear unitary maps m : ( I L 2 (B)) (L 2 (B) J ) I (L 2 (B) L 2 (B)) J I L 2 (B) J, where = or. We set I m J = (m ) m : ( I L 2 (B)) (L 2 (B) J ) ( I L 2 (B)) (L 2 (B) J ), whih is a unitary isomorphism in MI (B)M MJ (B). For A X B and B Y C, a unitary isomorphism m X,Y : X Y X Y is defined, with the help of projetive-module realizations u : X B p I L 2 (B) B and v : B Y B L 2 (B) J q as B-modules, by the ommutativity of the diagram ( ) u v X Y p ( I L 2 (B)) (L 2 (B) J ) q m X,Y I m J X Y u v. ( ) p ( I L 2 (B)) (L 2 (B) J ) q ( ) Note here that (p I L 2 (B)) (L 2 (B) J q) = p ( I L 2 (B)) (L 2 (B) J ) q for = or. By the bimodule linearity of I m J, m X,Y is A-C linear and independent of the hoie of projetive-module realizations. Furthermore, m X,Y is natural in X and Y as well: For f Hom(X, X ) and g Hom(Y, Y ), the diagram X Y f g X Y m X,Y m X,Y X Y f g X Y is ommutative. The following is then immediate from the definition of m X,Y.

7 biategory of bimodules 7 Let X = A X B be a W*-bimodule. Then the following diagrams om- Lemma 3.1. mute. L 2 (A) X m L 2 (A),X m X,L 2 (B) L 2 (A) X, X L 2 (B) X L 2 (B). l X l X r X r X X X Theorem 3.2. The identity funtor gives a monoidal equivalene between M and M with respet to the multipliativity isomorphisms {m X,Y }, i.e., the diagram (X Y ) Z m X,Y 1 (X Y ) Z a m X Y,Z (X Y ) Z a X (Y Z) 1 m Y,Z X (Y Z) m X,Y Z X (Y Z) is ommutative for any omposable triplets X, Y, Z of W*-bimodules. Proof. By projetive-module realizations of X and Z together with the naturality of relevant morphisms, the problem is redued to the ase X = L 2 (B) and Z = L 2 (C) with Y = B Y C, whose validity an be seen from the following division of the diagram (I B Y ) I C (I B Y ) I C (I B Y ) I C 1 I B Y 2 Y I C Y 3 I B Y Y I C I B (Y I C ) I B (Y I C ) I B (Y I C ), where diagrams around 1 ommute by the triangle identity for unit isomorphisms, diagrams around 2 ommute by the naturality of unit isomorphisms and diagrams around 3 ommute by Lemma Unitary Involutions Given a W*-bimodule A X B, the dual Hilbert spae X is naturally a B W* A bimodule so that the operation of taking duals gives a ontravariant funtor A M B B M A with the operation on morphisms given by taking the transposed t f : Y X of f Hom(X, Y ). With the notation ξ (ξ X) to stand for a linear form X ξ (ξ ξ ) (the inner produt being linear in the seond variable), t f is desribed by t fη, ξ = (η fξ)

8 8 Y. Sawada and S. Yamagami The operation is then involutive (so-alled self-duality on Hilbert spaes) in the sense that, if we denote the anonial isomorphism (X ) = X by d X : X ξ ξ = (ξ ) X, it is natural in X, satisfies t d X = d 1 X and gives an equivalene between the iterated involution and the identity funtor. As to the monoidal strutures in M, the dualizing funtor gives an anti-multipliative equivalene between M and M. To see this, we begin with some preparatory disussions. For x XB( 1/2) = Hom(L 2 (B) B, X B ) with X = A X B a W*-bimodule in AM B, define its onjugate x Hom( B L 2 (B), B X ) by x(β) = (x(β )) and set x = x B( 1/2)X = Hom( B L 2 (B), B X ). Reall that β denotes the natural *-operation on L 2 (B). Then the orrespondene x x gives a onjugate-linear isometri isomorphism between XB( 1/2) and B( 1/2)X in suh a way that (axb) = b x a and [x, x] B = B[x, (x ) ] for a A, b B and x, x XB( 1/2). We now introdue a natural (ovariant) family of unitary morphisms X,Y : Y X (X Y ) in C M A (Y = B Y C ) by X,Y (η B x ) = (x B η) (x XB( 1/2), η Y ). Note that the unitarity of X,Y is ensured by the fat that X,Y has a dense range and the equality η B x 2 = (η η B[x, x ]) = (η η [x, x] B) = x B η 2. We laim that X,Y is anti-multipliative in the sense that the following hexagon diagram ommutes. (Z Y ) X 1 (Y Z) X a Z (Y X ) 1 Z (X Y ) (X (Y Z)) t a. ((X Y ) Z) To see this, let x XB( 1/2), y Y C( 1/2) and ζ Z. Then the above diagram is traed by (ζ C y ) B x 1 (y C ζ) B x a (x B (y C ζ)) t a ζ C (y B x ) ζ C (x B y) ((x B y) C ζ) and the hexagonal ommutativity is redued to the equality (x B y) = y B x in ((X Y )C( 1/2)), whih is in turn heked by γ(x y) = ((x y)(γ )) y(γ ) x = γ(y x ) for γ L 2 (C). Being prepared, we define anti-multipliativity isomorphisms X,Y : Y X (X Y ) in M to be the omposition X,Y = t m 1 X,Y X,Y, whih together with the duality isomorphisms {d X } onstitute a unitary involution on M : As a omposition of anti-multipliative X,Y and multipliative t m 1 X,Y, X,Y is anti-multiative, i.e., the

9 biategory of bimodules 9 hexagon identity holds. For the ommutativity of X Y d d d (X Y ) t X Y, (Y X ) we first desribe in terms of standard spaes. Given a projetive-module realization u : p I L 2 (B) B = XB, its transposed map is omposed with the row-vetor extension of the anonial isomorphism L 2 (B) = L 2 (B) to get the aompanied isomorphim v : BX = B L 2 (B) I p. These are then ombined with X,Y and t m X,Y to form a ommutative diagram Y X X,Y (X Y ) t m X,Y (X Y ), Y L 2 (B) I p (p I L 2 (B) Y ) (p I (L 2 (B) Y )) where the bottom line is desribed by the orrespondenes η B (b i )p (p(b i ) B η) (p(β i B y)) with the relation p(b i η) = p(β i y) in I Y assumed at the seond one. Thus, replaing η B (b i )p with ( i y β i B 1 i )p, X,Y is speified by the ommutativity of the diagram Y X 1 v X,Y (X Y ) t (u 1) with the bottom line given by Y L 2 (B) I p (p I L 2 (B) Y ) ( i y β i 1 i) p (p(β i ) B y). Here y B( 1/2)Y, β i L 2 (B) and 1 i = δ i B I = B( 1/2)L 2 (B) I denotes the anonial row basis. By symmetry, X,Y is also desribed in terms of a projetive-module realization BY = BL 2 (B) J q by the orrespondene q J L 2 (B) X q(β j ) B x ( j (xβ j 1 j )q) ((X L 2 (B)) J q). Now the square identity for duality isomorphisms takes the form p I L 2 (B) Y d 1 d (p I L 2 (B) Y ) t p I L 2 (B) Y. (Y L 2 (B) I p)

10 10 Y. Sawada and S. Yamagami Here anonial isomorphisms (p I L 2 (B)) = p I L 2 (B), (p I L 2 (B)) = L 2 (B) I p are used at the right orners with and t modified aordingly. The diagram is then traed by p(β i ) B y p(β i ) B y (p(β i ) B y) ((y β i B 1)p) and the ommutativity holds. In this way, we have heked that { X,Y } defines a unitary involution on M. Likewise X,Y = X,Y m Y,X gives a unitary involution on M so that {m X,Y } intertwines these. As a onlusion, we have Theorem 4.1. Anti-multipliativity isomorphisms { X,Y } and { X,Y } define unitary involutions on M and M respetively so that they are equivalent through the monoidal equivalene {m X,Y }. Referenes [1] M. Baillet, Y. Denizeau and J.F. Havet, Indie d une espérane onditionnelle, Compositio Math. 66, (1988), [2] J.W. Barrett and B.W. Westbury, Spherial ategories, Adv. Math. 143, (1999), [3] A. Connes, Nonommutative Geometry. Aademi Press, [4] P. Ghez, R. Lima and J.E. Roberts, W*-ategories, Paifi J. Math., 120, (1985), [5] U. Haagerup, The standard form of von Neumann algebras, Math. Sand. 37, (1975), [6] S. MaLane, Categories for the working mathematiian. Seond edition. Springer-Verlag, [7] J.L. Sauvageot, Sur le produit tensoriel relatif d espaes de Hilbert, J. Operator Theory 9, (1983), [8] M. Takesaki, Theory of Operator Algebras. II, Enylopaedia of Mathematial Sienes, 125. Springer-Verlag, [9] A. Thom, A remark about Connes fusion tensor produt, Theory Appl. Categ., 25, (2011), [10] S. Yamagami, Algebrai aspets in modular theory, Publ. Res. Inst. Math. Si. 28, (1992), [11], Modular theory for bimodules, J. Funt. Anal. 125, (1994), [12], Frobenius duality in C -tensor ategories, J. Operator Theory 52, (2004), [13], Notes on operator ategories, J. Math. So. Japan, 59, (2007), [14], Around trae formulas in non-ommutative integration, arxiv: Yusuke Sawada Graduate Shool of Mathematis Nagoya University m14017@math.nagoya-u.a.jp Shigeru Yamagami Graduate Shool of Mathematis Nagoya University yamagami@math.nagoya-u.a.jp