Noncommutative Orlicz Spaces Over W*-Algebras Ryszard Paweł Kostecki

Noncommutative Orlicz spaces over W*-algebras Ryszard Paweł Kostecki

Perimeter Institute of Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada

May 11, 2019

Abstract Using the Falcone–Takesaki theory of noncommutative integration and Kosaki’s canonical representation, we construct a family of noncommutative Orlicz spaces that are associated to an arbitrary W*-algebra 풩 without any choice of weight involved, and we show that this construction is functorial over the category of W*-algebras with *-isomorphisms as arrows. Under a choice of representation, these spaces are isometrically isomorphic to Kunze’s noncommutative Orlicz spaces over crossed products 풩 o휎휓 R.

MSC2010: 46L52, 46L51, 46M15. Keywords: noncommutative Orlicz spaces, W*-algebras.

1 Introduction

The construction of a standard representation by Connes, Araki and Haagerup and its further canoni- * cal refinement by Kosaki allows one to assign a canonical 퐿2(풩 ) space to every W -algebra 풩 , without any choice of a weight on 풩 involved. This leads to a question: is it possible to develop the theory of * noncommutative integration and 퐿푝(풩 ) spaces for arbitrary W -algebras 풩 , analogously to consideration of the Hilbert–Schmidt space as a member G2(ℋ) of the family of von Neumann–Schatten spaces G푝(ℋ), and consideration of a Hilbert space 퐿2(풳 , f(풳 ), 휇˜) as a member of a family of Riesz–Radon– Dunford spaces 퐿푝(풳 , f(풳 ), 휇˜)? Falcone and Takesaki [13] answered this question in the affirmative by an explicit construction, showing that to any W*-algebra 풩 there is a corresponding noncommutative 퐿푝(풩 ) space, and furthermore, this assignment is functorial. In this paper we extend their result by constructing a family of noncommutative Orlicz spaces * 퐿ϒ(풩̃︀) that is associated to every W -algebra 풩 for any choice of an Orlicz function Υ. The key aspect of this construction is its functoriality: every *-isomorphism of W*-algebras induces a corresponding isometric isomorphism of noncommutative Orlicz spaces. We also infer basic properties of these spaces. These spaces, under a choice of any faithful normal semi-finite weight 휓 on 풩 , are isometrically isomorphic to Kunze’s [20] noncommutative Orlicz spaces 퐿ϒ(풩 o휎휓 R), which is induced by a weight- ∼ dependent unitary isomorphism 풩̃︀ = 풩 o휎휓 R. Hence, 퐿ϒ(풩̃︀) are not the canonical noncommutative Orlicz spaces over 풩 , similarly as 퐿푝(풳 × R, f(풳 × R), 휇˜) is not the canonical 퐿푝 space over a corresponding boolean algebra. Yet, we hope that our method of approaching the problem, based on the functorial properties of the Falcone–Takesaki noncommutative flow of weights, can be further arXiv:1409.0189v3 [math.OA] 25 Jul 2018 extended using the ideas in [21]. Section2 provides an overview of the Falcone–Takesaki integration theory over arbitrary W *- algebras. A brief discussion of noncommutative Orlicz spaces is contained in Section3. In Section4 we introduce our construction and show its functoriality. * For a W algebra 풩 , we denote: the set of semi-finite faithful nomal weights on 풩 as 풲0(풩 ); 휓 Connes’ cocycle of 휓, 휑 ∈ 풲0(풩 ) as [휓 : 휑]푡; Connes’ spatial quotient as 휑 ; the Tomita–Takesaki 휓 modular automorphism of 휓 ∈ 풲0(풩 ) as 휎 . For an extended discussion of all notions related to W*-algebras and noncommutative integration we refer to [18] as a recent overview (close to the spirit of [43]) which has precisely the same notation and terminology as we use here, as well as to [8, 46] as standard references. 2 Falcone–Takesaki integration on arbitrary W*-algebras

The Falcone–Takesaki approach to noncommutative integration relies on the construction and properties of the core algebra 풩̃︀ and on Masuda’s [27] reformulation of Connes’ noncommutative Radon– Nikodým type theorem. We will first briefly recall the key notions from the relative modular theory, and then move to the construction of the core algebra and integration over it. * Consider a W -algebra 풩 and a relation ∼푡 on 풩 × 풲0(풩 ) defined by [13]

(푥, 휓) ∼푡 (푦, 휑) ⇐⇒ 푦 = 푥[휓 : 휑]푡 ∀푥, 푦 ∈ 풩 ∀휓, 휑 ∈ 풲0(풩 ). (1)

The cocycle property of [휓 : 휑]푡 implies that ∼푡 is an equivalence relation in 풩 × 풲0(풩 ). The i푡 equivalence class (풩 × 풲0(풩 ))/ ∼푡 is denoted by 풩 (푡), and its elements are denoted by 푥휓 . The operations

푥휓i푡 + 푦휓i푡 := (푥 + 푦)휓i푡, (2) i푡 i푡 휆(푥휓 ) := (휆푥)휓 ∀휆 ∈ C, (3) ⃒⃒ i푡⃒⃒ ⃒⃒푥휓 ⃒⃒ := ||푥||, (4) equip 풩 (푡) with the structure of the Banach space, which is isometrically isomorphic to 풩 , considered as a Banach space. By definition, 풩 (0) a W*-algebra that is trivially *-isomorphic to 풩 . However, for 푡 ̸= 0 the spaces 풩 (푡) are not W*-algebras. The operations

· : 풩 (푡 ) × 풩 (푡 ) ∋ (푥휓i푡1 , 푦휓i푡2 ) ↦→ 푥휎휓 (푦)휓i(푡1+푡2) ∈ 풩 (푡 + 푡 ), (5) 1 2 푡1 1 2 * i푡 휓 * −i푡 : 풩 (푡) ∋ 푥휓 ↦→ 휎−푡(푥 )휓 ∈ 풩 (−푡), (6) equip the disjoint sum F(풩 ) := ∐︀ 풩 (푡) over 풩 × with the structure of *-algebra. The bijections 푡∈R R

i푡 풩 (푡) ∋ 푥휓 ↦→ (푥, 푡) ∈ 풩 × R (7) allow to endow F(풩 ) with the topology induced by (7) from the product topology on 풩 × R of the * weak-⋆ topology on 풩 and the usual topology on R. This provides the Fell’s Banach -algebra bundle structure on F(풩 ). One can consider the Fell bundle F(풩 ) as a natural algebraic structure which enables to translate between elements of 풩 (푡) at different 푡 ∈ R. In order to recover an element of F(풩 ) at a given 푡 ∈ R, one has to select a section 푥̃︀ : R → F(풩 ) of F(풩 ): i푡 푡 ↦→ 푥(푡)휓 =: 푥̃︀(푡). (8) Consider the set Γ1(F(풩 )) of such cross-sections of F(풩 ) that are integrable in the following sense:

i) for any 휖 > 0 and any bounded interval 퐼 ⊆ R there exists a compact subset 푌 ⊆ 퐼 such that |퐼 − 푌 | < 휖 and the restriction 푌 ∋ 푡 ↦→ 푥(푡) ∈ F(풩 ) is continuous relative to the topology induced in F(풩 ) by (7),

ii) ∫︀ d푟 ||푥(푟)|| < ∞. R The set Γ1(F(풩 )) can be endowed with a multiplication, involution, and norm, ∫︁ (︂∫︁ )︂ 휓 i푡 (푥̃︀푦̃︀)(푡) := d푟 푥(푟)푦(푡 − 푟) = d푟 푥(푟)휎푟 (푦(푡 − 푟)) 휓 , (9) R R * * 휓 * i푡 푥̃︀ (푡) := 푥̃︀(−푡) = 휎푡 (푥(−푡) )휓 , (10) ∫︁ ||푥̃︀|| := d푟 ||푥(푟)||, (11) R thus forming a Banach *-algebra, denoted by B(풩 ).

2 Falcone and Takesaki [12, 13] constructed also a suitably defined ‘bundle of Hilbert spaces’ over * R. Let 풩 = 휋(풞) be a von Neumann algebra representing a W -algebra 풞 in terms of a standard representation (ℋ, 휋, 퐽, ℋ♮). The space ℋ can be considered as a 풩 -(풩 ∙)표 bimodule1, with the left action of 풩 on ℋ given by ordinary multiplication from the left, and with the right action of (풩 ∙)표 on ℋ defined by 휉푥표 := 푥휉 ∀휉 ∈ ℋ ∀푥표 ∈ (풩 ∙)표. (12) Thus, the left action of 풩 on ℋ is just an action of a standard representation of the underlying W*- algebra 풞, while the right action of (풩 ∙)표 is provided by the corresponding antirepresentation of 풞 표 (that is, by commutant of a standard representation of 풞 ). Given arbitrary 푟1, 푟2 ∈ R, 휁1, 휁2 ∈ ℋ, ∙ 표 휑1, 휑2 ∈ 풲0(풩 ), and 휙1, 휙2 : 풲0((풩 ) ) the condition

(︂ )︂i푟1 (︂ )︂i푟2 휑1 휑2 휁1 = 휁2[휙2 : 휙1]푡, (13) 휙1 휙2 defines an equivalence relation

(푟1, 휑1, 휁1, 휙1) ∼푡 (푟2, 휑2, 휁2, 휙2) (14)

∙ 표 on the set R×풲0(풩 )×ℋ ×풲0((풩 ) ). The equivalence class of the relation (14) is denoted by ℋ(푡), and its elements have the form (︂ 휑 )︂i푡 휑i푡휉 = 휉휙i푡, (15) 휙 which is equivalent to (︂ 휑 )︂i푡 휑i푡휉휙−i푡 = 휉. (16) 휙

Falcone and Takesaki show that ℋ(푡) is a Hilbert space independent of the choice of weights 휑1, 휑2, 휙1, 휙2 and of the choice of 푟 ∈ . This enables to form the Hilbert space bundle over , ∐︀ ℋ(푡), and to R R 푡∈R form the Hilbert space of square-integrable cross-sections of this bundle, (︃ )︃ ∐︁ ℋ̃︀ := Γ2 ℋ(푡) . (17) 푡∈R The Hilbert space bundle ∐︀ ℋ(푡) is homeomorphic to ℋ × for any choice of weight 휓 ∈ 풲 (풩 ). 푡∈R R 0 The left action of B(풩 ) on ℋ̃︀ generates a von Neumann algebra 풩̃︀ called standard core [13]. For type III1 factors 풩 the standard core 풩̃︀ is a type II∞ factor, but in general case 풩̃︀ is not a factor. The structure of 풩̃︀ is independent of the choice of weight on 풩 . However, for any choice of 휓 ∈ 풲0(풩 ) there exists a unitary map (︃ )︃ 2 ∐︁ ∼ u휓 : ℋ̃︀ = Γ ℋ(푡) → 퐿2(R, d푡; ℋ) = ℋ ⊗ 퐿2(R, d푡), (18) 푡∈R such that −i푡 u휓(휉)(푡) = 휓 휉(푡) ∈ ℋ ∀휉 ∈ ℋ̃︀. (19) It satisfies

* 휓 (u휓푥u휓)(휉)(푡) = 휎−푡(푥)휉(푡), (20) i푠 * (u휓휓 u휓)(휉)(푡) = 휉(푡 − 푠), (21) (︂ 휓 )︂i푠 (u 휑−i푠u* )(휉)(푡) = . (22) 휓 휓 휑∙

1 That is, a Hilbert space ℋ equipped with a normal representation 휋1 : 풩 → B(ℋ) and a normal representation 표 표 휋2 : 풩 → B(ℋ) such that 휋1(풩 ) and 휋2(풩 ) commute.

3 for all 휉 ∈ 퐿2(R, d푡; ℋ), 푥 ∈ 풩 , 휑, 휓 ∈ 풲0(풩 ), 푠, 푡 ∈ R. This map provides a *-isomorphism between 2 the standard core 풩̃︀ on ℋ̃︀ and the crossed product 풩 o휎휓 R on ℋ ⊗ 퐿2(R, d푡), * u휓풩̃︀u휓 = 풩 o휎휓 R. (23) Using the uniqueness of the standard representation up to unitary equivalence, Falcone and Takesaki [13] proved that the map 풩 ↦→ 풩̃︀ extends to a functor VNCore from the category VNIso of von Neumann algebras with *-isomorphisms to its own subcategory VNsf Iso of semi-finite von Neumann algebras with *-isomorphisms. The one-parameter automorphism group of F(풩 ), i푡 −i푡푠 i푡 i푡 휎˜푠(푥휑 ) := e 푥휑 ∀푥휑 ∈ 풩 (푡), (24) corresponding to the unitary group 푢˜(푠) on ℋ̃︀ given by −i푠푡 (˜푢(푠)휉)(푡) = e 휉(푡) ∀푡, 푠 ∈ R ∀휉 ∈ ℋ̃︀, (25) extends uniquely to a group of automorphisms 휎˜푠 : 풩̃︀ → 풩̃︀. The automorphism 휎˜푡 provides a weight- 휓 independent replacement for a dual automorphism 휎ˆ푡 used in Haagerup’s theory [16, 47]. The triple * (풩˜ , R, 휎˜) is a W -dynamical system. There exist canonical isomorphisms ∼ 풩̃︀ o휎˜ R = 풩 ⊗ B(퐿2(R, d휆)), (26) ∼ 풩̃︀휎˜ = 풩 . (27)

The action of 휎˜푠 on 풩̃︀ is integrable over 푠 ∈ R, and ∫︁ + ext 푇휎˜ : 풩̃︀ ∋ 푥 ↦→ 푇휎˜(푥) := d푠 휎˜푠(푥) ∈ 풩 , (28) R ∼ is an operator valued weight from 풩̃︀ to 풩̃︀휎˜ = 풩 . For any 휑 ∈ 풲(풩 ), its dual weight over 풩̃︀ is given by ˆ ˜ 휑 := 휑 ∘ 푇휎˜ ∈ 풲(풩̃︀). (29)

Every 휑 ∈ 풲0(풩 ) can be considered as an analytic generator of the one parameter group of unitaries i푡 {휑 | 푡 ∈ R} ⊆ 풩̃︀ acting on ℋ̃︀ from the right, given by (︂ d )︂ 휑 = exp −i (︀휑i푡)︀ | . (30) d푡 푡=0 + This allows to equip 풩̃︀ with a faithful normal semi-finite trace τ̃︀휑 : 풩̃︀ → [0, ∞], ˆ −1 −1 −1 1/2 −1 −1 −1 1/2 τ휑(푥) : = lim 휑((휑 (1 + 휖휑 ) ) 푥((휑 (1 + 휖휑 ) ) ) ̃︀ 휖→+0 = lim 휑ˆ(휑−1/2(1 + 휖휑−1)−1/2푥휑−1/2(1 + 휖휑−1)−1/2) 휖→+0 = lim 휑ˆ((휑 + 휖)−1/2푥(휑 + 휖)−1/2). (31) 휖→+0 This definition is independent of the choice of weight (e.g., τ̃︀휙 = τ̃︀휓 ∀휙, 휓 ∈ 풲0(풩 )), which follows from the fact that [︁ ˜]︁ [︁ ˜ ]︁ −i푡 i푡 −i푡 i푡 −i푡 i푡 [τ휑 : τ휙] = [τ휑 :휙 ˜] 휙˜ : 휓 휓 : τ휙 = 휙 [휙 : 휓]푡휓 = 휙 휙 휓 휓 = 1 (32) ̃︀ ̃︀ 푡 ̃︀ 푡 푡 ̃︀ 푡 for all 휑, 휙 ∈ 풲0(풩 ) and for all 푡 ∈ R. This allows to write τ̃︀ instead of τ̃︀휙. Moreover, τ̃︀ has the scaling property −푠 τ̃︀ ∘ 휎˜푠 = e τ̃︀ ∀푠 ∈ R. (33) This allows to call τ̃︀ a canonical trace of 풩̃︀. It will play the role analogous to a Haagerup’s trace ˆ 휏˜휓 on a crossed product algebra 풩 = 풩 o휎휓 R [16, 47]. Nevertheless, the definition (31) is not a straightforward generalisation of Haagerup’s trace. It is another type of ‘perturbed’ construction of a weight, which is designed in this case for the purpose of direct elimination of the dependence of τ̃︀휑 on 휑. 2See [18] for the notions of, and further references on, crossed product, operator valued weight, dual weight, and 풩 ext.

4 3 Noncommutative Orlicz spaces

In this section we will first briefly recall the key notions from the theory of commutative Orlicz spaces [36, 37], and then we will move to discussion of the noncommutative version of this theory. For a detailed treatment of the theory of commutative Orlicz spaces, as well as the associated theory of modular spaces, see [33, 35, 34, 55, 19, 23, 31, 25, 39,6, 40]. If 푋 is a vector space over K ∈ {R, C}, and Υ: 푋 → [0, ∞] is a convex function satisfying 1) Υ(0) = 0,

2) Υ(휆푥) = 0 ∀휆 > 0 ⇒ 푥 = 0,

3) |휆| = 1 ⇒ Υ(휆푥) = Υ(푥), then Υ is called a pseudomodular function [32]. If 2) is replaced by

2’) Υ(푥) = 0 ⇒ 푥 = 0, then Υ is called a modular function [33, 35, 34]. If 2) is replaced by

2”) 푥 ̸= 0 ⇒ lim휆→+∞ Υ(휆푥) = +∞, then Υ is called a Young function [54,4,5]. A Young function 푓 on R is said to satisfy: local △2 condition iff [4,5] ∃휆 > 0 ∃푥0 ≥ 0 ∀푥 ≥ 푥0 푓(2푥) ≤ 휆푓(푥); (34) + global △2 condition iff 푥0 in (34) is set to 0. A convex function 푓 : R → R is called N-function 푓(푥) 푓(푥) [5] iff lim푥→+0 푥 = 0 and lim푥→+∞ 푥 = +∞. Every Young function 푓 allows do define a Young– Birnbaum–Orlicz dual [5, 26]

Y Y 푓 : R ∋ 푦 ↦→ 푓 (푦) := sup{푥|푦| − 푓(푥)} ∈ [0, ∞]. (35) 푥≥0

Every YBO dual is a nondecreasing Young function, and each pair (푓, 푓 Y) satisfies Young’s inequal- ity [54] Y 푥푦 ≤ 푓(푥) + 푓 (푦) ∀푥, 푦 ∈ R. (36) If 푓 is also an N-function, then 푓 YY = 푓. Every modular function Υ: 푋 → [0, ∞] determines a modular space [33, 35] 푋ϒ := {푥 ∈ 푋 | lim Υ(휆푥) = 0} (37) 휆→+0 and the Morse–Transue–Nakano–Luxemburg norm on 푋ϒ [28, 35, 24, 51],

−1 + ||·||ϒ : 푋ϒ ∋ 푥 ↦→ ||푥||ϒ := inf{휆 > 0 | Υ(휆 푥) ≤ 1} ∈ R , (38) which allows to define a Banach space

||·||ϒ 퐿ϒ(푋) := 푋ϒ . (39)

The noncommutative Orlicz spaces associated with the algebra B(ℋ) of bounded operators on a Hilbert space ℋ were implicitly introduced by Schatten [44] as ideals in B(ℋ) generated by the so- called symmetric gauge functions, and were studied in more details by Gokhberg and Kre˘ın[14] (see also [15]). First explicit study of those ideals which are direct noncommutative analogues of Orlicz spaces is due to Rao [38, 39], where Υ is assumed to be a continuous modular function, Υ(|푥|) for 푥 ∈ B(ℋ) is understood in terms of the spectral representation, an analogue of the MNTL norm reads

{︂ (︂ (︂|푥|)︂)︂ }︂ B(ℋ) ∋ 푥 ↦→ ||푥|| := inf 휆 > 0 | tr Υ ≤ 1 , (40) ϒ 휆

5 while the noncommutative Orlicz space is defined as

Gϒ(ℋ) := {푥 ∈ B(ℋ) | ||푥||ϒ < ∞}. (41) The generalisation of Orlicz spaces to semi-finite* W -algebras 풩 equipped with a faithful normal semi- finite trace 휏 were proposed by Muratov [29, 30], Dodds, Dodds, and de Pagter [9], and Kunze [20]. Two latter constructions are based on the results of Fack and Kosaki [11]. Given any 푦 ∈ M (풩 , 휏), the rearrangement function is defined as [15] (see also [52])

휏 휏 |푥| R푦 : [0, ∞[ ∋ 푡 ↦→ R푦(푡) := inf{푠 ≥ 0 | 휏(푃 (]푠, +∞[) ≤ 푡} ∈ [0, ∞]. (42)

If 푥 ∈ M (풩 , 휏)+ and 푓 : [0, ∞[→ [0, ∞[ is a continuous nondecreasing function, then [11] ∫︁ ∞ 휏 휏(푓(푥)) = d푡푓(R푥(푡)), (43) 0 휏 휏 + R푓(푥)(푡) = 푓(R푥(푡)) ∀푡 ∈ R . (44) Using this result, Kunze [20] defined a noncommutative Orlicz space associated with an arbitrary Orlicz function Υ as

퐿ϒ(풩 , 휏) := spanC{푥 ∈ M (풩 , 휏) | 휏(Υ(|푥|)) ≤ 1}, (45) equipped with a quantum version of a MNTL norm,

−1 ||·||ϒ : M (풩 , 휏) ∋ 푥 ↦→ inf{휆 > 0 | 휏(Υ(휆 |푥|)) ≤ 1}, (46) under which, as he proves, (45) is a Banach space. From linearity, it follows that

퐿ϒ(풩 , 휏) = {푥 ∈ M (풩 , 휏) | ∃휆 > 0 휏(Υ(휆|푥|)) < ∞}. (47)

On the other hand, Dodds, Dodds, and de Pagter [9] defined (implicitly) a noncommutative Orlicz space associated with (풩 , 휏) and an Orlicz function Υ as

휏 + + 퐿ϒ(풩 , 휏) := {푥 ∈ M (풩 , 휏) | R푥 ∈ 퐿ϒ(R , fBorel(R ), d휆)}. (48) By means of (43) and (44), these two definitions are equivalent3. Kunze [20] showed that, for Υ satisfying global △2 condition,

퐿ϒ(풩 , 휏) = {푥 ∈ M (풩 , 휏) | 휏(Υ(|푥|)) < ∞}, (49)

||·||ϒ 퐿ϒ(풩 , 휏) = 퐸ϒ(풩 , 휏) := 풩 ∩ 퐿ϒ(풩 , 휏) , (50) B ∼ (퐿ϒ(풩 , 휏)) = 퐿ϒY (풩 , 휏). (51)

* In [3] it is shown that if 휏1 and 휏2 are faithful normal semi-finite traces on a semi-finite W -algebra 풩 , and Υ is an Orlicz function satisfying global △2 condition, then 퐿ϒ(풩 , 휏1) and 퐿ϒ(풩 , 휏2) are isometrically isomorphic. Further analysis of the structure of 퐿ϒ(풩 , 휏) spaces in the context of modular function was provided by Sadeghi [41], who showed that the map M (풩 , 휏) ∋ 푥 ↦→ 휏(Υ(|푥|)) ∈ [0, ∞] is a modular function for any Orlicz function Υ. He also notes that the results of [7] and [10] allow to infer, respectively, the uniform convexity and reflexivity of the spaces (퐿ϒ(풩 , 휏), ||·||ϒ) from the + + corresponding properties of the commutative Orlicz spaces (퐿ϒ(R , fBorel(R ), d휆), ||·||ϒ). This leads to4

Corollary 3.1. (퐿ϒ(풩 , 휏), ||·||ϒ) is: uniformly convex if Υ is uniformly convex and satisfies global △2 Y condition; reflexive if Υ and Υ satisfy global △2 condition.

3 See a discussion in [22, 21] of the case when continuity of ϒ is relaxed to continuity on [0, 푥ϒ[ and left continuity at 푥ϒ with 푥ϒ ̸= +∞. 4The statement of the sufficient condition for reflexivity in Collorary 4.3of[41] is missing the requirement of the Y global △2 condition for ϒ .

6 Al-Rashed and Zegarliński [1,2] proposed a construction of a family of noncommutative Orlicz spaces associated with a faithful normal state on a countably finite* W -algebra. Ayupov, Chilin and Abdullaev [3] proposed the construction of a family of noncommutative Orlicz spaces 퐿ϒ(풩 , 휓) for a semi-finite* W -algebra 풩 , a faithful normal locally finite weight 휓, and an Orlicz function Υ satisfying global △2 condition. This construction extends Trunov’s theory of 퐿푝(풩 , 휏) spaces [48, 49, 56]. Labuschagne [21] provided a construction of the family of noncommutative Orlicz spaces 퐿ϒ(풩 , 휓) associated with an arbitrary W*-algebra and a faithful normal semi-finite weight 휓. The problem of the canonical (weight-independent and functorial) construction of noncommutative Orlicz spaces over arbitrary W*-algebras remains still open.

4 Orlicz spaces over standard core

Definition 4.1. For an arbitrary W*-algebra 풩 and arbitrary Orlicz function Υ, let a noncommutative core Orlicz space be defined as a vector space

퐿ϒ(풩̃︀) := {푥 ∈ M (풩̃︀, τ̃︀) | ∃휆 > 0 τ̃︀(Υ(휆|푥|)) < ∞}, (52) equipped with the norm

−1 ||·||ϒ : M (풩̃︀, τ̃︀) ∋ 푥 ↦→ inf{휆 > 0 | τ̃︀(Υ(휆 |푥|)) ≤ 1}. (53) In addition, we define ||·||ϒ 퐸ϒ(풩̃︀) := 풩 ∩ 퐿ϒ(풩̃︀) . (54)

Because 풩̃︀ is a semi-finite von Neumann algebra, while τ̃︀ is a faithful normal semi-finite trace on 풩̃︀, all above results on the Banach space structure of 퐿ϒ(풩 , 휏) immediately apply to 퐿ϒ(풩̃︀). B ∼ In particular, if Υ satisfies global △2 condition, then 퐿ϒ(풩̃︀) = 퐸ϒ(풩̃︀) and 퐿ϒ(풩̃︀) = 퐿ϒY (풩̃︀). Moreover: if Υ is also uniformly convex, then 퐿ϒ(풩̃︀) is uniformly convex and 퐿ϒY (풩̃︀) is uniformly Y Fréchet differentiable; if Υ also satisfies global △2 condition, then 퐿ϒ(풩̃︀) is reflexive. For any choice of a normal semi-finite weight 휓 on 풩 , 풩̃︀ can be represented as 풩 o휎휓 R by means of (23). Hence, for any choice of 휓, 퐿ϒ(풩̃︀) can be represented as isometrically isomorphic to Kunze’s noncommutative Orlicz spaces 퐿ϒ(풩 o휎휓 R). In what follows, will first recall Kosaki’s construction of canonical representation ofW*-algebra. Then we will use it to extend functoriality of the Falcone–Takesaki noncommutative flow of weights. (While Kosaki’s construction of canonical 퐿2(풩 ) spaces uses normal states instead of elements of 풲0(풩 ) applied in the Falcone–Takesaki 퐿2(풩 ), the algebraic character of its structure allows for using it for the purpose of canonical representation of any W*-algebra as a von Neumann algebra, which is the starting point required for the Falcone–Takesaki construction.) Finally, we will show that * *-isomorphisms of W -algebras are canonically mapped to isometric isomorphisms of 퐿ϒ(풩̃︀) spaces. + Following Kosaki [17], consider new addition and multiplication structure on 풩⋆ ,

√︀ √︀ 2 + + 휆 휑 = 휆 휑 ∀휆 ∈ R ∀휑 ∈ 풩⋆ , (55) √︀ √︀ √︀ * + 휑 + 휓 = (휑 + 휓)(푦 · 푦) ∀휑, 휓 ∈ 풩⋆ , (56) where

푦 := [휑 :(휑 + 휓)]−i/2 + [휓 :(휑 + 휓)]−i/2, (57) √ + and 휑 is understood as a symbol denoting the element 휑 of 풩⋆ whenever it is subjected to the above + operations instead of ‘ordinary’ addition and multiplication on 풩⋆ . A ‘noncommutative Hellinger + integral’ on 풩⋆ ,

(︁√︀ √︀ )︁ (︁ * )︁ 휑| 휓 := (휑 + 휓) [휓 :(휑 + 휓)]−i/2 [휑 :(휑 + 휓)]−i/2 (58)

7 + is a positive bilinear symmetric form on 풩⋆ with√ respect√ to the operations defined by (55) and (56). √ + + Consider an equivalence relation ∼ on pairs ( 휑, 휓) ∈ 풩⋆ × 풩⋆ ,

√︀ √︀ √ √︀ √︀ √︀ √︀ √︀ √︀ ( 휑1, 휓1) ∼ ( 휑2, 휓2) ⇐⇒ 휑1 + 휓2 = 휑2 + 휓1. (59)

+ + √ The set of equivalence classes 풩⋆ × 풩⋆ / ∼ can be equipped with a real vector space structure, provided by

√︀ √︀ √ √︀ √︀ √ √︀ √︀ √︀ √︀ √ ( 휑1, 휑2) + ( 휓1, 휓2) := ( 휑1 + 휓1, 휑2 + 휓2) , (60) √ √ {︂ (휆 휑, 휆 휓)√ : 휆 ≥ 0 휆 · (√︀휑, √︀휓)√ := √ √ (61) ((−휆) 휓, (−휆) 휑)√ : 휆 < 0, √ √ √ + + √ + + √ where ( 휑, 휓) denotes an element of 풩⋆ × 풩⋆ / ∼ . The real vector space (풩⋆ × 풩⋆ / ∼ , +, ·) will be denoted 푉 . The map + √︀ √ 풩⋆ ∋ 휑 ↦→ ( 휑, 0) ∈ 푉 (62) is injective, positive and preserves addition and multiplication by positive scalars. Its image in 푉 will + be denoted by 퐿2(풩 ) . A function

⟨·, ·⟩√ : 푉 × 푉 → R, (63)

⟨ √︀ √︀ √ √︀ √︀ √⟩ (︁√︀ √︀ )︁ (︁√︀ √︀ )︁ ( 휑1, 휑2) , ( 휓1, 휓2) √ : = 휑1| 휓1 + 휑1| 휓2 (︁√︀ √︀ )︁ (︁√︀ √︀ )︁ + 휑2| 휓1 + 휑2| 휓2 , (64)

√ is an inner product on 푉 , and (푉, ⟨·, ·⟩ ) is a real Hilbert space with respect to it, denoted 퐿2(풩 ; R). The canonical Hilbert space is defined as a complexification of the Hilbert space 퐿2(풩 ; R),

퐿2(풩 ) := 퐿2(풩 ; R) ⊗R C. (65)

+ + The space 퐿2(풩 ) is a self-polar convex cone in 퐿2(풩 ), and, by (62), it is an embedding of 풩⋆ + 1/2 + into 퐿2(풩 ). The elements of 퐿2(풩 ) will be denoted 휑 , where 휑 ∈ 풩⋆ . Every element of 퐿2(풩 ) + can be expressed as a linear combination of four elements of 퐿2(풩 ) . The antilinear conjugation 퐽풩 : 퐿2(풩 ) → 퐿2(풩 ) is defined by

퐽풩 (휉 + i휁) = 휉 − i휁 ∀휉, 휁 ∈ 퐿2(풩 ; R). (66)

+ The quadruple (퐿2(풩 ), 풩 , 퐽풩 , 퐿2(풩 ) ) is a standard form of 풩 , called a canonical standard form of 풩 . d (︀ 푡푥 )︀ A bounded generator ð풩 (푥) := d푡 푓(e ) |푡=0 of the norm continuous one parameter group of automorphisms 푡푥 R ∋ 푡 ↦→ 푓(e ) ∈ B(퐿2(풩 )) ∀푥 ∈ 풩 , (67) where 풩 ∋ 푥 ↦→ 푓(푥) ∈ B(퐿2(풩 )) is defined as a unique extension of the bounded linear function

1/2 + 1/2 (︁ (︁ 휑 * 휑 * )︁)︁ + + 퐿2(풩 ) ∋ 휑 ↦→ 휑 휎+i/2(푥)푥 · 푥휎−i/2(푥 ) ∈ 퐿2(풩 ) ∀푥 ∈ 풩 ∀휑 ∈ 풩⋆ , (68) determines a map ð풩 : 풩 ∋ 푥 ↦→ ð풩 (푥) ∈ B(퐿2(풩 )), (69) which is a homomorphism of real Lie algebras: for all 푥, 푦 ∈ 풩 and for all 휆 ∈ R,

[ð풩 (푥), ð풩 (푦)] = ð풩 ([푥, 푦]), (70) 휆ð풩 (푥) = ð풩 (휆푥). (71)

8 The faithful normal representation 휋풩 : 풩 → B(퐿2(풩 )),

1 휋풩 (푥) := 2 (ð풩 (푥) − ið풩 (i푥)), (72)

+ * determines a standard representation (퐿2(풩 ), 휋풩 , 퐽풩 , 퐿2(풩 ) ) of a W -algebra 풩 , called a canonical * representation of 풩 . Every *-isomorphism 휍 : 풩1 → 풩2 of W -algebras 풩1, 풩2 determines a unique + + * unitary equivalence 푢휍 : 퐿2(풩2) → 퐿2(풩1) satisfying 푢휍 (퐿2(풩2) ) = 퐿2(풩1) , 푢휍 퐽풩2 푢휍 = 퐽풩1 , * and such that Ad(푢휍 ) is a unitary implementation of 휍. This means that Kosaki’s construction of canonical representation defines a functor CanRep from the category W*Iso of W*-algebras with *- isomorphisms to the category StdRep of standard representations with standard unitary equivalences. It also allows to define a functor CanVN from the category W*n of W*-algebras with normal *- homomorphisms to the category VNn of von Neumann algebras with normal *-homomorphisms. The −1 functor CanVN assigns 휋풩 (풩 ) to each 풩 , and normal *-homomorphism 휋풩 ∘ 휍 ∘ 휋풩 : 풩1 → 풩2 to each normal *-homomorphism 휍 : 풩1 → 풩2 (which is well defined due to faithfulness of 휋풩 ). Let FrgHlb : VNn → W*n be the forgetful functor which forgets about Hilbert space structure that underlies von Neumann algebras and their normal *-homomorphisms. Due to Sakai’s theorem [42], CanVN and FrgHlb form the equivalence of categories, ∼ ∼ FrgHlb ∘ CanVN = idW*n, CanVN ∘ FrgHlb = idVNn. (73)

* * Consider the category W CovRTr of quadruples (풩 , R, 훼, 휏), where 풩 is a semi-finite W -algebra, * 휏 is a faithful normal semi-finite trace on 풩 , and (풩 , R, 훼) is a W -dynamical system, with morphisms

1 2 (풩1, R, 훼 , 휏1) → (풩2, R, 훼 , 휏2) (74) given by such *-isomorphisms 휍 : 풩1 → 풩2 which satisfy

1 2 휍 ∘ 훼푡 = 훼푡 ∘ 휍 ∀푡 ∈ R, (75) 휏1 = 휏2 ∘ 휍. (76)

Falcone and Takesaki call the quadruple (풩̃︀, R, 휎,˜ τ̃︀) a noncommutative flow of weights, and prove that every *-isomorphism 휍 : 풩1 → 풩2 of von Neumann algebras extends to a *-isomorphism 휍̃︀ : 풩̃︀1 → 풩̃︀2 satisfying (75) and (76). This defines a functor * FTﬂow : VNIso → W CovRTr, (77) where VNIso is a category of von Neumann algebras with *-isomorphisms. The restriction of 휎˜ to the center Z is the Connes–Takesaki flows of weights (Z , , 휎˜|Z ) [13]. Hence, the relationship between 풩̃︀ 풩̃︀ R 풩̃︁ the Falcone–Takesaki noncommutative flow of weights and the Connes–Takesaki flow of weights can be summarised in terms of the commutative diagram

CanVN FTflow W*Iso / VNIso / W*Cov Tr (78) O O R Z∘ForgTr O O W*facIso / VNfacIso / W*Cov , III CanVN III CTflow R

*fac fac * where W III Iso (respectively, VNIII Iso) is a category of type III factor W -algebras (respectively, von Neumann algebras) with *-isomorphisms, ForgTr denotes the forgetful functor that forgets about * * traces, while Z : W CovR → W CovR is an endofunctor that assigns an object (Z풩 , R, 훼|Z풩 ) to each 12 (풩 , R, 훼), and assigns a morphism 휍Z such that

휍12 ∘ 훼1| = 훼2| ∘ 휍12 (79) Z 푡 Z풩1 푡 Z풩2 Z

1 2 to each 휍 :(풩1, R, 훼 ) → (풩2, R, 훼 ).

9 * Proposition 4.2. Every *-isomorphism 휍 : 풩1 → 풩2 of W -algebras gives rise to a corresponding isometric isomorphism 퐿ϒ(풩̃︀1) → 퐿ϒ(풩̃︀2). Proof. By the Falcone–Takesaki construction, and its composition (78) with Kosaki’s construction, 휍 : 풩1 → 풩2 induces a *-isomorphism 휍̃︀ : 풩̃︀1 → 풩̃︀2 of semi-finite von Neumann algebras and a 1 2 mapping (풩̃︀1, R, 휎˜ , τ̃︀1) → (풩̃︀2, R, 휎˜ , τ̃︀2) satisfying 1 2 휍̃︀∘ 휎˜푡 =휎 ˜푡 ∘ 휍̃︀ ∀푡 ∈ R, (80) τ̃︀1 = τ̃︀2 ∘ 휍.̃︀ (81) By Collorary 38 in [47], every *-isomorphism of semi-finite von Neumann algebras satisfying (80) and (81) extends to a topological *-isomorphism of corresponding spaces of 휏-measurable operators affiliated with these algebras, and this extension preserves the property(81). The *-isomorphism 휍̃︀ * extends to 휍¯ : M (풩̃︀1, τ̃︀1) → M (풩̃︀2, τ̃︀2) by 휍¯(·) = 푢(·)푢 , where 푢 is a unitary operator implementing * 휍̃︀(·) = 푢(·)푢 . It remains to show that 휍¯ is an isometric isomorphism. Using (43), we can rewrite (53) as ∫︁ ∞ −1 τ̃︀ ||푥||ϒ = inf{휆 > 0 | d푡Υ(휆 R|푥|(푡)) < ∞}, (82) 0 where τ̃︀ |푥| R|푥|(푡) = inf{푠 ≥ 0 | τ̃︀(푃 (]푠, +∞[)) ≤ 푡}. (83) −1 For M (풩̃︀2, τ̃︀2) = M (휍̃︀(풩̃︀1), τ̃︀1 ∘ 휍̃︀ ) and 푥 ∈ M (풩̃︀1, τ̃︀1) we have

−1 (︁ |휍(푥)| )︁ −1 |푥| |푥| τ̃︀1 ∘ 휍̃︀ 푃 ̃︀ (푠, +∞[) = τ̃︀1 ∘ 휍̃︀ ∘ 휍̃︀(푃 (]푠, +∞[)) = τ̃︀1(푃 (]푠, +∞[)). (84)

Hence, 휍¯ : 퐿ϒ(풩̃︀1) → 퐿ϒ(풩̃︀2) is an isometric isomorphism.

Corollary 4.3. Denoting the category of noncommutative Orlicz spaces 퐿ϒ(풩̃︀) with isometric isomorphisms by ncL̃︀ϒIso, we conclude that our construction determines a functor

* ncL̃︀ϒ : W Iso → ncL̃︀ϒIso. (85)

Following the results of Sherman [45] (generalising earlier results of [53] and [50]), we end this section with an interesting problem: for which Orlicz functions Υ there exists a functor from ncL̃︀ϒIso to the category of W*-algebras with surjective Jordan *-isomorphisms? And, furthermore, for which Υ these functorial relationships can be extended to category ncLϒIso of all noncommutative Orlicz spaces with isometric isomorphisms?

Acknowledgments

I am very indebted to Professor Stanisław L. Woronowicz for introducing me to Tomita–Takesaki and Falcone–Takesaki theories. I would like also to thank Władysław Adam Majewski, Mustafa Muratov, Dmitri˘ıPavlov, and David Sherman for discussion and correspondence, as well as two anonymous referees of the previous version of this paper for their comments. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

References

All Cyrillic titles and names were transliterated from the original papers and books. For the Latin transliteration of the Cyrillic script (in references and surnames) we use the following modification of the system GOST 7.79-2000B: c = c, q = ch, h = kh, = zh, x = sh, w = shh, = yu, = ya, ¨e = ë, = ‘, ~ = ’, = è, $i = ˘ı,with an exception that names beginning with H are transliterated to H. For Russian texts: y = y, i = i; for Ukrainian: i = y, i = i, ï = ï.

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