A remark on non-commutative $L^p$-spaces

We explicitly describe the Haagerup and the Kosaki non-commutative $L^p$-spaces associated with a tensor product von Neumann algebra $M_1\bar{\otimes}M_2$ in terms of those associated with $M_i$ and usual tensor products of unbounded operators. The descriptions are then shown to be useful in the quantum information theory based on operator algebras.


Introduction
Quantum information theory (QIT for short) can be developed in the infinite-dimensional (even non-type I) setup with the help of operator algebras (such a general framework is necessary for quantum field theory for example), although QIT is usually discussed in the finitedimensional setup.In the finite-dimensional setup, the primary objects in QIT are density matrices, which no longer make sense in the non-type I setup.However, Haagerup's theory of non-commutative L p -spaces (see [19]) allows us to have a certain counterpart of density matrices; actually, the so-called Haagerup correspondence ϕ → h ϕ (the operator h ϕ is sometimes denoted by ϕ itself) between the normal functionals and a class of τ -measurable operators gives a correct counterpart of density matrices in the non-type I setting.
In QIT, tensor products of systems (i.e., systems consisting of independent subsystems) naturally emerges, and hence it is desirable to clarify how Haagerup non-commutative L pspaces behave under von Neumann algebra tensor products.In the commutative setup, the answer is simply L p (µ 1 ⊗ µ 2 ) = L p (µ 1 , L p (µ 2 )) = L p (µ 2 , L p (µ 1 )) with natural identifications by utilizing the concept of vector-valued L p -spaces.However, the concept of vector-valued L p -spaces has not been established yet in full generality of non-commutative setting.
The purpose of this short note is to give some descriptions of the Haagerup and the Kosaki non-commutative L p -spaces associated with a tensor product von Neumann algebra; see Theorem 6 and Corollary 7.Those descriptions are indeed rather natural but has not been given so far to the best of our knowledge.We remark that a similar but abstract result based on interpolation method was given by Junge [10] before.On the other hand, the descriptions we will give depend upon a technology of the so-called Takesaki duality [17] and are provided by means of tensor products of unbounded operators.Consequently, our descriptions are really concrete with multiplicativity of norm.An immediate consequence of our descriptions is a natural proof of the additivity of sandwiched Rényi divergences in the non-type I setup due to Berta et al. [3] and Jenčová [8,9].Remark that the additivity was claimed by Berta et al., in a different approach to non-commutative L p -spaces (see [3, page 1860]) and also confirmed by Hiai and Mosonyi [6,equation (3.16)] in the injective or AFD von Neumann algebra case.
JP18H01122.The lectures motivated us to investigate non-commutative L p -spaces associated with tensor product von Neumann algebras.We also thank him for his comments to a draft of this note.

Preliminaries
The basic references of this short note are [16] (on modular theory), [19] (on Haagerup noncommutative L p -spaces), [13] (on Kosaki non-commutative L p -spaces), but the reader can find concise expositions on those topics in [4, Appendix A] and its expansion [5].
Let M be a von Neumann algebra.Choose a faithful semifinite normal weight ϕ on M .The continuous core of M is the crossed product M := M ⋊σ ϕ R. Let θ M : R M be the dual action, which is characterized by ) = e −its λ ϕ (t) for all s, t ∈ R, where π ϕ : M → M ⋊σ ϕ R and λ ϕ : R → M denote the canonical injective normal * -homomorphism from M and the canonical unitary representation of R into M that generated by the π ϕ (a) and the λ ϕ (t) as a von Neumann algebra.In what follows, we will identify a = π ϕ (a) and M = π ϕ (M ) unless no confusions are possible.Note that the covariant relation holds.We remark that ( M , θ M ) is known to be independent of the choice of ϕ up to conjugacy.
The canonical trace τ M on M is a faithful semifinite normal tracial weight uniquely determined by (3) [D ϕ : where [D ϕ : Dτ M ] t is Connes's Radon-Nikodym cocycle of ϕ with respect to τ M .Here, ϕ is the dual weight of ϕ defined by (4) where ϕ is the canonical extension of ϕ to the extended positive part M + (see e.g., [16, §11]) and T M : M + → M + is the operator-valued weight In what follows, we denote by s(ψ) the support projection of a semifinite normal weight ψ.We also use Connes's Radon-Nikodym cocycle with general (not necessarily faithful) semifinite normal weight on the left-hand side, see [16, §3].
Lemma 1.Let ψ be a semifinite normal weight on M and ψ ′ be another semifinite normal weight on M such that s(ψ ′ ) = 1 − s(ψ).Then χ := ψ + ψ ′ is a faithful semifinite normal weight on M and [Dψ : The Haagerup correspondence ϕ → h ϕ is a bijection from the set of all semifinite normal weights on M onto the positive self-adjoint operators h affiliated with M satisfying that θ M s (h) = e −s h for every s ∈ R.
where h it ψ is the functional calculus f t (h ψ ) with function . By Lemma 1 we observe that for every t ∈ R. By the chain rule of Connes's Radon-Nikodym cocycles, we have holds for every t ∈ R. By [16,Theorem 11.9] we observe for every t ∈ R. Consequently, we have for every t ∈ R.
The Haagerup non-commutative L p -space L p (M ), 0 < p ≤ ∞, is defined to be all τ Mmeasurable operators h affiliated with M such that θ M t (h) = e −t/p h for all t ∈ R. The details on the space are referred to [19].
Here is another lemma, which is probably a known fact, but we give its proof for the sake of completeness.
Lemma 3. Assume that p ≥ 1 and ϕ is a faithful normal positive linear functional so that M must be σ-finite.Let A ⊂ M be a σ-weakly dense * -subalgebra.Then Ah Proof.We will use the (left) Kosaki non-commutative L p -space L p (M, ϕ) with norm • p,ϕ , which is the complex interpolation space C 1/p (M h ϕ , L 1 (M )), where the embedding a ∈ M → ah ϕ ∈ L 1 (M ) gives a compatible pair with norm For a given a ∈ M the Kaplansky density theorem enables us to choose a net a λ ∈ A in such a way that a λ M ≤ a M for all λ and a λ → a in the σ-weak topology.By means of complex interpolation theory, we obtain that ϕ , because so is M h ϕ thanks to a general fact on complex interpolation spaces.Hence, for each x ∈ L p (M ) there exists a sequence a n ∈ A so that a n h ϕ − xh and since [13, equation ( 21)] (with η = 0 there), we conclude that a n h

Main Results
Let M i , i = 1, 2, be von Neumann algebras.For each i = 1, 2, we choose a faithful semifinite normal weight ϕ i on M i .Let be the continuous cores of M i , i = 1, 2, and M 1 ⊗M 2 together with the dual actions θ (i) := θ Mi : R M i , i = 1, 2, and θ := θ M1 ⊗M2 : R M 1 ⊗M 2 .The next fact is known in the structure analysis of type III factors.Especially, the fact is known among specialists on type III factors as a key tool to compute invariants such as flows of weights for tensor product type III factors.
Let G := R 2 > H := {(t, t); t ∈ R}, a closed subgroup, and define σ g := σ ϕ1 t1 ⊗σ ϕ2 t2 for g = (t 1 , t 2 ) ∈ G. Then we have an action σ : G M 1 ⊗M 2 , and its restrication to H is the modular action σ ϕ1 ⊗ϕ2 t = σ ϕ1 t ⊗σ ϕ2 t .Thus, we have on the first identity, the second inclusion is the natural one, and on the third identity.Here, π σ : M 1 ⊗M 2 → (M 1 ⊗M 2 ) ⋊σ G and λ σ : G → (M 1 ⊗M 2 ) ⋊σ G denote the canonical injective normal * -homomorphism and the canonical unitary representation, respectively.We have G = G with the dual pairing (t between G and its copy, and H becomes {(−t, t); t ∈ R} in G.Moreover, the dual action σg with g = (t 1 , t 2 ) ∈ G is given by θ t1 ⊗θ (2) t2 via the above identification.Hence, the desired first assertion immediately follows by [16,Theorem 21.8].Then the desired identity of the dual action θ t can easily be confirmed by investigating its behavior on the canonical generators.
In what follows, we use the description of the continuous core of M 1 ⊗M 2 equipped with the dual action θ in Lemma 4.
Remark that τ M1 ⊗M2 cannot be identified with a restriction of the tensor product trace τ M1 ⊗τ M2 .However, τ M1 ⊗M2 is characterized by in the description of Lemma 4. This is indeed a key fact in the discussion below.
, whose entries can be regarded as unbounded operators on Hilbert spaces H i , i = 1, 2, on which M i are constructed.Let Here is a lemma.Lemma 5.The following hold true: (1) ) is the polar decomposition, where the tensor product of τ -measurable operators is understood as that on (3) h ψ1 ⊗h ψ2 = h ψ1 ⊗ψ2 and (h ψ1 ⊗h ψ2 ) it = h it ψ1 ⊗ψ2 .
By Lemma 5 we have Consequently, we have the first part of the following theorem: Theorem 6.For any pair (x 1 , x 2 ) ∈ L p (M 1 ) × L p (M 2 ) the unbounded operator tensor product x 1 ⊗x 2 affiliated with M 1 ⊗ M 2 actually gives an element of L p (M 1 ⊗M 2 ), and then holds.
The mapping (x 1 , x 2 ) → x 1 ⊗x 2 is clearly bilinear, and induces a natural map from the vector space tensor product L p (M 1 ) ⊗ alg L p (M 2 ) into L p (M 1 ⊗M 2 ), which has dense image when both M i are σ-finite and p ≥ 1.
Proof.Let us prove the second part.We can assume that both ϕ i are faithful normal states.Then (M ⊗ alg N )h 1/p ϕ1 ⊗ϕ2 is dense in L p (M ⊗N ) by Lemma 3. Therefore, the (ah . Hence we are done. We do not know whether or not the second assertion (the density of the induced map) in the above theorem holds without σ-finiteness.However, we think that an approximation by σ-finite projections might give the same assertion without σ-finiteness.We leave this question to the interested reader.
Here is a corollary on the Kosaki non-commutative L p -space L p (M, ϕ) η with 1 ≤ p ≤ ∞ and 0 ≤ η ≤ 1, which is defined as the complex interpolation space Assume that both M i are σ-finite, and both ϕ i are faithful normal positive linear functionals.For each 1 ≤ p ≤ ∞ and 0 ≤ η ≤ 1, the mapping in Theorem 6 induces a bilinear map from the vector space tensor product L p (M 1 , ϕ 1 ) η ⊗ alg L p (M 2 , ϕ 2 ) η into L p (M 1 ⊗M 2 , ϕ 1 ⊗ϕ 2 ) η with dense image, and then , respectively, where q is the dual exponent of p, that is, 1/p + 1/q = 1.

A Sample of Application in QIT
Here we illustrate how our description of non-commutative L p -spaces with ψ = 0 allows several definitions, one of which is , where This formulation is mainly due to Jenčová.See [4, 3.3].
The sandwiched α-Rényi divergence Q α (ψ||ϕ) admits a two parameter extension, called the α-z-Rényi divergence, in the finite-dimensional or more generally the infinite-dimensional type I setup.See [7], [2], [14] in historical order.Here we propose a possible definition of its non-type I extension, for we want to explain how the present description of non-commutative L p -spaces associated with tensor product von Neumann algebras works even for the extension.Let α, z > 0 with α = 1 be arbitrarily given.For each pair ϕ, ψ ∈ M + * with ψ = 0 we define .Since all the τ -measurable operators form a * -algebra, we have Moreover, h ϕ is a τ -measurable operator and non-singular affiliated with s(ϕ) M s(ϕ).In addition, f (t) = t (α−1)/2z is a continuous strictly monotone increasing function on [0, ∞) with f (0) = 0 if α > 1.Hence, h (α−1)/2z ϕ is also τ -measurable (see e.g., [5,Proposition 4.19]; but this fact is implicitly utilized in the general theory of Haagerup non-commutative L p -spaces that we have employed) and non-singular.Thus, x − y = 0 by [12, Lemma 2.1], and hence x = y.
should be called the α-z-Rényi divergence.
The next fact was claimed for the sandwiched α-Rényi divergence Q α (ψ||ϕ) in [3, page 1860] without detailed proof.Then, its detailed proof when both M i are injective or AFD was given by Hiai and Mosonyi [6, equation (3.16)] by using the finite-dimensional result and also the martingale convergence property that they established.We believe that the proof below is more natural than those.
Proof.We first consider the case when α < 1.We have, by Lemma 5(2)(3) together with Lemma 11, (h as unbounded operators on H 1 ⊗H 2 , on which M 1 ⊗ M 2 naturally act.Since both the tensor components of the above right-most side fall into L 1 (M i ), i = 1, 2, respectively, we conclude, by Theorem 6, that the desired multiplicativity of Q α,z holds true.
Let x ′ and y ′ be τ P -and τ Q -measurable operators, respectively.It is known that the usual products xx ′ and yy ′ are densely defined closable, and then the closures xx ′ and yy ′ become τ P -and τ Q -measurable again.By [15,Lemma 7.22] we have (xx ′ ) ⊗(yy ′ ) = (xx ′ ) ⊗(yy ′ ), whose right-hand side coincides with the closure of (x ⊗y)(x ′ ⊗y ′ ).Thus, the following holds: Lemma 11.All the linear combinations of 'simple tensors' x ⊗y with τ P -and τ Q -measurable xηP and yηQ form a * -algebra with strong sum and strong product.We simply understand (x ⊗y)(x ′ ⊗y ′ ) as the strong product of x ⊗y and x ′ ⊗y ′ without the use of closure sign.With this notational rule, (x ⊗y)(x ′ ⊗y ′ ) = (xx ′ ) ⊗(yy ′ ) holds for any τ P -and τ Q -measurable x, x ′ ηP and y, y ′ ηQ, where xx ′ and yy ′ are understood as the strong product.

Lemma 2 .
Let ψ be a semifinite normal weight on M and ψ := ψ • T M be its dual weight.Then [D ψ : Dτ M ] t = [Dψ : Dϕ] t λ ϕ (t) holds for every t ∈ R, and the Haagerup correspondence h ψ is uniquely determined by(6)