MATRICIAL STRUCTURE OF THE HAAGERUP L p -SPACES (WORK IN PROGRESS) and describe the matricial structure of these spaces. Note that we consider the

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1 MATICIAL STUCTUE OF THE HAAGEUP L p -SPACES (WOK IN POGESS) DENIS POTAPOV AND FEDO SUKOCHEV Here, we review the construction of the Haagerup L p -spaces for the special algebra M n of all n n complex matrices (n is fixed for the rest of the section) and describe the matricial structure of these spaces. Note that we consider the algebra M n as the algebra of all bounded linear operators on l 2 n. Fix a n.s.f. weight φ on M n. Suppose that the weight φ is given by φ(x) Tr ( e Φ x ), Φ diag φ α } n α1 M n, φ α, 1 α n. (1) Let σ t be the modular group for the weight φ, i.e., the group of -automorphisms on M n such that (i) φ(σ t (x)) φ(x), x M n ; (ii) for every x, y M n, there is a function f x,y (z) holomorphic in the strip 0 < Iz < 1 and continuous in the closure of that strip such that φ (σ t (x)y) f x,y (t) and φ (yσ t (x)) f x,y (i + t), t. (2) It is not very difficult to see that for the weight φ introduced in (1) the corresponding modular group is given by σ t (x) e itφ xe itφ. Indeed, it is clear that the group σ introduced above is φ-invariant, furthermore, if x, y M n, then, the identities (2) hold with the following holomorphic function ( ) f x,y (z) Tr e (1+iz)Φ xe izφ y. The Haagerup L p spaces construction starts with the crossed product of the algebra M n with the action σ σ t } t. The crossed product algebra is the subalgebra of B(L 2 (, l 2 n)) which is the minimal algebra containing the collections π(x), x M n } and λ t } t, 1

2 2 D. POTAPOV AND F. SUKOCHEV where and π(x)ξ(t) σ t (x)ξ(t), x M n λ t (ξ)(s) ξ(s t), t, s, ξ L 2 (, l 2 n). The construction of continuous crossed product is very abstract in its nature. The first step in our discussion is to present a constructive matricial description of the algebra. Let us consider the linear space of matrix-valued functions ˆ } x(t) [x αβ (t)] n α,β1, x αβ L () equipped with the following product [xy] αβ (t φ β ) x αγ (t φ γ )y γβ (t φ β ) and involution γ1 [x ] αβ (t φ β ) x βα (t φ α ), t, x, y ˆ. Together with introduced product and involution, the space ˆ becomes a -algebra. We shall embed this algebra into B(L 2 (, l 2 n)) as follows: if η x(ξ) and η (η α ) n α1, ξ (ξ β) n β1, then η α (t φ α ) x αβ (t φ β )ξ β (t φ β ). (3) β1 Observe that the algebra ˆ has a subalgebra of constant matrix functions which is isomorphic to M n. Furthermore, observe that identity (3) gives a (twisted) representation for the algebra M n in L 2 (, l 2 n). Observe also that when the matrix Φ diag φ α } n α1 is null, i.e., the weight φ coincides with the standard trace Tr, then the algebra ˆ clearly coincides with the tensor product von Neumann algebra L () M n. Proposition 1. The crossed product is isomorphic to the algebra ˆ. Proof. Let F be the Fourier transform on L 2 (, l 2 n) and F 1 is the inverse Fourier transform, i.e., ˆξ(t) Fξ(t) 1 ξ(s) e its ds, ξ L 2 (, l 2 n)

3 MATICIAL HAAGEUP L p -SPACES 3 and ξ(t) F 1ˆξ(t) 1 ˆξ(s) e its ds, ˆξ L 2 (, l 2 n). The mappings F and F 1 are unitary transformations of L 2 (, l 2 ). We shall show that ˆ FF 1. Let t x(t) [x αβ ] n α,β1 M n be a Schwartz matrix function, i.e., a matrix function with every entry being a Schwartz scalar function. The function x(t) defines an operator T by the following formula T π(x(t)) λ t dt. The collection of all such operators T is weakly dense in (see [Ta, Ch. X, Lemma 1.8]). In order to describe the algebra FF 1, let us compute the operator ˆT FTF 1. To this end let us fix a vector function ξ L 2 (, l 2 n) and represent the functions x(t) and ξ(t) in terms of their Fourier transforms ˆx(t) and ˆξ(t), i.e., ξ(t) ˆξ(s) e its ds and x(t) ˆx(s) e its ds.

4 4 D. POTAPOV AND F. SUKOCHEV Furthermore, assuming that ˆx [ˆx αβ ] n α,β1 and ˆξ ) n (ˆξα α1, Tξ(t) π(x(k)) λ k (t)ξ(t) dk σ t (x(k)) ξ(t k) dk e itφ x(k)e itφ ξ(t k) dk [ ] [ e itφ ˆx(s) e iks ds e itφ ˆξ(m) e im(t k) dm ] dk e itφˆx(s)e itφˆξ(m) e iks+im(t k) ds dk dm 3 e itφαˆx αβ (s)e itφ β ˆξβ (m) e iks+im(t k) ds dk dm β1 3 [ ] ˆx αβ (s) ˆξ β (m) e it(m φ α+φ β ) e ik(s m) dk ds dm β1 2 ˆx αβ (s) ˆξ β (m) e it(m φ α+φ β ) δ(s m) ds dm β1 2 ˆx αβ (s)ˆξ β (s)e it(s φ α+φ β ) ds β1 β1 Consequently, if η Tξ and ˆη (ˆη α ) n α1, then ˆη α (s φ α ) ˆx αβ (s φ β ) ˆξ β (s φ β ) e it(s φ α) ˆx αβ (s φ β ) ˆξ β (s φ β ). β1 ds. Thus, we showed that the operator FTF 1 belongs to ˆ. The proof of the proposition is finished. From now on we shall identify the algebras and ˆ. The crossed product ˆ possesses a distinguished trace τ. Indeed, for an element x ˆ, introduce the functional τ by τ(x) φ(x(t)) e t dt x αα (t φ α ) e t dt, x [x αβ ] n α,β1 ˆ. α1 Proposition 2. The functional τ is a normal semi-finite faithful trace. Proof. Clearly, the functional τ is semi-finite. Let us show that it trace, i.e., τ (xy) τ (yx).

5 MATICIAL HAAGEUP L p -SPACES 5 Due to the polarization identity, it is sufficient to check that τ(x x) τ(xx ). Let us fix x [x αβ ] n α,β1 and compute τ(x x). We have [x x] αβ (t φ β ) Consequently, τ (x x) [x ] αγ (t φ γ )x γβ (t φ β ) γ1 [x x] αα (t φ α ) e t dt α1 α,γ1 x γα (t φ α )x γβ (t φ β ). γ1 x γα (t φ α )x γα (t φ α ) e t dt α,γ1 x γα (t φ α ) 2 e t ds. Clearly, if we replace x with x, i.e., x γα (t φ α ) with x αγ (t φ γ ) the latter quantity does not change. In other words, τ(x x) τ(xx ) and therefore τ is a trace. The expression for τ(x x) above also clearly implies that the trace τ is faithful. Thus, the proposition is proved. Consider the group of translations on ˆ, i.e., the group θ θ t } t such that θ t (x)(s) x(s + t), t, s, x ˆ. The action θ θ t } t is called the dual action on ˆ. It is not very difficult to see that τ (θ t (x)) e t τ(x), x ˆ. Let us denote by the algebra of all τ-measurable operators with respect to the couple (, τ). Clearly, the -algebra is alternatively described as follows } x(t) [x αβ ] α,β1, x αβ S(e t dt), where S(e t dt) is the algebra of all measurable (with respect to the trace e t dt) functions on.

6 6 D. POTAPOV AND F. SUKOCHEV The Haagerup L p -space is the subspace of of all elements x ˆ such that θ t (x) e t/p x, t, i.e., L p (M n ) x : } θ t (x) e t/p x. Theorem 3. (i) The space L p (M n ) admits the following description L p (M n ) x ] } : x [e (t+φβ)/p ψ αβ nα,β1, [ψ αβ] M n. (ii) If µ(x) is the decreasing rearrangement of a element x (with respect to the trace τ), then µ t (x) const t 1/p, t, x Lp (M n ). (iii) If an element x L p (M n ) has the representation x [ e (t+φ β)/p ψ αβ ] n α,β1 for some matrix ψ [ψ αβ ] n α,β1, then the constant from the statement above is equal to ψ S p, where S p is the Schatten-von Neumann class. Proof. Clearly every matrix function x [ ] e (t+φβ)/p n ψ αβ satisfies the equa- α,β1 tion Conversely, if x [x αβ ] n α,β1 satisfies (4), then Setting ψ x(0) e Φ/p M n yields that Thus, (i) follows. θ t (x) e t/p x. (4) x(t) θ t (x)(0) e t/p x(0). x(t) [ e (t+φ β)/p ψ αβ ] n α,β1. Let us show (ii) and (iii). Fix the element x [ e (t+φ β) ψ αβ ] n α,β1 Lp (M n ). Without loss of generality, we may assume that the matrix ψ is positive and diagonal. In this case x is also positive operator. Let ψ αα δ α > 0 and ψ αβ 0 if α β. ecall that µ t (x) inf s > 0 : τ ( χ (s,+ ) (x) ) t },

7 MATICIAL HAAGEUP L p -SPACES 7 where χ(x) is the spectral measure of x. Observe that e (t+φ α)/p δ α > s t + φ α p > log s δ α t < p log s δ α φ α. Consequently, χ (s,+ ) (x) diag χ (,p log s δα φ α) } n α1 ˆ. Furthermore, τ(χ (s,+ ) (x)) The instantly implies that α1 [ χ(s,+ ) (x) ] αα (t φ α) e t dt α1 χ (,p log s p log s δα α1 δα ) (t) et dt e t dt µ t (x) const t 1/p, 1 s p δ p α. α1 where const ( n α1 δ p α ) 1 p ψ S p. Thus, (ii) and (iii) are proved. The proof of the theorem is finished. At the end of this section, let us note that a similar argument may be given in order to describe Haagerup L p -spaces with respect to the algebra L (). We shall state the result without the proof. We leave details to the audience. Theorem 4. Let M L (), L ( ) and let S(dtds) (the space of all measurable functions on the plane ). (i) The space L p (M) admits the following description L p (M) x : } x(t, s) e t/p ψ(s), ψ L p (ds).

8 8 D. POTAPOV AND F. SUKOCHEV (ii) If µ(x) is the decreasing rearrangement of an element x (with respect to the trace e t dtds), then µ t (x) const t 1/p, t, x Lp (M). (iii) If an element x L p (M) has the representation x(t, s) e t/p ψ(s) for some function ψ L p (ds), then the constant from the statement above is equal to ψ L p (ds). address: denis.potapov@flinders.edu.au address: sukochev@infoeng.flinders.edu.au