Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Matricial structure of the Haagerup Lp-spaces(work in progress)

Denis Potapov Fedor Sukochevsukochev@infoeng.flinders.edu.au

denis.potapov@flinders.edu.au

Flinders University

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Contents

1 Matrix algebra Mn

2 Modular group

3 Continuous crossed product

4 The trace

5 The dual action

6 Haagerup Lp-spaces, 1 6 p 6 ∞

7 Proof of the main theorem

8 Commutative version

9 Appendix (construction of the isomorphism between R and R̂)

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Matrix algebra Mn

Let n be a fixed integer. Let Mn is the algebra of all n × ncomplex matrices

Mn ={

x =[

xαβ

]n

α,β=1, x ∈ C

}

.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Matrix algebra Mn

Let n be a fixed integer. Let Mn is the algebra of all n × ncomplex matrices

Mn ={

x =[

xαβ

]n

α,β=1, x ∈ C

}

.

Fix a n.s.f. weight φ on Mn. Suppose that φ is given by

φ(x) = Tr(

eΦx)

, Φ = diag {φα}nα=1 ∈ Mn,

φα ∈ R, 1 6 α 6 n. (1)

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Matrix algebra Mn

Let n be a fixed integer. Let Mn is the algebra of all n × ncomplex matrices

Mn ={

x =[

xαβ

]n

α,β=1, x ∈ C

}

.

Fix a n.s.f. weight φ on Mn. Suppose that φ is given by

φ(x) = Tr(

eΦx)

, Φ = diag {φα}nα=1 ∈ Mn,

φα ∈ R, 1 6 α 6 n. (1)

The talk shall present the Haagerup construction of Lp-spacesassociated with the couple (Mn,φ).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Modular group

Every n.s.f. weight φ on a von Neumann algebra possesses theunique modular group σ, i.e., a group of ∗-automorphisms suchthat

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Modular group

Every n.s.f. weight φ on a von Neumann algebra possesses theunique modular group σ, i.e., a group of ∗-automorphisms suchthat

1 φ(σt(x)) = φ(x), x ∈ Mn;

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Modular group

Every n.s.f. weight φ on a von Neumann algebra possesses theunique modular group σ, i.e., a group of ∗-automorphisms suchthat

1 φ(σt(x)) = φ(x), x ∈ Mn;

2 ∀x , y ∈ Mn, ∃fx ,y (z), holomorphic in 0 < Im z < 1 andcontinuous in 0 6 Im z 6 1 such that

φ (σt(x)y) = fx ,y (t) and

φ (yσt(x)) = fx ,y (i + t), t ∈ R. (2)

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Modular group

Every n.s.f. weight φ on a von Neumann algebra possesses theunique modular group σ, i.e., a group of ∗-automorphisms suchthat

1 φ(σt(x)) = φ(x), x ∈ Mn;

2 ∀x , y ∈ Mn, ∃fx ,y (z), holomorphic in 0 < Im z < 1 andcontinuous in 0 6 Im z 6 1 such that

φ (σt(x)y) = fx ,y (t) and

φ (yσt(x)) = fx ,y (i + t), t ∈ R. (2)

Remark

If the weight φ is tracial, then the modular group is trivial andthe function fx ,y (z) is constantly φ(xy).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Modular group (cont.)

Lemma

The modular group σ for the weight φ defined in (1) is given by

σt(x) = e itΦxe−itΦ, t ∈ R.

Proof.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Modular group (cont.)

Lemma

The modular group σ for the weight φ defined in (1) is given by

σt(x) = e itΦxe−itΦ, t ∈ R.

Proof.

The group σt is φ-invariant:

φ(σt(x)) = Tr(

eΦσt(x))

= Tr(

eΦe itΦxe−itΦ)

= Tr(eΦx) = φ(x).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Modular group (cont.)

Lemma

The modular group σ for the weight φ defined in (1) is given by

σt(x) = e itΦxe−itΦ, t ∈ R.

Proof.

If x , y ∈ Mn and if fx ,y (z) = Tr(

e(1+iz)Φxe−izΦy)

, then

1 fx ,y is holomorphic in 0 < Im z < 1 and continuousin 0 6 Im z 6 1;

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Modular group (cont.)

Lemma

The modular group σ for the weight φ defined in (1) is given by

σt(x) = e itΦxe−itΦ, t ∈ R.

Proof.

If x , y ∈ Mn and if fx ,y (z) = Tr(

e(1+iz)Φxe−izΦy)

, then

1 fx ,y is holomorphic in 0 < Im z < 1 and continuousin 0 6 Im z 6 1;

2 fx ,y (t) = Tr(

e(1+it) Φxe−it Φy)

= φ (σt(x) y)

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Modular group (cont.)

Lemma

The modular group σ for the weight φ defined in (1) is given by

σt(x) = e itΦxe−itΦ, t ∈ R.

Proof.

If x , y ∈ Mn and if fx ,y (z) = Tr(

e(1+iz)Φxe−izΦy)

, then

1 fx ,y is holomorphic in 0 < Im z < 1 and continuousin 0 6 Im z 6 1;

2 fx ,y (t) = Tr(

e(1+it) Φxe−it Φy)

= φ (σt(x) y)

3 fx ,y (i + t) = Tr(

e it Φxe(1−it) Φy)

= φ (yσt(x)).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (abstract approach)

Let R is the continuous crossed product of Mn

and σ = {σt }t∈R.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (abstract approach)

Let R is the continuous crossed product of Mn

and σ = {σt }t∈R.

Algebra R ⊆ B(L2(R, ℓ2n)) is the minimal von Neumann algebrainduced by

{π(x), x ∈ Mn} and {λt }t∈R,

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (abstract approach)

Let R is the continuous crossed product of Mn

and σ = {σt }t∈R.

Algebra R ⊆ B(L2(R, ℓ2n)) is the minimal von Neumann algebrainduced by

{π(x), x ∈ Mn} and {λt }t∈R,

whereπ(x)ξ(t) = σ−t(x)ξ(t), x ∈ Mn

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (abstract approach)

Let R is the continuous crossed product of Mn

and σ = {σt }t∈R.

Algebra R ⊆ B(L2(R, ℓ2n)) is the minimal von Neumann algebrainduced by

{π(x), x ∈ Mn} and {λt }t∈R,

whereπ(x)ξ(t) = σ−t(x)ξ(t), x ∈ Mn

andλt(ξ)(s) = ξ(s − t), t, s ∈ R, ξ ∈ L2(R, ℓ2n).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (abstract approach)

Let R is the continuous crossed product of Mn

and σ = {σt }t∈R.

Algebra R ⊆ B(L2(R, ℓ2n)) is the minimal von Neumann algebrainduced by

{π(x), x ∈ Mn} and {λt }t∈R,

whereπ(x)ξ(t) = σ−t(x)ξ(t), x ∈ Mn

andλt(ξ)(s) = ξ(s − t), t, s ∈ R, ξ ∈ L2(R, ℓ2n).

Remark

The mapping π : Mn 7→ R is a ∗-representation of thealgebra M on L2(R, ℓ2n).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (constructiveapproach)

R̂ ={

x(t) =[

xαβ(t)]n

α,β=1, xαβ ∈ L∞(R)

}

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (constructiveapproach)

R̂ ={

x(t) =[

xαβ(t)]n

α,β=1, xαβ ∈ L∞(R)

}

[xy ]αβ (t − φβ) =

n∑

γ=1

xαγ(t − φγ)yγβ(t − φβ)

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (constructiveapproach)

R̂ ={

x(t) =[

xαβ(t)]n

α,β=1, xαβ ∈ L∞(R)

}

[xy ]αβ (t − φβ) =

n∑

γ=1

xαγ(t − φγ)yγβ(t − φβ)

[x∗]αβ (t − φβ) = x̄βα(t − φα), t ∈ R, x , y ∈ R̂

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (constructiveapproach)

R̂ ={

x(t) =[

xαβ(t)]n

α,β=1, xαβ ∈ L∞(R)

}

[xy ]αβ (t ) =

n∑

γ=1

xαγ(t )yγβ(t )

[x∗]αβ (t ) = x̄βα(t ), t ∈ R, x , y ∈ R̂

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (constructiveapproach)

R̂ ={

x(t) =[

xαβ(t)]n

α,β=1, xαβ ∈ L∞(R)

}

[xy ]αβ (t ) =

n∑

γ=1

xαγ(t )yγβ(t )

[x∗]αβ (t ) = x̄βα(t ), t ∈ R, x , y ∈ R̂

Remark

If Φ = diag {φα}nα=1 is null, i.e., φ coincides with Tr ,

then R̂ = L∞(R)⊗̄Mn.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (constructiveapproach)

R̂ ={

x(t) =[

xαβ(t)]n

α,β=1, xαβ ∈ L∞(R)

}

[xy ]αβ (t − φβ) =

n∑

γ=1

xαγ(t − φγ)yγβ(t − φβ)

[x∗]αβ (t − φβ) = x̄βα(t − φα), t ∈ R, x , y ∈ R̂

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (constructiveapproach)

R̂ ={

x(t) =[

xαβ(t)]n

α,β=1, xαβ ∈ L∞(R)

}

[xy ]αβ =

n∑

γ=1

xαγ yγβ

[x∗]αβ = x̄βα , t ∈ R, x , y ∈ R̂

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (constructiveapproach)

R̂ ={

x(t) =[

xαβ(t)]n

α,β=1, xαβ ∈ L∞(R)

}

[xy ]αβ =

n∑

γ=1

xαγ yγβ

[x∗]αβ = x̄βα , t ∈ R, x , y ∈ R̂

Remark

The algebra R̂ has a subalgebra of constant matrix functionsisomorphic to Mn.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (constructiveapproach, cont.)

Proposition

The crossed product R is isomorphic to R̂. The isomprphism isimplemented via the Fourier transform, i.e., themapping T ∈ R 7→ T̂ ∈ R̂ = FTF−1, where F is the Fouriertransform on L2(R, ℓ2n).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Continuous crossed product (constructiveapproach, cont.)

Proposition

The crossed product R is isomorphic to R̂. The isomprphism isimplemented via the Fourier transform, i.e., themapping T ∈ R 7→ T̂ ∈ R̂ = FTF−1, where F is the Fouriertransform on L2(R, ℓ2n).

The von Neumann algebra R̂ is represented on L2(R, ℓ2n) asfollows: if η = x(ξ) and η = (ηα)

nα=1, ξ =

(

ξβ

)n

β=1, then

ηα(t − φα) =

n∑

β=1

xαβ(t − φβ)ξβ(t − φβ). (3)

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

The distinguished trace on R̂

For x =[

xαβ

]n

α,β=1∈ R̂, introduce τ by

τ(x) =

∫

R

φ(x(t)) etdt =

n∑

α=1

∫

R

xαα(t − φα) etdt.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

The distinguished trace on R̂

For x =[

xαβ

]n

α,β=1∈ R̂, introduce τ by

τ(x) =

∫

R

φ(x(t)) etdt =

n∑

α=1

∫

R

xαα(t − φα) etdt.

Proposition

The functional τ is a normal semi-finite faithful trace on R̂.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

The trace (cont.)

Proof.

τ is semi-finite.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

The trace (cont.)

Proof.

τ is tracial, i.e., τ(x∗x) = τ(xx∗).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

The trace (cont.)

Proof.

τ is tracial, i.e., τ(x∗x) = τ(xx∗).

τ (x∗x) =

n∑

α=1

∫

R

[x∗x ]αα (t − φα) etdt

=

n∑

α,γ=1

∫

R

x̄γα(t − φα)xγα(t − φα) etdt

=

n∑

α,γ=1

∫

R

|xγα(t − φα)|2 etds

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

The trace (cont.)

Proof.

τ is tracial, i.e., τ(x∗x) = τ(xx∗).

τ (xx∗) =

n∑

α=1

∫

R

[xx∗]αα (t − φα) etdt

=

n∑

α,γ=1

∫

R

xαγ(t − φγ) x̄αγ(t − φγ) etdt

=

n∑

α,γ=1

∫

R

|x̄αγ(t − φγ)|2 etds

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

The trace (cont.)

Proof.

τ is tracial, i.e., τ(x∗x) = τ(xx∗).

τ (xx∗) =

n∑

α=1

∫

R

[xx∗]αα (t − φα) etdt

=

n∑

α,γ=1

∫

R

xαγ(t − φγ) x̄αγ(t − φγ) etdt

=

n∑

α,γ=1

∫

R

|x̄αγ(t − φγ)|2 etds

τ is faithful

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

The dual action

θ = {θt }t∈Ris the group of translations on R̂:

θt(x)(s) = x(s + t), t, s ∈ R, x ∈ R̂.

The group θ is called the dual group.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

The dual action

θ = {θt }t∈Ris the group of translations on R̂:

θt(x)(s) = x(s + t), t, s ∈ R, x ∈ R̂.

The group θ is called the dual group.

Proposition

The group θ = {θt }t∈Ris dual (via the Fourier transform) to

the group λ = {λt }t∈R.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Haagerup Lp-spaces, 1 6 p 6 ∞ (definition)

R̃ the ∗-algebra of all τ-measurable operators for (R̂, τ). R̃ isdescribed as follows

R̃ ={

x(t) =[

xαβ

]∞

α,β=1, xαβ ∈ S(etdt)

}

,

where S(etdt) is the algebra of all measurable (with respect tothe trace etdt) functions on R.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Haagerup Lp-spaces, 1 6 p 6 ∞ (definition)

R̃ the ∗-algebra of all τ-measurable operators for (R̂, τ). R̃ isdescribed as follows

R̃ ={

x(t) =[

xαβ

]∞

α,β=1, xαβ ∈ S(etdt)

}

,

where S(etdt) is the algebra of all measurable (with respect tothe trace etdt) functions on R.

Definition

The Haagerup Lp-space is the subspace of R̃ of allelements x ∈ R̂ such that θt(x) = e−t/px , t ∈ R, i.e.,

Lp(Mn) ={

x ∈ R̃ : θt(x) = e−t/px}

.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Haagerup Lp-spaces, 1 6 p 6 ∞ (description)

Theorem

1 The space Lp(Mn) admits the following description

Lp(Mn) =

{[

e−(t+φβ)/pψαβ

]n

α,β=1,[

ψαβ

]

∈ Mn

}

.

2 If µ is the decreasing rearrangement (wrt τ), then

µt(x) =k(x)

t1/p, t ∈ R, x ∈ Lp(Mn).

3 If x =[

e−(t+φβ)/pψαβ

]n

α,β=1∈ Lp(Mn) for

some ψ =[

ψαβ

]n

α,β=1, then k(x) = ‖ψ‖

Sp .

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Proof of the main theorem

Clearly every matrix function x =[

e−(t+φβ)/pψαβ

]n

α,β=1

satisfies the equation

θt(x) = e−t/px . (4)

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Proof of the main theorem

Clearly every matrix function x =[

e−(t+φβ)/pψαβ

]n

α,β=1

satisfies the equation

θt(x) = e−t/px . (4)

Conversely, if x =[

xαβ

]n

α,β=1satisfies (4), then

x(t) = θt(x)(0) = e−t/px(0).

Setting ψ = x(0) eΦ/p ∈ Mn yields that

x(t) =[

e−(t+φβ)/pψαβ

]n

α,β=1.

Thus, (1) follows.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Proof of the main theorem

Fix x =[

e−(t+φβ)ψαβ

]n

α,β=1∈ Lp(Mn).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Proof of the main theorem

Fix x =[

e−(t+φβ)ψαβ

]n

α,β=1∈ Lp(Mn). We may assume

that ψ is positive and diagonal, i.e., let ψαα = δα > 0and ψαβ = 0 if α 6= β.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Proof of the main theorem

Fix x =[

e−(t+φβ)ψαβ

]n

α,β=1∈ Lp(Mn). We may assume

that ψ is positive and diagonal, i.e., let ψαα = δα > 0and ψαβ = 0 if α 6= β.

µt(x) = inf{s > 0 : τ

(

χ(s,+∞)(x))

6 t}

,

where χ(x) is the spectral measure of x .

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Proof of the main theorem

Fix x =[

e−(t+φβ)ψαβ

]n

α,β=1∈ Lp(Mn). We may assume

that ψ is positive and diagonal, i.e., let ψαα = δα > 0and ψαβ = 0 if α 6= β.

µt(x) = inf{s > 0 : τ

(

χ(s,+∞)(x))

6 t}

,

where χ(x) is the spectral measure of x .

e−(t+φα)/pδα > s ⇐⇒ −t + φα

p> log

s

δα

⇐⇒ t < −p logs

δα− φα.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Proof of the main theorem

Fix x =[

e−(t+φβ)ψαβ

]n

α,β=1∈ Lp(Mn). We may assume

that ψ is positive and diagonal, i.e., let ψαα = δα > 0and ψαβ = 0 if α 6= β.

µt(x) = inf{s > 0 : τ

(

χ(s,+∞)(x))

6 t}

,

where χ(x) is the spectral measure of x .

e−(t+φα)/pδα > s ⇐⇒ −t + φα

p> log

s

δα

⇐⇒ t < −p logs

δα− φα.

χ(s,+∞)(x) = diag{

χ(−∞,−p log sδα

−φα)

}n

α=1∈ R̂.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Proof of the main theorem

Fix x =[

e−(t+φβ)ψαβ

]n

α,β=1∈ Lp(Mn). We may assume

that ψ is positive and diagonal, i.e., let ψαα = δα > 0and ψαβ = 0 if α 6= β.

µt(x) = inf{s > 0 : τ

(

χ(s,+∞)(x))

6 t}

,

where χ(x) is the spectral measure of x .

τ(χ(s,+∞)(x)) =

n∑

α=1

∫

R

[

χ(s,+∞)(x)]

αα(t − φα) etdt

=

n∑

α=1

∫

R

χ(−∞,−p log sδα

)(t) etdt

=

n∑

α=1

∫−p log sδα

−∞

et dt =1

sp

n∑

α=1

δpα.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Proof of the main theorem

Fix x =[

e−(t+φβ)ψαβ

]n

α,β=1∈ Lp(Mn). We may assume

that ψ is positive and diagonal, i.e., let ψαα = δα > 0and ψαβ = 0 if α 6= β.

µt(x) = inf{s > 0 : τ

(

χ(s,+∞)(x))

6 t}

,

where χ(x) is the spectral measure of x .

τ(χ(s,+∞)(x)) =1

sp

n∑

α=1

δpα.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Proof of the main theorem

Fix x =[

e−(t+φβ)ψαβ

]n

α,β=1∈ Lp(Mn). We may assume

that ψ is positive and diagonal, i.e., let ψαα = δα > 0and ψαβ = 0 if α 6= β.

µt(x) = inf{s > 0 : τ

(

χ(s,+∞)(x))

6 t}

,

where χ(x) is the spectral measure of x .

τ(χ(s,+∞)(x)) =1

sp

n∑

α=1

δpα.

µt(x) =k(x)

t1/p, where k(x) =

(

n∑

α=1

δpα

)1p

= ‖ψ‖Sp .

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Commutative version

A similar argument describes Lp-spaces for M = L∞(R).

Theorem

1 The space Lp(M) admits the following description

Lp(M) ={

x ∈ R̃ : x(t, s) = e−t/pψ(s), ψ ∈ Lp(ds)}

.

2 If µ is the decreasing rearrangement (wrt etdtds), then

µt(x) =‖ψ‖Lp(ds)

t1/p, t ∈ R,

x(t, s) = e−t/pψ(s) ∈ Lp(M).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Let F be the Fourier transform on L2(R, ℓ2n) and F−1 is the inverseFourier transform, i.e.,

ξ̂(t) = Fξ(t) =1√2π

∫

R

ξ(s) e−its ds, ξ ∈ L2(R, ℓ2n)

and

ξ(t) = F−1ξ̂(t) =1√2π

∫

R

ξ̂(s) e its ds, ξ̂ ∈ L2(R, ℓ2n).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Let F be the Fourier transform on L2(R, ℓ2n) and F−1 is the inverseFourier transform, i.e.,

ξ̂(t) = Fξ(t) =1√2π

∫

R

ξ(s) e−its ds, ξ ∈ L2(R, ℓ2n)

and

ξ(t) = F−1ξ̂(t) =1√2π

∫

R

ξ̂(s) e its ds, ξ̂ ∈ L2(R, ℓ2n).

The mappings F and F−1 are unitary transformations of L2(R, ℓ2).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

We shall show thatR̂ = FRF−1.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

We shall show thatR̂ = FRF−1.

Let t 7→ x(t) = [xαβ]n

α,β=1 ∈ Mn be a Schwartz function.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

We shall show thatR̂ = FRF−1.

Let t 7→ x(t) = [xαβ]n

α,β=1 ∈ Mn be a Schwartz function.x(t) defines an operator T ∈ R by

T =

∫

R

π(x(t)) λt dt.

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

We shall show thatR̂ = FRF−1.

Let t 7→ x(t) = [xαβ]n

α,β=1 ∈ Mn be a Schwartz function.x(t) defines an operator T ∈ R by

T =

∫

R

π(x(t)) λt dt.

The collection of all such operators T is weakly dense in R (see [Ta,Ch. X, Lemma 1.8]).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

∫

R

σ−t(x(k))ξ(t − k) dk

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

∫

R

σ−t(x(k))ξ(t − k) dk

=

∫

R

e−itΦx(k)e itΦ ξ(t − k) dk

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

∫

R

e−itΦx(k)e itΦ ξ(t − k) dk

=

∫

R

e−itΦ

[∫

R

x̂(s) e iks ds

]

e itΦ

[∫

R

ξ̂(m) e im(t−k) dm√2π

]

dk

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

∫

R

e−itΦ

[∫

R

x̂(s) e iks ds

]

e itΦ

[∫

R

ξ̂(m) e im(t−k) dm√2π

]

dk

=

∫

R3

e−itΦx̂(s)e itΦξ̂(m) e iks+im(t−k) ds dkdm√2π

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

∫

R3

e−itΦx̂(s)e itΦξ̂(m) e iks+im(t−k) ds dkdm√2π

=

n∑

β=1

∫

R3

e−itφα x̂αβ(s)e itφβ ξ̂β(m) e iks+im(t−k) ds dkdm√2π

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

n∑

β=1

∫

R3

e−itφα x̂αβ(s)e itφβ ξ̂β(m) e iks+im(t−k) ds dkdm√2π

=

n∑

β=1

∫

R2

x̂αβ(s) ξ̂β(m) e it(m−φα+φβ)

[∫

R

e ik(s−m) dk

]

dsdm√2π

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

n∑

β=1

∫

R2

x̂αβ(s) ξ̂β(m) e it(m−φα+φβ)

[∫

R

e ik(s−m) dk

]

dsdm√2π

=

n∑

β=1

∫

R2

x̂αβ(s) ξ̂β(m) e it(m−φα+φβ)δ(s − m) dsdm√2π

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

n∑

β=1

∫

R2

x̂αβ(s) ξ̂β(m) e it(m−φα+φβ)δ(s − m) dsdm√2π

=

n∑

β=1

∫

R

x̂αβ(s)ξ̂β(s)e it(s−φα+φβ) ds√2π

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

n∑

β=1

∫

R

x̂αβ(s)ξ̂β(s)e it(s−φα+φβ) ds√2π

=

n∑

β=1

∫

R

x̂αβ(s − φβ) ξ̂β(s − φβ) e it(s−φα) ds√2π

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

n∑

β=1

∫

R

x̂αβ(s − φβ) ξ̂β(s − φβ) e it(s−φα) ds√2π

Consequently, if η = Tξ and η̂ = (η̂α)nα=1, then

η̂α(s − φα) =

n∑

β=1

x̂αβ(s − φβ) ξ̂β(s − φβ).

Matricial

Haagerup

Lp -spaces

D. Potapov,

F. Sukochev

Matrix

algebra Mn

Modular group

Continuous

crossed

product

The trace

The dual

action

Haagerup

Lp -spaces,

1 6 p 6 ∞

Proof of the

main theorem

Commutative

version

Appendix

Appendix (construction of the isomorphismbetween R and R̂)

Fix ξ ∈ L2(R, ℓ2n) and let

ξ(t) =

∫

R

ξ̂(s) e its ds√2π

and x(t) =

∫

R

x̂(s) e its ds

let also x̂ = [x̂αβ]n

α,β=1 and ξ̂ =(

ξ̂α

)n

α=1

Tξ(t) =

∫

R

π(x(k)) λk(t)ξ(t) dk

=

n∑

β=1

∫

R

x̂αβ(s − φβ) ξ̂β(s − φβ) e it(s−φα) ds√2π

Thus, we showed that the operator FTF−1 belongs to R̂. The proofof the proposition is finished.