Measure Preserving Isomorphisms

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Measure Preserving Isomorphisms

M. Gheytaran Marzrood Department of Mathematics

University of Tabriz, 5166617766, Tabriz, Iran E-mail: m [email protected]

(Received: 6-5-15 / Accepted: 30-6-15) Abstract

In this note we study the relationship between the isomorphic and unitar- ily isomorphic measure preserving mappings. Also, we show that the concept of zero-product preserving mappings and unitarily isomorphic mappings are equivalent.

Keywords: Measure preserving transformation, unitarily equivalent, iso- morphic, unitarily isomorphic, zero-product.

1 Introduction

Let (X,Σ, µ) be a probability measure space and letA be a sub-sigma algebra of Σ. All comparisons between two functions or two sets are to be interpreted as holding up to aµ-null set. We denote the linear space of all complex-valued Σ-measurable functions on X by L⁰(Σ). The support of f ∈ L⁰(Σ) is de- fined by σ(f) = {x ∈ X : f(x) 6= 0}. Let ϕ : X → X be a measurable transformation such that µ◦ϕ⁻¹ is absolutely continuous with respect to µ, that is, ϕ is non-singular. It is assumed that the Radon-Nikodym derivative h_ϕ = dµ◦ϕ⁻¹/dµ is finite-valued. In the setting of L^p-spaces the so called conditional expectation operatorE^ϕ−1(Σ) with respect to ϕ⁻¹(Σ) plays an im- portant role. If there is no possibility of confusion, for each 0≤f ∈L⁰(Σ) or f ∈L^p(Σ), we writeE_ϕf in place ofE^ϕ−1(Σ)f. For a deep study of conditional expectation operator we refer the reader to the monograph [7]. For a finite valued function u ∈ L⁰(Σ), the weighted composition operator W on L²(Σ) induced byuand non-singular measurable functionϕis given byW =M_u◦C_ϕ where M_u is a multiplication operator and C_ϕ is a composition operator on

L²(Σ) defined byM_uf =uf andC_ϕf =f◦ϕ, respectively. It is a classical fact that W ∈ B(L²(Σ)), the algebra of all bounded linear operators on L²(Σ), if and only if J := hE(|u|²)◦ϕ⁻¹ ∈ L^∞(Σ) and W ∈ B(L^∞(Σ)) if and only if u∈L^∞(Σ) (see [3]).

We recall that the measure preserving transformations ϕ1, ϕ2 : X → X are said to be isomorphic if there is a bi-measurable, measure preserving bijection φ : X → X such that ϕ₁ ◦φ = φ◦ϕ₂ (see [5]). If φ is not necessarily mea- sure preserving, we say thatϕ1 and ϕ2 are pseudo-isomorphic (see [8]). Also, the bounded linear operators C_ϕ1 and C_ϕ2 are said to be unitarily equivalent if there is a unitary transformation U such that U C_ϕ1 = C_ϕ2U( in this case ϕ1 and ϕ2 are not necessarily measure preserving). Note that, if ϕ1 and ϕ2

are isomorphic then kC_ϕ1k = kC_ϕ2k = 1 and ϕ₁ ◦ φ = φ ◦ ϕ₂. Hence for each f ∈ L²(Σ), C_φC_ϕ1f = f ◦ϕ₁ ◦φ = f ◦φ◦ϕ₂ = C_ϕ2C_φf. Also, since hφ= 1 thenC_φ^∗f =f◦φ⁻¹ =C_φ⁻¹f, and so ϕ1 and ϕ2 are unitarily equivalent.

Hence, isomorphic transformations are unitarily equivalent. For a fix measure preserving mappingϕ:X→X, define

Wϕ ={uCϕ :Eϕ(|u|²)◦ϕ⁻¹ ∈L^∞(X)}, K_ϕ ={u∈L⁰(Σ) :uC_ϕ ∈ W_ϕ}.

Foru∈ K_ϕ,putkukKϕ =kE_ϕ(|u|²)◦ϕ⁻¹k^1/2. It is easy to show that (K_ϕ,k·kKϕ) is a norm space ([4]). Let Λ :A → B be an additive surjective map between some operator algebras. The mapping Λ is said to be a zero-product preserving if Λ(A)Λ(B) = 0 whenever AB = 0 (see [9]). In this note we study the relationship between the isomorphic (pseudo-isomorphic), unitarily isomorphic measure preserving and zero-product preserving mappings.

2 Main Results

Proposition 2.1 W_ϕ is a closed subspace of B(L²(Σ)).

Proof. ClearlyW_ϕ is a subspace of B(L²(Σ)). Let{u_nC_ϕ} ⊆ W_ϕ and u_nC_ϕ → T for some T ∈ B(L²(Σ)). We show that T ∈ W_ϕ. Since u_n = u_nC_ϕ(1) → T(1) =:u, then for every f ∈L²(Σ) we have

ku_nC_ϕ(f)−uC_ϕ(f)k ≤ ku_n−ukkC_ϕkkfk ≤ ku_n−ukkfk.

ThusT =uC_ϕ ∈ W_ϕ, and soW_ϕ ⊆ B(L²(Σ)) is close.

Proposition 2.2 (K_ϕ,k · kKϕ) is a Banach space. In particular, K_ϕ is an order ideal.

Proof. Define Λ : K_ϕ −→ W_ϕ by Λ(u) = uC_ϕ. Then for each u ∈ K_ϕ, kΛ(u)k² = kE_ϕ(|u|²)◦ϕ⁻¹k = kuk²_Kϕ. Hence Λ is an isometry isomorphism and so, by Proposition 2.1, Kϕ is also a Banach space. Now, if u2 ∈ Kϕ and u₁ ≤u₂, thenE_ϕ(|u₁|²)◦ϕ⁻¹ ≤E_ϕ(|u₂|²)◦ϕ⁻¹ <∞, and hence u₁ ∈ K_ϕ. The measure preserving transformations ϕ₁ and ϕ₂ are said to be unitarily isomorphic if there is a unitary transformationV onL²(Σ) such thatVWϕ1 = W_ϕ2V (see [1, 2, 5]).

Theorem 2.3 Ifϕ₁ andϕ₂ are isomorphic, then they are unitarily isomor- phic.

Proof. LetuCϕ1 ∈ Wϕ1. Sinceϕ1◦φ=φ◦ϕ2 andφis a bijection, bi-measurable and measure preserving transformation, thenC_φ is a unitary operator and for eachf ∈L²(Σ),

Cφ(uCϕ1)(f) = (u◦φ)(f ◦ϕ1 ◦φ) = (u◦φ)(f ◦φ◦ϕ2) = ((u◦φ)Cϕ2)Cφf.

Now, let uC_ϕ1 ∈ W_ϕ1. Then kE_ϕ1(|u|²)◦ ϕ⁻¹₁ k < ∞. Since E_φ = I and kC_φ⁻¹k=h_φ⁻¹ = 1, then for each f ∈L²(Σ) we get that

k(u◦φ)C_ϕ2(f)k² =

Z

|u|²|f|²◦ϕ₂◦φ⁻¹dµ=

Z

|u|²|f|²◦φ⁻¹◦ϕ₁dµ

=

Z

E_ϕ1(|u|²)◦ϕ⁻¹₁ |f|²◦φ⁻¹dµ≤ kE_ϕ1(|u|²◦ϕ⁻¹₁ )k_∞kC_φ⁻¹k²kfk² <∞.

Hence (u ◦φ)C_ϕ2 is in W_ϕ2 for each u in K_ϕ1, and consequently C_φW_ϕ1 ⊆ W_ϕ2C_φ. Now, if υ is in K_ϕ2 thenυ◦φ⁻¹ is in K_ϕ1, thusυ = (υ◦φ⁻¹)◦φ is in Kϕ2. It follows that each element ofWϕ2 can be written as (u◦φ)Cϕ2 for some uin K_ϕ1. Thus W_ϕ2C_φ ⊆C_φW_ϕ1, and so ϕ₁ and ϕ₂ are unitarily isomorphic.

We recall that the measure preserving transformations ϕ₁, ϕ₂ are said to be pseudo-isomorphic if there is a bi-measurable bijection φ such that ϕ₁ ◦φ = φ◦ϕ₂ . Note that φ is not necessarily measure preserving (see[8]). In [5] A.

Lambert proved that unitarily isomorphic implies pseudo isomorphic. In the following theorem we give a simple proof for the converse of this fact.

Theorem 2.4 If the measure preserving transformations ϕ₁ and ϕ₂ are pseudo-isomorphic, then they are unitarily isomorphic.

Proof. Letϕ₁◦φ =φ◦ϕ₂, whereφis a bi-measurable bijection. Puth= ^dµ◦φ_dµ⁻¹ and w = √¹

h◦φ

. Define V : L²(Σ) → L²(Σ) by V f = w(f ◦φ). Then for eachf ∈L²(Σ) we have

kV fk² =

Z

X

1

h◦φ|f|²◦φdµ=

Z

X

1

h|f|²dµ◦φ⁻¹ dµ =

Z

X

|f|²dµ=kfk²

HenceV is an isometry. Now, for eachg ∈L²(Σ), put f = (w◦φ⁻¹)⁻¹g◦φ⁻¹ =

√hg ◦φ⁻¹. Then V f = g. Thus V is unitary. Now we show V(uC_ϕ1) = (u◦φ)C_ϕ2V , for any u∈ K_ϕ1. Set υ = (^qh◦ϕ_h¹.u)◦φ. Thenυ ∈ K_ϕ2, because

V(uC_ϕ1)V⁻¹g =V(uC_ϕ1)((w◦φ⁻¹)⁻¹g◦φ⁻¹)

= 1

√h◦φ(u◦φ)((w◦φ⁻¹)⁻¹◦ϕ1◦φ)(g◦φ⁻¹◦ϕ1◦φ)

= 1

√h◦φ(u◦φ)(w◦ϕ₂)⁻¹(g◦ϕ₂)

=w(w◦ϕ2)⁻¹(u◦φ)(g◦ϕ2) = υ(g◦ϕ2) = υCϕ2g, and

kυC_ϕ2fk² =

Z

X

(h◦ϕ₁

h |u|²)◦φ(|f|²◦ϕ₂)dµ

=

Z

X

h◦ϕ₁◦φ

h◦φ (|u|²◦φ)(|f|²◦φ⁻¹◦ϕ₁◦φ)dµ

=

Z

X

h◦ϕ1

h |u|²(|f|²◦φ⁻¹◦ϕ₁)dµ◦φ⁻¹

=

Z

X

(h◦ϕ₁)E_ϕ1(|u|²)(|f|²◦φ⁻¹◦ϕ₁)dµ

=

Z

X

hE_ϕ1(|u|²)◦ϕ⁻¹₁ (|f|²◦φ⁻¹)dµ

≤ kEϕ1(|u|²)◦ϕ⁻¹₁ k∞

Z

X

h|f|²◦φ⁻¹dµ

≤ kE_ϕ1(|u|²)◦ϕ⁻¹₁ k∞kfk² <∞.

ThuskυC_ϕ2k<∞, and so VW_ϕ1 =W_ϕ2V.

Corollary 2.5 Let Λ :W_ϕ1 −→ W_ϕ2 be linear and surjection map.Then Λ zero-prouduct preserving if and only if ϕ₁ and ϕ₂ are pseudo-isomorphic.

Proof. Let Λ be a zero-product preserving map. Then there exists an invertible bounded linear operatorV such that Λ(uC_ϕ1) =V(uC_ϕ1)V⁻¹, by [6]. Since Λ is surjection soW_ϕ2 = Λ(W_ϕ1) =V(W_ϕ1)V⁻¹. Consequently VW_ϕ1 =W_ϕ2V. It follows thatϕ₁ and ϕ₂ are pseudo-isomorphic.

Conversely, assume thatϕ₁andϕ₂are pseudo-isomorphic. So there is a unitary transformationV onL²(Σ) such thatVW_ϕ1 =W_ϕ2V. Now define Λ :W_ϕ1 → W_ϕ2 by Λ(uC_ϕ1) =V(uC_ϕ1)V⁻¹. Thus, if (u₁C_ϕ1)(u₂C_ϕ1) = 0, we get that

Λ(u₁C_ϕ1)Λ(u₁C_ϕ1) = (V(u₁C_ϕ1)V⁻¹)(V(u₂C_ϕ1)V⁻¹) = 0 and hence Λ is a zero-product preserving map.

Acknowledgements: The author would like to thank to the Prof. Jab- barzadeh for his comments and suggestions improving the contents of the pa- per. Also, the author would like to thank referee for their useful comments.

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