Gen. Math. Notes, Vol. 29, No. 1, July 2015, pp.1-5
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Measure Preserving Isomorphisms
M. Gheytaran Marzrood
Department of Mathematics
University of Tabriz, 5166617766, Tabriz, Iran
E-mail: m gheytaran91@ms.tabrizu.ac.ir
(Received: 6-5-15 / Accepted: 30-6-15)
Abstract
In this note we study the relationship between the isomorphic and unitarily 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, isomorphic, unitarily isomorphic, zero-product.
1
Introduction
Let (X, Σ, µ) be a probability measure space and let A be a sub-sigma algebra
of Σ. All comparisons between two functions or two sets are to be interpreted
as holding up to a µ-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 defined 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
hϕ = dµ ◦ ϕ−1 /dµ is finite-valued. In the setting of Lp -spaces the so called
−1
conditional expectation operator E ϕ (Σ) with respect to ϕ−1 (Σ) plays an important role. If there is no possibility of confusion, for each 0 ≤ f ∈ L0 (Σ) or
−1
f ∈ Lp (Σ), we write Eϕ f in place of E ϕ (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 by u and non-singular measurable function ϕ is given by W = Mu ◦Cϕ
where Mu is a multiplication operator and Cϕ is a composition operator on
2
M. Gheytaran Marzrood
L2 (Σ) defined by Mu f = uf and Cϕ f = 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 measure preserving, we say that ϕ1 and ϕ2 are pseudo-isomorphic (see [8]). Also,
the bounded linear operators Cϕ1 and Cϕ2 are said to be unitarily equivalent
if there is a unitary transformation U such that U Cϕ1 = Cϕ2 U ( in this case
ϕ1 and ϕ2 are not necessarily measure preserving). Note that, if ϕ1 and ϕ2
are isomorphic then kCϕ1 k = kCϕ2 k = 1 and ϕ1 ◦ φ = φ ◦ ϕ2 . Hence for
each f ∈ L2 (Σ), Cφ Cϕ1 f = f ◦ ϕ1 ◦ φ = f ◦ φ ◦ ϕ2 = Cϕ2 Cφ f . Also, since
hφ = 1 then Cφ∗ f = f ◦ φ−1 = Cφ−1 f , and so ϕ1 and ϕ2 are unitarily equivalent.
Hence, isomorphic transformations are unitarily equivalent. For a fix measure
preserving mapping ϕ : X → X, define
Wϕ = {uCϕ : Eϕ (|u|2 ) ◦ ϕ−1 ∈ L∞ (X)},
Kϕ = {u ∈ L0 (Σ) : uCϕ ∈ Wϕ }.
For u ∈ Kϕ , put kukKϕ = kEϕ (|u|2 )◦ϕ−1 k1/2 . It is easy to show that (Kϕ , 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 Wϕ is a closed subspace of B(L2 (Σ)).
Proof. Clearly Wϕ is a subspace of B(L2 (Σ)). Let{un Cϕ } ⊆ Wϕ and un Cϕ →
T for some T ∈ B(L2 (Σ)). We show that T ∈ Wϕ . Since un = un Cϕ (1) →
2
T (1) =: u, then for every f ∈ L (Σ) we have
kun Cϕ (f ) − uCϕ (f )k ≤ kun − ukkCϕ kkf k ≤ kun − ukkf k.
Thus T = uCϕ ∈ Wϕ , and so Wϕ ⊆ B(L2 (Σ)) is close.
Proposition 2.2 (Kϕ , k · kKϕ ) is a Banach space. In particular, Kϕ is an
order ideal.
3
Measure Preserving Isomorphisms
Proof. Define Λ : Kϕ −→ Wϕ by Λ(u) = uCϕ . Then for each u ∈ Kϕ ,
kΛ(u)k2 = kEϕ (|u|2 ) ◦ ϕ−1 k = kuk2Kϕ . 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 , then Eϕ (|u1 |2 ) ◦ ϕ−1 ≤ Eϕ (|u2 |2 ) ◦ ϕ−1 < ∞, and hence u1 ∈ Kϕ .
The measure preserving transformations ϕ1 and ϕ2 are said to be unitarily
isomorphic if there is a unitary transformation V on L2 (Σ) such that V Wϕ1 =
Wϕ2 V (see [1, 2, 5]).
Theorem 2.3 If ϕ1 and ϕ2 are isomorphic, then they are unitarily isomorphic.
Proof. Let uCϕ1 ∈ Wϕ1 . Since ϕ1 ◦φ = φ◦ϕ2 and φ is a bijection, bi-measurable
and measure preserving transformation, then Cφ is a unitary operator and for
each f ∈ L2 (Σ),
Cφ (uCϕ1 )(f ) = (u ◦ φ)(f ◦ ϕ1 ◦ φ) = (u ◦ φ)(f ◦ φ ◦ ϕ2 ) = ((u ◦ φ)Cϕ2 )Cφ f.
Now, let uCϕ1 ∈ Wϕ1 . Then kEϕ1 (|u|2 ) ◦ ϕ−1
1 k < ∞. Since Eφ = I and
2
−1
−1
kCφ k = hφ = 1, then for each f ∈ L (Σ) we get that
2
k(u ◦ φ)Cϕ2 (f )k =
=
Z
Z
2
2
−1
|u| |f | ◦ ϕ2 ◦ φ dµ =
Z
|u|2 |f |2 ◦ φ−1 ◦ ϕ1 dµ
2
−1
2
−1
2
2
Eϕ1 (|u|2 ) ◦ ϕ−1
1 |f | ◦ φ dµ ≤ kEϕ1 (|u| ◦ ϕ1 )k∞ kCφ−1 k kf k < ∞.
Hence (u ◦ φ)Cϕ2 is in Wϕ2 for each u in Kϕ1 , and consequently Cφ Wϕ1 ⊆
Wϕ2 Cφ . Now, if υ is in Kϕ2 then υ ◦ φ−1 is in Kϕ1 , thus υ = (υ ◦ φ−1 ) ◦ φ is in
Kϕ2 . It follows that each element of Wϕ2 can be written as (u ◦ φ)Cϕ2 for some
u in Kϕ1 . Thus Wϕ2 Cφ ⊆ Cφ Wϕ1 , 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.
−1
Proof. Let ϕ1 ◦φ = φ◦ϕ2 , where φ is a bi-measurable bijection. Put h = dµ◦φ
dµ
1
2
2
√
. Define V : L (Σ) → L (Σ) by V f = w(f ◦ φ). Then for
and w =
h◦φ
2
each f ∈ L (Σ) we have
2
kV f k =
Z
X
Z
1
1 2 dµ ◦ φ−1 Z
2
|f | ◦ φdµ =
|f |
=
|f |2 dµ = kf k2
h◦φ
dµ
X h
X
4
M. Gheytaran Marzrood
Hence
V is an isometry. Now, for each g ∈ L2 (Σ), put f = (w ◦φ−1 )−1 g ◦φ−1 =
√
hg ◦ φ−1 . Then V f = g. Thus V is qunitary. Now we show V (uCϕ1 ) =
1
.u) ◦ φ. Then υ ∈ Kϕ2 , because
(u ◦ φ)Cϕ2 V , for any u ∈ Kϕ1 . Set υ = ( h◦ϕ
h
V (uCϕ1 )V −1 g = V (uCϕ1 )((w ◦ φ−1 )−1 g ◦ φ−1 )
1
(u ◦ φ)((w ◦ φ−1 )−1 ◦ ϕ1 ◦ φ)(g ◦ φ−1 ◦ ϕ1 ◦ φ)
h◦φ
1
=√
(u ◦ φ)(w ◦ ϕ2 )−1 (g ◦ ϕ2 )
h◦φ
= w(w◦ϕ2 )−1 (u◦φ)(g◦ϕ2 ) = υ(g◦ϕ2 ) = υCϕ2 g,
=√
and
h ◦ ϕ1 2
|u| ) ◦ φ(|f |2 ◦ ϕ2 )dµ
h
X
Z
h ◦ ϕ1 ◦ φ
(|u|2 ◦ φ)(|f |2 ◦ φ−1 ◦ ϕ1 ◦ φ)dµ
=
h◦φ
X
Z
h ◦ ϕ1 2
=
|u| (|f |2 ◦ φ−1 ◦ ϕ1 )dµ ◦ φ−1
h
ZX
kυCϕ2 f k2 =
Z
=
=
ZX
X
(
(h ◦ ϕ1 )Eϕ1 (|u|2 )(|f |2 ◦ φ−1 ◦ ϕ1 )dµ
2
−1
hEϕ1 (|u|2 ) ◦ ϕ−1
1 (|f | ◦ φ )dµ
≤ kEϕ1 (|u|2 ) ◦ ϕ−1
1 k∞
Z
h|f |2 ◦ φ−1 dµ
X
2
≤ kEϕ1 (|u|2 ) ◦ ϕ−1
1 k∞ kf k < ∞.
Thus kυCϕ2 k < ∞, and so V Wϕ1 = Wϕ2 V.
Corollary 2.5 Let Λ : Wϕ1 −→ Wϕ2 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 operator V such that Λ(uCϕ1 ) = V (uCϕ1 )V −1 , by [6]. Since Λ
is surjection so Wϕ2 = Λ(Wϕ1 ) = V (Wϕ1 )V −1 . Consequently V Wϕ1 = Wϕ2 V .
It follows that ϕ1 and ϕ2 are pseudo-isomorphic.
Conversely, assume that ϕ1 and ϕ2 are pseudo-isomorphic. So there is a unitary
transformation V on L2 (Σ) such that V Wϕ1 = Wϕ2 V . Now define Λ : Wϕ1 →
Wϕ2 by Λ(uCϕ1 ) = V (uCϕ1 )V −1 . Thus, if (u1 Cϕ1 )(u2 Cϕ1 ) = 0, we get that
Λ(u1 Cϕ1 )Λ(u1 Cϕ1 ) = (V (u1 Cϕ1 )V −1 )(V (u2 Cϕ1 )V −1 ) = 0
and hence Λ is a zero-product preserving map.
Acknowledgements: The author would like to thank to the Prof. Jabbarzadeh for his comments and suggestions improving the contents of the paper. Also, the author would like to thank referee for their useful comments.
Measure Preserving Isomorphisms
5
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