Disjointness preserving operators between little Lipschitz algebras
Given a real number α ∈ ( 0 , 1 ) and a metric space ( X , d ) , let Lip α ( X ) be the algebra of all scalar-valued bounded functions f on X such that p α ( f ) = sup { | f ( x ) − f ( y ) | / d ( x , y ) α : x , y ∈ X , x ≠ y } < ∞ , endowed with any one of the norms ‖ f ‖ = max { p α ( f ) , ‖...
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Published in | Journal of mathematical analysis and applications Vol. 337; no. 2; pp. 984 - 993 |
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Main Author | |
Format | Journal Article |
Language | English |
Published |
San Diego, CA
Elsevier Inc
15.01.2008
Elsevier |
Subjects | |
Online Access | Get full text |
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Summary: | Given a real number
α
∈
(
0
,
1
)
and a metric space
(
X
,
d
)
, let
Lip
α
(
X
)
be the algebra of all scalar-valued bounded functions
f on
X such that
p
α
(
f
)
=
sup
{
|
f
(
x
)
−
f
(
y
)
|
/
d
(
x
,
y
)
α
:
x
,
y
∈
X
,
x
≠
y
}
<
∞
,
endowed with any one of the norms
‖
f
‖
=
max
{
p
α
(
f
)
,
‖
f
‖
∞
}
or
‖
f
‖
=
p
α
(
f
)
+
‖
f
‖
∞
. The little Lipschitz algebra
lip
α
(
X
)
is the closed subalgebra of
Lip
α
(
X
)
formed by all those functions
f such that
|
f
(
x
)
−
f
(
y
)
|
/
d
(
x
,
y
)
α
→
0
as
d
(
x
,
y
)
→
0
. A linear mapping
T
:
lip
α
(
X
)
→
lip
α
(
Y
)
is called disjointness preserving if
f
⋅
g
=
0
in
lip
α
(
X
)
implies
(
T
f
)
⋅
(
T
g
)
=
0
in
lip
α
(
Y
)
. In this paper we study the representation and the automatic continuity of such maps
T in the case in which
X and
Y are compact. We prove that
T is essentially a weighted composition operator
T
f
=
h
⋅
(
f
○
φ
)
for some nonvanishing little Lipschitz function
h and some continuous map
φ. If, in addition,
T is bijective, we deduce that
h is a nonvanishing function in
lip
α
(
Y
)
and
φ is a Lipschitz homeomorphism from
Y onto
X and, in particular, we obtain that
T is automatically continuous and
T
−1
is disjointness preserving. Moreover we show that there exists always a discontinuous disjointness preserving linear functional on
lip
α
(
X
)
, provided
X is an infinite compact metric space. |
---|---|
ISSN: | 0022-247X 1096-0813 |
DOI: | 10.1016/j.jmaa.2007.04.045 |