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Abstract
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Let
and
be locally compact Hausdorff
spaces with
being a normal
space. We prove that if
is a linear isomorphism from an extremely regular subspace
of
onto an extremely
regular subspace
of
with
, then there exists exactly
one homeomorphism
from
onto
such that for some
continuous function
we have
for every
with
,
and
,
where
and
if
is the real
numbers, and
and
if
is the
complex numbers.
This result provides a full extension of Banach–Stone theorem
for small isomorphisms between extremely regular subspaces of
spaces
that unifies and improves some classical results of Amir (1966), Cambern (1967),
Cengiz (1973), and Cutland and Zimmer (2005).
Moreover, for the real scalar case, our result is optimal in the sense that
cannot be replaced by any other number.
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Keywords
Banach–Stone theorem, Amir–Cambern theorem, Cengiz theorem,
weighted composition operators, extremely regular subspaces
of $C_{0}(K)$ space, small isomorphisms, uniqueness of
homeomorphisms
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Mathematical Subject Classification
Primary: 46B03, 46E15
Secondary: 46B25
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Milestones
Received: 12 September 2020
Accepted: 18 November 2020
Published: 26 January 2021
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