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Simultaneously continuous retraction and Bishop-Phelps-Bollob\'as type theorem
We study the existence of a retraction from the dual space of a (real
or complex) Banach space onto its unit ball which is uniformly
continuous in norm topology and continuous in weak- topology. Such a
retraction is called a uniformly simultaneously continuous retraction.
It is shown that if has a normalized unconditional Schauder basis with
unconditional basis constant 1 and is uniformly monotone, then a
uniformly simultaneously continuous retraction from onto
exists. It is also shown that if is a family of separable Banach
spaces whose duals are uniformly convex with moduli of convexity
such that and or
for , then a uniformly simultaneously continuous retraction
exists from onto .
The relation between the existence of a uniformly simultaneously continuous
retraction and the Bishsop-Phelps-Bollob\'as property for operators is
investigated and it is proved that the existence of a uniformly simultaneously
continuous retraction from onto its unit ball implies that a pair has the Bishop-Phelps-Bollob\'as property for every locally compact
Hausdorff spaces . As a corollary, we prove that has the
Bishop-Phelps-Bollob\'as property if and are the spaces of
all real-valued continuous functions vanishing at infinity on locally compact
metric space and locally compact Hausdorff space respectively.Comment: 15 page
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