Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Simultaneously continuous retraction and Bishop-Phelps-Bollob\'as type
  theorem   We study the existence of a retraction from the dual space $X^*$ of a (real
or complex) Banach space $X$ onto its unit ball $B_{X^*}$ 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 $X$ has a normalized unconditional Schauder basis with
unconditional basis constant 1 and $X^*$ is uniformly monotone, then a
uniformly simultaneously continuous retraction from $X^*$ onto $B_{X^*}$
exists. It is also shown that if $\{X_i\}$ is a family of separable Banach
spaces whose duals are uniformly convex with moduli of convexity
$\delta_i(\varepsilon)$ such that $\inf_i \delta_i(\varepsilon)>0$ and $X=
\left[\bigoplus X_i\right]_{c_0}$ or $X=\left[\bigoplus X_i\right]_{\ell_p}$
for $1\le p<\infty$, then a uniformly simultaneously continuous retraction
exists from $X^*$ onto $B_{X^*}$.
  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 $X^*$ onto its unit ball implies that a pair $(X,
C_0(K))$ has the Bishop-Phelps-Bollob\'as property for every locally compact
Hausdorff spaces $K$. As a corollary, we prove that $(C_0(S), C_0(K))$ has the
Bishop-Phelps-Bollob\'as property if $C_0(S)$ and $C_0(K)$ are the spaces of
all real-valued continuous functions vanishing at infinity on locally compact
metric space $S$ and locally compact Hausdorff space $K$ respectively.

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The categories of this paper are: hep-ex
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The categories of this paper are: stat
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: nlin
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int