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.Comment: 15 page

Properties of Microlensing Central Perturbations by Planets in Binary Stellar Systems under the Strong Finite-Source Effect

We investigate high-magnification events caused by planets in wide binary stellar systems under the strong finite-source effect, where the planet orbits one of the companions. From this, we find that the pattern of central perturbations in triple lens systems commonly appears as a combination of individual characteristic patterns of planetary and binary lens systems in a certain range where the sizes of the caustics induced by a planet and a binary companion are comparable, and the range changes with the mass ratio of the planet to the planet-hosting star. Specially, we find that because of this central perturbation pattern, the characteristic feature of high-magnification events caused by the triple lens systems appears in the residual from the single-lensing light curve despite the strong finite-source effect, and it is discriminated from those of the planetary and binary lensing events and thus can be used for the identification of the existence of both planet and binary companion. This characteristic feature is a simultaneous appearance of two features. First, double negative-spike and single positive-spike features caused by the binary companion appear together in the residual, where the double negative spike occurs at both moments when the source enters and exits the caustic center and the single positive spike occurs at the moment just before the source enters into or just after the source exits from the caustic center. Second, the magnification excess before or after the single positive-spike feature is positive due to the planet, and the positive excess has a remarkable increasing or decreasing pattern depending on the source trajectory.Comment: 12 pages, 3 figures, accepted for publication in Ap