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

Ultrastructural Changes of the Vestibular Sensory Organs After Streptomycin Application on the Lateral Canal

Early changes in the vestibular sense organs resulting from the application of a streptomycin sulfate soaked Gelfoam® pledget on the fenestra of the lateral semicircular canal were studied by transmission and scanning electron microscopy. Three days after the application, lesions were present in the central part of the lateral crista. The type I sensory cells were more affected than the type JI cells. These sensory cells showed mitochondrial swelling, cytoplasm protrusion at the cell apex, inclusion of multiple vacuoles, fusion or loss of stereocilia, and pyknotic nuclei. Seven days after the drug application, the sensory cell damage extended to all three cristae and macula utriculi. The lesions were very extensive after ten days and the sensory cells had almost equally disappeared in all three cristae; the lesion in the macula utriculi was smaller and the macula sacculi was unaffected. At fourteen days, the lesions appeared less severe. Thus, a single application of a small amount of streptomycin on the lateral canal fenestra affected all vestibular sense organs, except the saccule, in a short time. The strong affinity of aminoglycosides for the cristae suggests possible entrapment of the drug at the ampullae. This local drug application technique to the canal will be useful in studying vestibular function in animals, and it is applicable to controlling severe vestibular symptoms in human patients