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April 2016

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Abstract of a paper by Jose Pedro Moreno and Rolf Schneider
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “Multiplication of convex sets in $C(K)$ spaces” by Jose Pedro Moreno and Rolf Schneider. Abstract: Let $C(K)$ denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of $C(K)$, one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. Our main interest is in correlations between properties of the product of closed order intervals in $C(K)$ and properties of the underlying space $K$. When $K$ is finite, the product of two intervals in $C(K)$ is always an interval. Surprisingly, the converse of this result is true for a wide class of compacta. We show that a first-countable space $K$ is finite whenever it has the property that the product of two nonnegative intervals is closed, or the property that the product of an interval with itself is convex. That some assumption on $K$ is needed, can be seen from the fact that, if $K$ is the Stone-Cech compactification of $N$, then the product of two intervals in $C(K)$ with continuous boundary functions is always an interval. For any $K$, it is proved that the product of two positive intervals is always an interval, and that the product of two nonnegative intervals is always convex. Finally, square roots of intervals are investigated, with results of similar type. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.01211

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Abstract of a Paper by Sheldon Dantas, Domingo Garcia, Manuel Maestre and Miguel Martin
by Bentuo Zheng (bzheng) 30 Apr '16

30 Apr '16

This is an announcement for the paper “The Bishop-Phelps-Bollobas property for compact operators” by Sheldon Dantas, Domingo Garcia, Manuel Maestre and Miguel Martin. Abstract: We study the Bishop-Phelps-Bollob$\'$as property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X, Y$ be Banach spaces. If $(c_0, Y)$ has the BPBp for compact operators, then so do $(C_0(L), Y)$ for every locally compact Hausdorff topological space $L$ and $(X, Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikod$\'$ym property and $(\ell_1(X), Y)$ has the BPBp for compact operators, then so does $(L_1(\mu, X), Y)$ for every positive measure $\mu$; as a consequence, $(L_1(\mu, X), Y)$ has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any positive measure $\nu$ and $1<p<\infty$. For $1\leq p<\infty$, if $(X, \ell_p(Y))$ has the BPBp for compact operators, then so does $(X, L_p(\mu, Y))$ for every positive measure $\mu$ such that $L_1(\mu)$ is infinite-dimensional. If $(X, Y)$ has the BPBp for compact operators, then so do $(X, L_{\infty}(\mu, Y))$ for every $\sigma$-finite positive measure $\mu$ and $(X, C(K, Y))$ for every compact Hausdorff topological space $K$. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1604.00618

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