arXiv 1902.092092019-02-25math.FA

切片与散布C的相对弱开凸组合$^*$ -代数

Relatively weakly open convex combinations of slices and scattered C$^*$-Algebras

Becerra Guerrero J., Fernández-Polo F. J

摘要：

我们证明了给定一个局部紧Hausdorff空间$K$和一个紧C-$^*$-代数$\mathcal{A}$，C-$^*$-代数$C(K, \mathcal{A})$满足闭单位球的每个切片的凸组合是闭单位球的相对弱开子集当且仅当$K$是离散的，并且$\mathcal{A}$是有限维C-$^*$-代数的$c_0$ -和.引入并研究了具有性质$(\overline{\hbox{P1}})$的Banach空间，即. e.对于Banach空间单位球的切片$C$的每一凸组合$X$和$x\in C$，存在包含$x$的$W$相对弱开集，使得$W\subseteq \overline{C}$ .在一般的C-$^*$-代数的背景下，我们得到了这一性质的一个刻划.事实上，一个C $^*$ -代数具有性质$(\overline{\hbox{P1}})$当且仅当它被有限维不可约表示离散。给出了满足$(\overline{\hbox{P1}})$性质的Banach空间的稳定性结果.作为这些结果的结果，我们证明了一个真正的$L_1$ -预对偶Banach空间不包含$\ell_1$同构副本当且仅当它有性质$(\overline{\hbox{P1}})$。

英文摘要：

We prove that given a locally compact Hausdorff space, $K$, and a compact C$^*$-algebra, $\mathcal{A}$, the C$^*$-algebra $C(K, \mathcal{A})$ satisfies that every convex combination of slices of the closed unit ball is relatively weakly open subset of the closed unit ball if and only if $K$ is scattered and $\mathcal{A}$ is the $c_0$-sum of finite-dimensional C$^*$-algebras. We introduce and study Banach spaces which have property $(\overline{\hbox{P1}})$, i. e. For every convex combination of slices $C$ of the unit ball of a Banach space $X$ and $x\in C$ there exists $W$ relatively weakly open set containing $x$, such that $W\subseteq \overline{C}$. In the setting of general C$^*$-algebras we obtain a characterization of this property. Indeed, a C$^*$-algebra has property $(\overline{\hbox{P1}})$ if and only if is scattered with finite dimensional irreducible representations. Some stability results for Banach spaces satisfying property $(\overline{\hbox{P1}})$ are also given. As a consequence of these results we prove that a real $L_1$-predual Banach space contains no isomorphic copy of $\ell_1$ if and only if it has property $(\overline{\hbox{P1}})$.

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