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Relatively Weakly Open Convex Combinations of Slices and Scattered C $$^*$$ ∗ -Algebras
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-06-24 , DOI: 10.1007/s00009-020-01539-x
Julio Becerra Guerrero , Francisco J. Fernández-Polo

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 \((\hbox {P1})\), namely that every convex combination of slices of the closed unit ball is a relatively weakly open subset of the closed unit ball, if and only if K is scattered and \(\mathcal {A}\) is some \(c_0\)-sum of finite-dimensional C\(^*\)-algebras. To obtain a similar characterization in the setting of general C\(^*\)-algebras, we consider a weaker property \((\overline{\hbox {P1}})\), namely For every convex combination of slices C of the unit ball of a Banach space X and \(x\in C\), there exists a relatively weakly open set W containing x, such that \(W\subseteq \overline{C}\). We prove that a C\(^*\)-algebra has property \((\overline{\hbox {P1}})\) if and only if it is scattered with finite-dimensional irreducible representations. We obtain some stability results for property \((\overline{\hbox {P1}})\). For instance, this property passes down from Banach spaces to its closed ideals. As a consequence, we prove that an \(L_1\)-predual Banach space contains no isomorphic copy of \(\ell _1\) if and only if it has property \((\overline{\hbox {P1}})\).

中文翻译:

切片和散射C $$ ^ * $$ ∗-代数的相对弱开放凸组合

我们证明给定一个局部紧Hausdorff空间K和一个紧C \(^ * \)-代数\(\ mathcal {A} \),C \(^ * \)-代数\(C(K,\ mathcal {A})\)满足\((\ hbox {P1})\),即,当且仅当K分散时,闭合单位球的每个片凸组合都是闭合单位球的相对弱开放子集和\(\ mathcal {A} \)是一些\(C_0 \)有限维C的-sum \(^ * \) -代数。为了在一般C \(^ * \)-代数的设置中获得相似的表征,我们考虑了较弱的性质\((\ overline {\ hbox {P1}})\),即对于 Banach空间X的单位球的切片C \(x \ in C \)的每个凸组合C,都存在一个相对弱的开放集W包含x例如\(W \ subseteq \ overline {C} \)。我们证明,当且仅当C \(^ * \)-代数具有属性\((\ overline {\ hbox {P1}})\)时,它才具有有限维不可约表示。我们获得属性\((\ overline {\ hbox {P1}})\)的一些稳定性结果 。例如,此属性从Banach空间传递到其封闭的理想。因此,我们证明\(L_1 \)-前Banach空间仅当具有属性\((\ overline {\ hbox {P1}})\\时,不包含\(\ ell _1 \)的同构副本。
更新日期:2020-06-24
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