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：

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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 和\（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。）。