论文标题
您可以参加Akemann-Weaver的$ \ diamondsuit _ {\ aleph_1} $离开吗？
Can you take Akemann--Weaver's $\diamondsuit_{\aleph_1}$ away?
论文作者
论文摘要
通过Glimm的二分法，是一种可分离的，简单的$ \ textrm {c}^*$ - 代数具有连续的单位不等式的不可减至的不可约性表示，并且只有当它是非类型i时，它是不可约束的，而所有不可修复的表示是iN type nim type，naim equival是否等于。 $ \ textrm {c}^*$ - 代数。 2004年，Akemann和Weaver使用Jensen的Diamond Axiom $ \ DiamondSuit _ {\ Aleph_1} $，对Naimark的问题做出了负面答案，这是一种强大的对角度化原则，暗示着连续假设（$ \ \ sathsf {Ch} $）。由于罗森伯格（Rosenberg）的结果，一个简单的简单$ \ textrm {c}^*$ - 具有独特的不可约代表的代数，我们表明，通过构建一个可分离的，简单的$ \ textrm {c}^*$ - algebra的示例，可以很好地表明这一结果，从而可以满足两种不可及的代表。二分法。我们的建筑使用Jensen的$ \ DiamondSuit _ {\ Aleph_1} $，表示为$ \ Diamondsuit^\ Mathsf {Cohen} $，该{Cohen} $，它以$ \ sathsf {Ch} $的否定为原始Cohen的模型。我们还证明，$ \ diamondsuit^\ mathsf {cohen} $足够对奈马克（Naimark）的问题给出负面答案。我们的主要技术工具是一个强迫概念，该概念通常添加给定的$ \ textrm {c}^*$ - 代数的自动形态，并在其纯状态下采取规定的诉讼。
By Glimm's dichotomy, a separable, simple $\textrm{C}^*$-algebra has continuum-many unitarily inequivalent irreducible representations if, and only if, it is non-type I while all of its irreducible representations are unitarily equivalent if, and only if, it is type I. Naimark asked whether the latter equivalence holds for all $\textrm{C}^*$-algebras. In 2004, Akemann and Weaver gave a negative answer to Naimark's problem, using Jensen's diamond axiom $\diamondsuit_{\aleph_1}$, a powerful diagonalization principle that implies the Continuum Hypothesis ($\mathsf{CH}$). By a result of Rosenberg, a separably represented simple $\textrm{C}^*$-algebra with a unique irreducible representation is necessarily of type I. We show that this result is sharp by constructing an example of a separably represented, simple $\textrm{C}^*$-algebra that has exactly two inequivalent irreducible representations, and therefore does not satisfy the conclusion of Glimm's dichotomy. Our construction uses a weakening of Jensen's $\diamondsuit_{\aleph_1}$, denoted $\diamondsuit^\mathsf{Cohen}$, that holds in the original Cohen's model for the negation of $\mathsf{CH}$. We also prove that $\diamondsuit^\mathsf{Cohen}$ suffices to give a negative answer to Naimark's problem. Our main technical tool is a forcing notion that generically adds an automorphism of a given $\textrm{C}^*$-algebra with a prescribed action on its space of pure states.
