论文标题
Whyburn定理的概括和Abelian C*Inclusions的多个
A Generalization of Whyburn's Theorem, and Aperiodicity for Abelian C*-Inclusions
论文作者
论文摘要
令$ j：y \ to x $是紧凑的度量空间的连续陈述。怀伯恩证明了$ j $是不可约的，这意味着$ j（f）\ subsetneq x $对于任何适当的封闭子集$ f \ subsetneq y $，且仅当$ j $几乎是一对一的情况下，从某种意义上说\ [ \ edline {\ {y \ in y：j^{ - 1}（j（y））= y \}}} = y。 \overline{\{x \in X: card(j^{-1}(x)) = 1\}} = X. \] Translated to the language of operator algebras, this says that if $A \subseteq B$ is a unital inclusion of separable abelian $C^*$-algebras, then there exists a unique pseudo-expectation (in the sense of Pitts) if只有当Nagy-Reznikoff的几乎扩展属性拥有。更普遍地，我们证明（不一定是可分离的）Abelian $ C^*$ - 代数在且仅当它是Aperiodic的情况下（从Kwaśniewski-Meyer的意义上）时，代数的包含（代数）独特。
Let $j:Y \to X$ be a continuous surjection of compact metric spaces. Whyburn proved that $j$ is irreducible, meaning that $j(F) \subsetneq X$ for any proper closed subset $F \subsetneq Y$, if and only if $j$ is almost one-to-one, in the sense that \[ \overline{\{y \in Y: j^{-1}(j(y)) = y\}} = Y. \] In this note we prove the following generalization: There exists a unique minimal closed set $K \subseteq Y$ such that $j(K) = X$ if and only if \[ \overline{\{x \in X: card(j^{-1}(x)) = 1\}} = X. \] Translated to the language of operator algebras, this says that if $A \subseteq B$ is a unital inclusion of separable abelian $C^*$-algebras, then there exists a unique pseudo-expectation (in the sense of Pitts) if and only if the almost extension property of Nagy-Reznikoff holds. More generally, we prove that a unital inclusion of (not necessarily separable) abelian $C^*$-algebras has a unique pseudo-expectation if and only if it is aperiodic (in the sense of Kwaśniewski-Meyer).
