不可对角化的纯态
A non-diagonalizable pure state
摘要：
我们在C* 代数$\mathcal B(\ell_2)$上构造了一个纯态，它是由$\ell_2$上所有不可对角化的有界线性算子构成的，即，对于$\ell_2$的任何规范正交基$(e_k)_{k\in \mathbb N}$和$\mathbb N$上的超滤波器$u$，它不是$\lim_u\langle T(e_k), e_k\rangle$的形式。这构成了Anderson猜想的一个反例，改进了C. Akemann，N.韦弗岛法拉和我。斯迈思通过附加的集合论假设构建了这样的状态。它是由J. Anderson的结果和A. Marcus，D. Spielman，N. Srivastava证明了我们的纯态对对角算子的任何原子masa $D((e_k)_{k\in \mathbb N})$的限制相对于规范正交基$(e_k)_{k\in \mathbb N}$在$D((e_k)_{k\in \mathbb N})$上不是乘法的。
英文摘要：
We construct a pure state on the C*-algebra $\mathcal B(\ell_2)$ of all bounded linear operators on $\ell_2$ which is not diagonalizable, i.e., it is not of the form $\lim_u\langle T(e_k), e_k\rangle$ for any orthonormal basis $(e_k)_{k\in \mathbb N}$ of $\ell_2$ and an ultrafilter $u$ on $\mathbb N$. This constitutes a counterexample to Anderson's conjecture without additional hypothesis and improves results of C. Akemann, N. Weaver, I. Farah and I. Smythe who constructed such states making additional set-theoretic assumptions. It follows from results of J. Anderson and the positive solution to the Kadison-Singer problem due to A. Marcus, D. Spielman, N. Srivastava that the restriction of our pure state to any atomic masa $D((e_k)_{k\in \mathbb N})$ of diagonal operators with respect to an orthonormal basis $(e_k)_{k\in \mathbb N}$ is not multiplicative on $D((e_k)_{k\in \mathbb N})$.
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