论文标题
不可用的纯净状态
A non-diagonalizable pure state
论文作者
论文摘要
我们在$ \ ell_2 $上的所有有界线性运算符的c*-algebra $ \ Mathcal b（\ ell_2）$上构建纯状态，该$ \ ell_2 $不可对数，即，它不是$ \ lim_u \ lim_u \ langle \ langle t（e_k \ e_k \ rangle $ y__________________________________） \ Mathbb n} $ \ ell_2 $和$ \ mathbb n $上的Ultrafilter $ u $。这构成了安德森（Anderson）的猜想的反例，而没有其他假设，并改善了C. Akemann，N。Weaver，I。Farah和I. Smythe的结果。 这是从J. Anderson的结果以及对Kadison-Singer问题的积极解决方案来看，D。Spielman，D。Spielman，N。Srivastava，我们纯净的状态限制对任何具有对角线运营商的原子MASA $ d（（E_K）_ {K \ in \ MATHBB N}）的限制，以对数的$ ___________________________________________________________________ n} $在$ d（（e_k）_ {k \ in \ mathbb n}）$上不是乘法。
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 $D((e_k)_{k\in \mathbb N})$.
