K-理论到算子系的一种推广
A generalization of K-theory to operator systems
AI总结：
本文将K-理论推广到算子系，通过构造矩阵尺寸索引的正向系统并取Grothendieck群定义新的$K_0$-群不变量，对$C^*$-代数退化为经典定义，并用谱局部化子加以说明。
摘要：
我们提出将K-理论推广到算子系。受非交换空间的谱截断（由$C^*$-代数描述）的启发，并受将$C^*$-代数的K-理论实现为埃尔米特型的Witt群的启发，我们引入了由相应矩阵尺寸索引的新算子系不变量。构造了一个正向系统，其正向极限具有半群结构，并将$K_0$-群定义为相应的Grothendieck群。这是单位算子系的一个不变量，更一般地，是算子系在Morita等价意义下的不变量。对于$C^*$-代数，它退化为通常的定义。我们借助谱局部化子来说明我们的不变量。
英文摘要：
We propose a generalization of K-theory to operator systems. Motivated by spectral truncations of noncommutative spaces described by $C^*$-algebras and inspired by the realization of the K-theory of a $C^*$-algebra as the Witt group of hermitian forms, we introduce new operator system invariants indexed by the corresponding matrix size. A direct system is constructed whose direct limit possesses a semigroup structure, and we define the $K_0$-group as the corresponding Grothendieck group. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. For $C^*$-algebras it reduces to the usual definition. We illustrate our invariant by means of the spectral localizer.
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