论文标题
$ k_1 $和$ k $ - 绝对矩阵订单单位空间
$K_1$ and $K$-groups of absolute matrix order unit spaces
论文作者
论文摘要
在本文中，我们分别描述了Grothendieck组$ k_1（v）$和$ k（v）$的绝对矩阵订单单位空间$ v $分别用于单一和部分统一元素。为此，我们研究了统一和部分统一元素的一些基本特性，并定义了它们的路径同性恋等效性。 $ k（v）$的构建几乎以$ k_1（v）的方式遵循。$我们证明$ k_1（v）$和$ k（v）$是订购的阿贝利安集团。我们还证明，$ k_1（v）$和$ k（v）$是来自具有形态的绝对矩阵单位空间类别的函子，因为unital $ \ vert \ vert \ cdot \ cdot \ vert $ - 保留订单类的亚伯利亚组类别。稍后，我们表明，在某些条件下，$ k（v）$的商与$ k_0（v）$和$ k_1（v）的直接和$ k_0（v）$是订单预测的Grothendieck组。
In this paper, we describe the Grothendieck groups $K_1(V)$ and $K(V)$ of an absolute matrix order unit space $V$ for unitary and partial unitary elements respectively. For this purpose, we study some basic properties of unitary and partial unitary elements, and define their path homotopy equivalence. The construction of $K(V)$ follows in a almost similar manner as that of $K_1(V).$ We prove that $K_1(V)$ and $K(V)$ are ordered abelian groups. We also prove that $K_1(V)$ and $K(V)$ are functors from the category of absolute matrix order unit spaces with morphisms as unital completely $\vert \cdot \vert$-preserving maps to the category of ordered abelian groups. Later, we show that under certain conditions, quotient of $K(V)$ is isomorphic to the direct sum of $K_0(V)$ and $K_1(V),$ where $K_0(V)$ is the Grothendieck group for order projections.
