离散群上带支配算子的有限截面
Finite sections of band-dominated operators on discrete groups
AI总结：
针对有限生成离散恰当群Γ上的带支配算子，利用极限算子刻画Fredholm性，研究有限截面方法的稳定性，并分析离散边界序列在拟交换子理想中的作用。
摘要：
设Γ为有限生成离散恰当群。我们考虑l^2(Γ)上的算子，它们由l^∞(Γ)中函数乘法算子和Γ中元素的左平移算子复合而成。这些算子生成L(l^2(Γ))的一个C*-子代数，其元素称为Γ上的带支配算子。我们研究关于Γ的给定生成系统的带支配算子的有限截面方法的稳定性。我们的方法基于序列稳定性与关联算子Fredholm性的等价性，以及Roe关于恰当离散群上带支配算子Fredholm性的准则（我们用极限算子表述）。特别关注由有限截面序列生成的代数的拟交换子理想以及该代数中序列的稳定性。对于这两个问题，离散边界序列起着关键作用。
英文摘要：
Let $Γ$ be a finitely generated discrete exact group. We consider operators on $l^2(Γ)$ which are composed by operators of multiplication by a function in $l^\infty (Γ)$ and by the operators of left-shift by elements of $Γ$. These operators generate a $C^*$-subalgebra of $L(l^2(Γ))$ the elements of which we call band-dominated operators on $Γ$. We study the stability of the finite sections method for band-dominated operators with respect to a given generating system of $Γ$. Our approach is based on the equivalence of the stability of a sequence and the Fredholmness of an associated operator, and on Roe's criterion for the Fredholmness of a band-dominated operator on a exact discrete group, which we formulate in terms of limit operators. Special emphasis is paid to the quasicommutator ideal of the algebra generated by the finite sections sequences and to the stability of sequences in that algebra. For both problems, the sequence of the discrete boundaries plays an essential role.
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