The inverse-closed subalgebra of $C^{*}(G,A)$

TL;DR

The paper addresses inverse-closed (spectrally invariant) subalgebras within the Roe algebra with coefficients $C^{*}(G,A)$ for a countable discrete group $G$ and a noncommutative coefficient algebra $A$. It constructs the Sobolev-type intersection $W_a^{\infty}(G,A)$ of Banach algebras $W_a(G,A)$ with weighted norms and proves this subalgebra is spectrally invariant and dense in $C^{*}(G,A)$ under polynomial-growth of the group action. The key technique uses a norm-controlled, Neumann-series-type argument applied to $T^{*}T$ to show invertibility remains inside $W_a^{\infty}(G,A)$, establishing spectral invariance. This yields $K$-theory invariance and provides a practical subalgebra for computing $K$-groups while preserving structural properties of the Roe algebra.$

Abstract

This paper studies the inverse-closed subalgebras of the Roe algebra with coefficients of the type \(l^2(G, A)\). The coefficient \(A\) is chosen to be a non-commutative \(C^*\)-algebra, and the object of study is \(C^*(G, A)\) generated by the countable discrete group \(G\). By referring to the Sobolev-type algebra, the intersection of a family of Banach algebras is taken. It is proved that the intersection \(W_a^{\infty}(G, A)\) of Banach spaces is a spectrally invariant dense subalgebra of \(C^*(G, A)\), and a sufficient condition for this is that the group action of \(G\) has polynomial growth.

Theorems & Definitions (10)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Lemma 3.4
• Theorem 3.5
• proof