We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute \(\ell ^2\)-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational \(\ell ^2\)-Betti numbers arising from the lamplighter group algebra \({\mathbb Q}[{\mathbb Z}_2 \wr {\mathbb Z}]\). This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational \(\ell ^2\)-Betti numbers from the algebras \({\mathbb Q}[{\mathbb Z}_n \wr {\mathbb Z}]\), where \(n \ge 2\) is a natural number. We also apply the techniques developed to the generalized odometer algebra \({\mathcal {O}}({\overline{n}})\), where \({\overline{n}}\) is a supernatural number. We compute its \(*\)-regular closure, and this allows us to fully characterize the set of \({\mathcal {O}}({\overline{n}})\)-Betti numbers.

我们应用作者在前一篇论文中开发的构造，以获得一个公式，该公式使我们能够计算\(\ell ^2\) -Betti 数，这些数来自可表示为交叉积代数的群代数族。作为一个应用，我们获得了由点灯者群代数\({\mathbb Q}[{\mathbb Z}_2 \wr {\mathbb Z}]产生的一整套无理数\(\ell ^2\) -Betti 数\)。这个过程是建设性的，因为人们对实现这种无理数的元素有明确的描述。这扩展了 Grabowski 所做的工作，他首先从代数中计算了无理数\(\ell ^2\) -Betti 数\({\mathbb Q}[{\mathbb Z}_n \wr {\mathbb Z}]\)， 在哪里\(n \ge 2\)是一个自然数。我们还将开发的技术应用于广义里程计代数\({\mathcal {O}}({\overline{n}})\)，其中\({\overline{n}}\)是一个超自然数。我们计算它的\(*\) -正则闭包，这使我们能够完全表征\({\mathcal {O}}({\overline{n}})\) -Betti 数的集合。