The complete metric space of the lattice of varieties and a fixed point theorem in complete metric spaces (Q2908025)

Varieties of Banach algebras were introduced by \textit{P. G. Dixon} [Q. J. Math., Oxf. II. Ser. 27, 481--487 (1976; Zbl 0338.46045)]. A class \(\mathcal{C}\) of Banach algebras is a variety if there exists a nonnegative real-valued function \(p\mapsto \| p\|_\mathcal{C}\) on the set of polynomials \(p(X_1,\dots, X_n)\) such that a Banach algebra \(A\) is in \(\mathcal{C}\) precisely when \(\| p(a_1,\dots, a_n) \| \leq \| p\|_\mathcal{C}\) for each \(a_1, \dots, a_n\) in the unit ball of \(A\).NEWLINENEWLINEIn the article under review, the authors define a metric on the space of varieties (as in their article [Int. J. Math. Sci. 6, No. 1, 51--62 (2007; Zbl 1169.46304)]). The primary goal of the present article appears to be the construction of a complete subspace \(\{v_a:0\leq a\leq 1\}\) of the metric space of varieties that is homeomorphic to \([0,1]\). The authors also consider equivalent metrics on the space of varieties, and conclude the article with a fixed point theorem that is, essentially, Banach's contraction mapping principle.NEWLINENEWLINEReviewer's remark. Unfortunately, this article contains a plethora of typographical errors, which makes it somewhat difficult to read and to discern the significance of the authors' results.