A new measure of growth for countable-dimensional algebras. II
A new dimension function, called bandwidth dimension, was introduced by \textit{J. Hannah} and the author in part I [Trans. Am. Math. Soc. 347, No. 1, 111-136 (1995; Zbl 0826.16018)] for countable-dimensional algebras over a field \(F\). According to a theorem of \textit{K. R. Goodearl, P. Menal} and \textit{J. Moncasi} [J. Algebra 156, No. 2, 407-432 (1993; Zbl 0780.16006)] any such algebra \(A\) can be embedded into the algebra \(B(F)\) of all row-finite and column-finite \(\omega \times \omega\) matrices over \(F\). The width of the band along the main diagonal that contains the nonzero entries of the matrix representing an element of \(A\) in \(B(F)\) gives rise to a new type of growth function for \(A\), leading to the bandwidth dimension of \(A\). In the earlier paper it had been shown that this dimension is always a real number in the interval [0,1], and the article under review shows that every real number in this interval actually occurs as the bandwidth dimension of some finitely generated algebra over \(F\). Although the proof of this result is quite long, it is surprisingly elementary in the sense that it does not require much beyond basic calculus and linear algebra.
scientific article; zbMATH DE number 763918