On \(C^*\)-algebras having linear, polynomial and subexponential growth
Answering q a question of \textit{D. Voiculescu} [On the existence of quasi- central approximate units relative to normed ideals, Problem 5.9, J. Funct. Anal. 91, 1-36 (1990)], we show that \(C^*\)-algebras having filtrations \((A_ n)_{n\in\mathbb{N}}\) satisfying the condition \(\limsup_{n\to\infty}ln \dim A_ n/n=0\) (in particular having subexponential growth), are nuclear. For the case of linear growth we obtain the following particular result: let \(X\) be a finite dimensional selfadjoint generating system of a \(C^*\)-algebra \(A\) such that \(\dim(span(X^{n+1}))\leq1+\dim(span(X^ n))\), then there exist a finite dimensional \(C^*\)-algebra \(C\) having only irreducible representations of dimension \(\leq1+\sqrt{\dim(span(X))}\) and a \(C^*\)-algebra \(B\), which is generated by a single self-adjoint element, such that \(A\simeq C\oplus B\). Some other results are given on linear growth and we show that there exist singly generated \(C^*\)-algebras such that the growth of the filtration \((span(X^ n))_{n\in\mathbb{N}}\) is polynomial, where \(X=\{x,x^*,\mathbf{1}\}\) is a generating system, and such that in every neighbourhood of \(x\) there exists an invertible \(y\) such that \(Y=\{\mathbf{1},y,y^*\}\) is a generating system whose associated filtration \((span(Y^ n))_{n\in\mathbb{N}}\) doesn't satisfy the previous condition of Voiculescu, and in particular does not have subexponential growth.
scientific article; zbMATH DE number 4216410