Generators for algebras dense in $L^{p}$-spaces

Alexander J. Izzo¹ ; Bo Li¹

¹Department of Mathematics and Statistics Bowling Green State University Bowling Green, OH 43403, U.S.A.

Studia Mathematica, Tome 217 (2013) no. 3, pp. 243-263

Résumé

For various $L^p$-spaces ($1\leq p\infty $) we investigate the minimum number of complex-valued functions needed to generate an algebra dense in the space. The results depend crucially on the regularity imposed on the generators. For $\mu $ a positive regular Borel measure on a compact metric space there always exists a single bounded measurable function that generates an algebra dense in $L^p(\mu )$. For $M$ a Riemannian manifold-with-boundary of finite volume there always exists a single continuous function that generates an algebra dense in $L^p(M)$. These results are in sharp contrast to the situation when the generators are required to be smooth. For smooth generators we prove a result similar to a known fact about algebras uniformly dense in continuous functions: for $M$ a smooth manifold-with-boundary of dimension $n$, at least $n$ smooth functions are required in order to generate an algebra dense in $L^p(M)$. We also show that on every smooth manifold-with-boundary there exists a bounded continuous real-valued function that is one-to-one on the complement of a set of measure zero.

Détail

DOI : 10.4064/sm217-3-3

Keywords: various p spaces leq infty investigate minimum number complex valued functions needed generate algebra dense space results depend crucially regularity imposed generators positive regular borel measure compact metric space there always exists single bounded measurable function generates algebra dense riemannian manifold with boundary finite volume there always exists single continuous function generates algebra dense these results sharp contrast situation generators required smooth smooth generators prove result similar known about algebras uniformly dense continuous functions smooth manifold with boundary dimension least smooth functions required order generate algebra dense every smooth manifold with boundary there exists bounded continuous real valued function one to one complement set measure zero