Generators for algebras dense in $L^{p}$-spaces
Tom 217 / 2013
Streszczenie
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.
