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

Alexander J. Izzo ; Bo Li

Studia Mathematica, Tome 215 (2013), p. 243-263 / Harvested from The Polish Digital Mathematics Library

Résumé

For various $L^{p}$-spaces (1 ≤ p < ∞) 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 μ a positive regular Borel measure on a compact metric space there always exists a single bounded measurable function that generates an algebra dense in $L^{p}(μ)$. 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

Publié le : 2013-01-01

Zbl 1291.46028

EUDML-ID : urn:eudml:doc:285809