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

Alexander J. Izzo, and Bo Li. "Generators for algebras dense in $L^{p}$-spaces." Studia Mathematica 217.3 (2013): 243-263. <http://eudml.org/doc/285809>.

@article{AlexanderJ2013,
abstract = {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.},
author = {Alexander J. Izzo, Bo Li},
journal = {Studia Mathematica},
keywords = {-space; generators for algebras; manifold; one-to-one almost everywhere function},
language = {eng},
number = {3},
pages = {243-263},
title = {Generators for algebras dense in $L^\{p\}$-spaces},
url = {http://eudml.org/doc/285809},
volume = {217},
year = {2013},
}

TY - JOUR
AU - Alexander J. Izzo
AU - Bo Li
TI - Generators for algebras dense in $L^{p}$-spaces
JO - Studia Mathematica
PY - 2013
VL - 217
IS - 3
SP - 243
EP - 263
AB - 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.
LA - eng
KW - -space; generators for algebras; manifold; one-to-one almost everywhere function
UR - http://eudml.org/doc/285809
ER -