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2013 | 217 | 3 | 243-263
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Generators for algebras dense in $L^{p}$-spaces

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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.
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  • Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, U.S.A.
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  • Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm217-3-3
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