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Exposed points in the set of representing measures for the disc algebra

100%

Izzo A.

Annales Polonici Mathematici

1995

tom 61

nr 1

59-62

It is shown that for each nonzero point x in the open unit disc D, there is a measure whose support is exactly ∂D ∪ {x} and that is also a weak*-exposed point in the set of representing measures for the origin on the disc algebra. This yields a negative answer to a question raised by John Ryff.

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

63%

Izzo A., Li B.

Studia Mathematica

2013

tom 217

nr 3

243-263

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Studia Mathematica

2 Izzo A.

1 Li B.

1 2013

1 1995

Wyniki wyszukiwania

Exposed points in the set of representing measures for the disc algebra

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