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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The algebra of log-summable functions


Author: Daniel O. Etter
Journal: Proc. Amer. Math. Soc. 25 (1970), 1-7
MSC: Primary 46.35
DOI: https://doi.org/10.1090/S0002-9939-1970-0253033-3
MathSciNet review: 0253033
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Abstract: The space ${L_0}$ consists of measurable functions $f$ on $[0,1]$ such that $\log (1 + |f(x)|)$ is summable on $[0,1]$, with functions equal almost everywhere identified. The integral defines a quasinorm on ${L_0}$. With this quasinorm, ${L_0}$ becomes a complete quasinormed linear space, the topology of which is not locally bounded. The quasinorm is plurisubharmonic (subharmonic on one-dimensional complex manifolds). ${L_0}$ is closed under multiplication, and multiplication is continuous. Inversion is not continuous, and the group of invertible elements is not open. There are no proper closed maximal ideals. The resolvent ${(\lambda - f)^{ - 1}}$ may exist for all complex $\lambda$, but it cannot be entire.


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Keywords: Quasinormed linear algebra, non-locally-bounded topology plurisubharmonic metric, plurisubharmonic functional, Lebesgue function space, log-summable modulus, analyticity of resolvent, closed maximal ideals, spectrum
Article copyright: © Copyright 1970 American Mathematical Society