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Proceedings of the American Mathematical Society

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

 
 

 

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