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Proceedings of the American Mathematical Society

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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
MathSciNet review: 0253033
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Abstract: The space $ {L_0}$ consists of measurable functions $ f$ on $ [0,1]$ such that $ \log (1 + \vert f(x)\vert)$ 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

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