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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Daniel O. Etter Jr., Vector-valued analytic functions, Trans. Amer. Math. Soc. 119 (1965), 352–366. MR 188750, DOI https://doi.org/10.1090/S0002-9947-1965-0188750-X
- P. Lelong, Les fonctions plurisousharmoniques, Ann. Sci. École Norm. Sup. (3) 62 (1945), 301–338 (French). MR 0018304
- M. E. Munroe, Introduction to measure and integration, Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1953. MR 0053186
- W. Żelazko, On the locally bounded and $m$-convex topological algebras, Studia Math. 19 (1960), 333–356. MR 126739, DOI https://doi.org/10.4064/sm-19-3-333-356
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46.35
Retrieve articles in all journals with MSC: 46.35
Additional Information
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