Absolutely continuous functions on idempotent semigroups in the locally convex setting

Katsaras, A.

doi:10.1090/S0002-9947-1975-0374904-2

Absolutely continuous functions on idempotent semigroups in the locally convex setting
HTML articles powered by AMS MathViewer

by A. Katsaras

Trans. Amer. Math. Soc. 206 (1975), 329-337

DOI: https://doi.org/10.1090/S0002-9947-1975-0374904-2

PDF | Request permission

Abstract:

Let $E$ be a locally convex space and let $T$ be a semigroup of semicharacters on an idempotent semigroup. It is shown that there exists an isomorphism between the space of $E$-valued functions on $T$ and the space of all $E$-valued finitely additive measures on a certain algebra of sets. The space of all $E$-valued functions on $T$ which are absolutely continuous with respect to a positive definite function $F$ is identified with the space of all $E$-valued measures which are absolutely continuous with respect to the measure ${m_F}$ corresponding to $F$. Finally a representation is given for the operators on the set of all $E$-valued finitely additive measures on an algebra of sets which are absolutely continuous with respect to a positive measure.

References

Richard A. Alò and André de Korvin, Functions of bounded variation on idempotent semigroups, Math. Ann. 194 (1971), 1–11. MR 293377, DOI 10.1007/BF01351816
Richard A. Alò, André de Korvin, and Richard J. Easton, Vector valued absolutely continuous functions on idempotent semigroups, Trans. Amer. Math. Soc. 172 (1972), 491–500. MR 310181, DOI 10.1090/S0002-9947-1972-0310181-3
Richard B. Darst, A decomposition of finitely additive set functions, J. Reine Angew. Math. 210 (1962), 31–37. MR 137808, DOI 10.1515/crll.1962.210.31
N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
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
J. R. Edwards and S. G. Wayment, Representations for transformations continuous in the $\textrm {BV}$ norm, Trans. Amer. Math. Soc. 154 (1971), 251–265. MR 274704, DOI 10.1090/S0002-9947-1971-0274704-4
T. H. Hildebrandt, Linear continuous functionals on the space $(BV)$ with weak topologies, Proc. Amer. Math. Soc. 17 (1966), 658–664. MR 193490, DOI 10.1090/S0002-9939-1966-0193490-3
Stephen E. Newman, Measure algebras and functions of bounded variation on idempotent semigroups, Trans. Amer. Math. Soc. 163 (1972), 189–205. MR 308686, DOI 10.1090/S0002-9947-1972-0308686-4
Stephen E. Newman, Measure algebras and functions of bounded variation on idempotent semigroups, Bull. Amer. Math. Soc. 75 (1969), 1396–1400. MR 251551, DOI 10.1090/S0002-9904-1969-12430-4