On the character space of Banach vector-valued function algebras

Document Type : Research Paper


School of Mathematics and Computer Sciences, ‎Damghan University‎, ‎Damghan‎, ‎P.O‎. ‎Box 36715-364‎, ‎Damghan‎, ‎Iran.


‎Given a compact space $X$ and a commutative Banach algebra‎ ‎$A$‎, ‎the character spaces of $A$-valued function algebras on $X$ are‎ ‎investigated‎. ‎The class of natural $A$-valued function algebras‎, ‎those whose characters can be described by means of characters of $A$ and‎ ‎point evaluation homomorphisms‎, ‎is introduced and studied‎. ‎For an‎ ‎admissible Banach $A$-valued function algebra $\mathscr{A}$ on $X$‎, ‎conditions under which the character space $\mathfrak{M}(\mathscr{A})$‎ ‎is homeomorphic to $\mathfrak{M}(\mathfrak{A})\times \mathfrak{M}(A)$ are presented‎, ‎where $\mathfrak{A}=C(X) \cap\mathscr{A}$‎ ‎is the subalgebra of $\mathscr{A}$ consisting of scalar-valued functions‎. ‎An illustration of the results is given by some examples.


Main Subjects

M. Abel, M. Abtahi, Description of closed maximal ideals in topological algebras of continuous vector-valued functions, Mediterr. J. Math. 11 (2014), no. 4, 1185--1193.
M. Abtahi, Vector-valued characters on vector-valued function algebras, Banach J. Math. Anal. 10 (2016), no. 3, 608--620.
M. Abtahi, Normed algebras of infinitely differentiable Lipschitz functions, Quaest. Math. 35 (2012), no. 2, 195--202.
M. Abtahi and S. Farhangi, Vector-valued spectra of Banach algebra valued continuous functions, Rev. R. Acad. Cienc. Exacts Fís. Nat. Ser. A Math. (2016) https://doi.org/10.1007/s13398-016-0367-2.
F.F. Bonsall, J. Duncan, Complete Normed Algebras, Springer-Verlag, Berlin, Heidelberg, New York, 1973.
H.G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monogr. 24, Clarendon Press, Oxford, New York, 2000.
K. Esmaeili and H. Mahyar, The character spaces and Silov boundaries of vector-valued Lipschitz function algebras, Indian J. Pure Appl. Math. 45 (2014), no. 6, 977--988.
T.W. Gamelin, Uniform Algebras, Prentice Hall Press, 1969.
A. Hausner, Ideals in a certain Banach algebra, Proc. Amer. Math. Soc. 8 (1957), 246--249.
T.G. Honary, Relations between Banach function algebras and their uniform closures, Proc. Amer. Math. Soc. 109 (1990), no. 2, 337--342.
T.G. Honary, A. Nikou and A.H. Sanatpour, On the character space of vector-valued Lipschitz algebras, Bull. Iranian Math. Soc. 40 (2014), no. 6, 1453--1468.
J. Horváth, Topological Vector Spaces and Distributions, Vol. I, Addison-Wesley, Reading, Mass., 1966.
E. Kaniuth, A Course in Commutative Banach Algebras, Grad. Texts in Math. 246, Springer, 2009.
G.M. Leibowitz, Lectures on Complex Function Algebras, Scott Foresman, 1970.
A. Nikou and A.G. O'Farrell, Banach algebras of vector-valued functions, Glasgow Math. J. 56 (2014), no. 2, 419--426.
E.L. Stout, The Theory of Uniform Algebras, Bogden & Quigley, 1971.
J. Tomiyama, Tensor products of commutative Banach algebras, T^ohoku Math. J. (2) 12 (1960), 147--154.
B. Yood, Banach algebras of continuous functions, Amer. J. Math. 73 (1951) 30--42.
W. Żelazko, Banach Algebras, Translated from the Polish by M.E. Kuczma. Elsevier Publishing Co. Amsterdam-London-New York; PWN--Polish Scientific Publishers, Warsaw, 1973.