Contemporary Mathematics Volume 341, 2004 Description of all closed maximal regular ideals in subalgebras of the algebra C(X A u) Mart Abel ABSTRACT. A description of all closed maximal regular left (right and two- sided) ideals in subalgebras of C(X, A u) (the algebra of all continuous A-valued functions on X, endowed with the topology of a-convergence) is given in case, when X is a completely regular Hausdorff space, u is a com- pact cover of X, closed with respect to finite unions, and A is a locally m-pseudoconvex Hausdorff algebra over IC, a locally pseudoconvex Wael- broeck Hausdorff algebra over IC or an exponentially galbed Hausdorff algebra over IC with bounded elements. 1. Introduction 1. Let C be the field of complex numbers, A a topological algebra over C with jointly continuous multiplication (in short, a topological algebra), radA the topo- logical radical1 of A and Z(A) the center of A. By mt(A) (mr(A) and mt(A)) we denote the set of all closed maximal regular left (right and two-sided, respectively) ideals of A, by m(A) the set of those closed regular two-sided ideals of A, which are maximal as left (or right) ideals in A, and by homA the set of all nontriv- ial continuous multiplicative linear functionals on A, endowed with the Gelfand topolog'Jl. A set U C homA is equicontinuous at ao E A, if for any c 0 there ex- ists a neighbourhood O(ao) of ao such that lf(a) - ¢(a0 ) I c for each ¢ E U and a E O(ao), and is equicontinuous, if U is equicontinuous at every a E A. Moreover, the set homA is locally equicontinuous, if every ¢0 E homA has an equicontinuous neighbourhood. 2. Let A be an algebra over C. A set U C A is balanced, if J.LU C U, whenever IJ.LI $ 1 convex, if .x + f..LY E U for each x, y E U, whenever . 2:: 0, J.L 2:: 0 and . + J.L = 1 pseudoconvex, if U + U C vU for a number v 0, and idempotent, if uucu. 1991 Mathematics Subject Classification. Primary 46H05 Secondary 46H20. Key words and phrases. Topological algebras, Gelfand-Mazur algebras, exponentially galbed algebras, locally pseudoconvex algebras, topological algebras of vector-valued functions, descrip- tion of all closed maximal regular ideals, description of all continuous characters. Research is in part supported by the Estonian Science Foundation grant 4514. 1As it is shown in [13], Theorem 1, the topological radical radA of a topological algebra A is equal to the intersection of all closed maximal regular· left (or right) ideals of A. 2In this topology a subbase of .neighbourhoods of element cfo E homA consists of sets {c/ E homA: lc/(a)- c/o(a)l e}, where a E A and e 0. © 2004 American Mathematical Society 1 http://dx.doi.org/10.1090/conm/341/06158
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