Closed maximal regular one-sided ideals in topological algebras

Abstract

Topoloogilise algebra ehituse uurimisel on kasulik tunda selle kinniseid ideaale. Sealjuures on eriti hea teada, millised maksimaalsetest ideaalidest on kinniseed. Üks lihtsamaid topoloogilisi algebraid on kompaktsel Hausdorffi ruumil defineeritud pidevate funktsioonide algebra, mis on varustatud ühtlase koondumise topoloogiaga. Selle kommutatiivse topoloogilise algebra kinnised ideaalid kirjeldati eelmise sajandi esimeses pooles. Hiljem on kirjeldatud kinniseid maksimaalseid ideaale paljudes teistes topoloogilistes algebrates. Väitekirjas uuritakse ühe- ja kahepoolseid ideaale üldiste topoloogiliste algebrate pere poolt defineeritud pidevate lõigete topoloogilses algebras. Kui pere indeksite hulk on täielikult regulaarne Hausdorffi ruum, saadakse kõigi kinniste ja kinniste maksimaalsete ühepoolsete ideaalide kirjeldus. Kui pere indeksiteks on topoloogise algebra A kõigi kahepoolsete ideaalide hulk, saadakse tingimused, mille korral saab A esitada selle pidevate lõigete algebra kõikjal tiheda alamalgebrana. Lisaks sellele esitatakse A ka teatud pidevate kujutuste algebra kõikjal tiheda alamalgebrana. Kui indeksid on topoloogilise algebra A kõikide kinniste maksimaalsete regulaarsete vasakpoolsete (parempoolsete) ideaalide poolt defineeritud primitiivsed ideaalid, saadakse A kõigi kinniste maksimaalsete regulaarsete ühepoolsete ideaalide kirjeldus. Nende tulemuste rakendusena saadakse vasakpoolse ja parempoolse topoloogilise Jacobsoni radikaali kirjeldus, mille abil antakse tingimused nende radikaalide võrdumiseks. See on osaline vastus B. Yoodi poolt 1964. aastal püstitatud probleemile.

To study the structure of a topological algebra, it is useful to know the description of the closed ideals in that topological algebra. In particular, it is important to know which maximal ideals are closed. One of the simplest topological algebras is the algebra of continuous functions on a compact Hausdorff space with the uniform topology. Closed ideals in that commutative topological algebra were described in the first half of the previous century. After that, closed maximal ideals are described in many other topological algebras. Ideals in the topological algebra of continuous sections, defined by a family of general topological algebras, are considered in the present Thesis. In case the family of topological algebras is indexed by a completely regular Hausdorff space, the description of all closed and closed maximal one-sided ideals is obtained. In case the family is indexed by all two-sided ideals of a topological algebra A, conditions when A is densely embedded in that topological algebra of continuous sections are given. In addition, A is densely embedded in a certain topological algebra of continuous maps. In case the indexes are the primitive ideals, defined by the closed maximal regular left (right) ideals of a topological algebra A, the description of all closed maximal regular one-sided ideals of A is obtained. As an application of these results, the descriptions of the left and right topological Jacobson radical are given, which are used to find the conditions when these radicals coincide, partly solving the problem posed by B. Yood in 1964.