Abstract
LetA be aC * — algebra for which all irrèducible representations are of dimensional n. Then ([F], [TT], [V]) algebraA is isomorphic to algebra of all continuous sections of an appropriate algebraic bundle ε A . The basisX of this bundle coincides with the compact of all maximal two-sided ideals ofA. We obtain some conditions which provide that ε A is trivial and this yields thatA is isomorphic to the algebra of alln×n matrix functions continuous onX. In the case whenX=S n is a sphere we describe the set of algebraic bundles overX and algebraic structures on this set. Some applications to algebras generated by idempotents are suggested.
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References
[A] Antonevich A.,Linear functional equations. Operator approach. Operator theory Vol. 83, Birkhauser, (1996).
[AH] Antonevich A., Nguen Tuan Hung,Operator algebras generated by quasirepresentations of groups, Doklady AN Belarusi, (1987). V. 31, n. 6, 489–492.
[At] Atiyah M.,K-theory. Harvard University, (1965).
[BGKKRSS] Bottcher A., Gohberg I., Karlovich Yu., Krupnik N., Roch S., Silbermann B., Spitkovsky I.,Banach algebras generated by n idempotents and applications, Operator Theory. Advances and Applications, Vol. 90, (1996), 19–54.
[B] Blackuadar B.,K-theory for Operator Algebras. Springer Verlag, (1986).
[D] Dixmier J.,Les C *—algebres et leurs representations. Paris, Gauthier-Villars Editeur, (1969).
[F] Fell J.M.G.,The structure of algebras of operator fields, Acta Math., 106, n. 3–4, (1961), 233–280.
[GK] Gohberg I., Krupnik N.:Extension theorems for invertibility symbols in Banach algebras, Integ. Equat. and Oper. Th. 15 (1992), 991–1010.
[Hu] Husemoller D.,Fibre bundles. McGraw-Hill Book Company, (1966).
[K1] Krupnik N.,Banach algebras with symbol and singular integral operators. Stiintsa, Kishinev, (1984); English translation: Birkhauser Verlag, Basel-Boston, (1987), 205 pp.
[K2] Krupnik N.,Symmetrization of the symbol in Banach algebras generated by idempotents, Integr. Equat. and Oper. Th., 31 (1998), 470–481.
[KS] Krupnik N., Silbermann B.,The structure of some Banach algebras fulfilling a standard identity, Math. Nachr., 142 (1989), 175–180.
[La] Lankaster P.,Theory of matrices. Academic Press, (1969).
[Le] Leng S.,Algebra. Addison-Wesley publ.comp., (1965).
[Mi] Mischenko A.,Vector bundles and applications, M., Nauka, (1984).
[Mu] Murphy G.J.,C *—algebras and Operator theory, Academic press, (1990).
[RS] Roch S., Silbermann B.,Algebras generated by idempotents and symbol calculus for singular integral operators, Integr. Equat. and Oper. Th., 11 (1988), 385–419.
[S] Seeley R.,Integro-differential operators on vector bundles, Trans. Amer. Math. Soc. 117 (1965), 167–204.
[TT] Tomiyama J., Takesaki M.,Application of fibre bundle to certain class of C *—algebras, Tohoku Math. Journ. 13. N 3, (1961), 498–522.
[V] Vasilev N.,C *—algebras with finite-dimensional irreducible representanions, Uspechi mat. nauk. XXI, n. 1, (1966). 136–154.