Homogeneous C^*-algebras
by
Mikhail Shchukin
Belarusian State University

Every n-homogeneous C^*-algebra corresponds one to one to the appropriate a lgebraic bundle over a compact space. We consider the algebraic bundles over sphere S². We show that every n-hom ogeneous C^*-algebra A with the set of primitive ideals Prim A=S² can be generated by three idemp otents. These algebras can not be generated by two idempotents. Also the next theorem describes the set of all n-homogene ous C^*-algebras that can be generated by idempotents. Let A denotes a n-homogeneous C^*-algebra. Let A be separable and non-c ommutative. Let A contains a non-trivial idempotent. Then A can be generated by a finit e set of idempotents. Let us to note that the conditions for n-homogeneous C^*-algebra are exact. It means that every C^*-algebra with finite number of generators is separabl e. An commutative C^*-algebra can be generated by a finite set of idempotents if  and only if the set of maximal ideals of the algebra is a finite set. These results are extend a result of the paper [1] and generalise some results  of the paper [2].

1

A. Antonevich, N. Krupnik, On trivial and non-trivial n-homogeneo us C^*-algebras.

2

N. Krupnik, S. Roch, B. Silbermann, On C^*-algebras generated by  idempotents. J. Funct. Anal., vol. 137, no. 2, 1996.

3

M. Shchukin, Non-trivial C^*-algebras generated by four idempot ents, Proceedings of the Institut of Mathematics of NAN of Belarus, no. 9 (2001), 161- 163.

4

M. Shchukin, Non-trivial C^*-algebras generated by three idempo tents, Proceedings of the international conference on non-linear operators, differenti al equations and applications, Cluj-Napoca, Romania, 2001.

Date received: June 10, 2002