SYMMETRIZATION OF THE SYMBOL IN BANACH-ALGEBRAS GENERATED BY IDEMPOTENTS

It is known [2] that a Banach algebra A generated by 2N idempotents wi th relations (1),(2) is an algebra with matrix symbol of order 2N. Thi s symbol is completely and explicitly defined via the spectrums of two (indicator) elements (3) and (5). But in the case when A, is a c-alg ebra, the symbol constructed in [2] is not symmetric. Moreover in orde r to construct a symmetric symbol, one needs some additional informati on about the algebra A, even for the case when p(j) = p(j) (j = 1,... ,2N). These additional conditions and the explicit form of symmetric s ymbol are described in this paper.