Symmetrization of the symbol in Banach algebras generated by idempotents

It is known [2] that a Banach algebra A generated by 2N idempotents with relations (1),(2) is an algebra with matrix symbol of order 2N. This symbol is completely and explicitly denned via the spectrums of two (indicator) elements (3) and (5). But in the case when double-struck A is a c*-algebra. the symbol constructed in [2] is not symmetric. Moreover in order to construct a symmetric symbol, one needs some additional information 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 symbol are described in this paper.