On balanced bases (Q2387840)

Let \(A\) be an associative semisimple algebra, \(\text{dim}_\mathbb{C} A<\infty\). A balanced \(t\)-system is a family of idempotents \( \mathcal{E} =\{e_1 , e_2, \ldots , e_v\}\) in \(A\) (where \(\text{dim} A >1\)) for which: 1. \(e_1 +\ldots + e_v=k\dot 1_A , k\in \mathbb{C}\); 2. the value of the trace function Tr\(_A e_{i_1} \ldots e_{i_s}=\lambda_s\) depends only on \(s\) for any \(s\) distinct idempotents \(e_{i_1} \ldots e_{i_s} \in \mathcal{E}\), where \(s\leq t\). The number \(\text{rank}\, \mathcal{E} =\lambda_1\) is referred to as the rank of a system \(\mathcal{E}\). If a balanced 2-system \(\mathcal{E}\) is a basis of the algebra \(A\), then \(\mathcal{E}\) is said to be a balanced basis and \(A\) a balanced algebra. The main theorem of the paper states that if \(\mathcal{E}\) is a balanced basis of the algebra \((n+1) M_1\oplus M_n\), than \(\text{rank}(\mathcal{E})\) is equal to either \(n+1\) or \(n^2\). From this fact there follows the equivalence of a \(WP\)-decomposition [\textit{D. N. Ivanov}, Sb. Math. 195, No.11, 1557--1574 (2004); translation from Mat. Sb. 195, No. 11, 13--30 (2004; Zbl 1109.16026)] in the matrix algebra \(M_n\) and the condition that algebra \((n+1) M_1\oplus M_n\) is balanced.