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In this paper, we prove that given finite elements $X_1,\\cdots X_n \\in \\M$, there is a finite decomposition of the identity into $N \\in \\NNN$ mutually orthogonal nonzero projections $E_j\\in\\M$, $I=\\sum_{j=1}^NE_j$, such that $E_jX_iE_j=\\tau(X_i) E_j$ for all $j=1,\\cdots,N$ and $i=1,\\cdots,n$. Equivalently, there is a unitary operator $U \\in \\M$ such that $\\frac{1}{N}\\sum_{j=0}^{N-1}{U^*}^jX_iU^j=\\tau(X_i)I$ for $i=1,\\cdots,n$. 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