Constructions of minimal Hermitian matrices related to a C*-subalgebra of M_n(\Bbb C).

This paper provides a constructive method using unitary diagonalizable elements to obtain all hermitian matrices A in M_n(\Bbb C) such that \begin{equation*} \|A\|=\min _{B\in \mathcal {B}}\|A+B\|, \end{equation*} where \mathcal {B} is a C*-subalgebra of M_n(\Bbb C), \|\cdot \| denotes the operator norm. Such an A is called \mathcal {B}-minimal. Moreover, for a C*-subalgebra \mathcal {B} determined by a conditional expectation from M_n(\Bbb C) onto it, this paper constructs \bigoplus _{i=1}^k\mathcal {B}-minimal hermitian matrices in M_{kn}(\Bbb C) through \mathcal {B}-minimal hermitian matrices in M_n(\Bbb C), and gets a dominated condition that the matrix \hat {A}\!=\!\operatorname {diag}(A_1,A_2,\cdots, A_k) is \bigoplus _{i=1}^k\mathcal {B}-minimal if and only if \|\hat {A}\|\leq \|A_s\| for some s\in \{1,2,\cdots,k\} and A_s is \mathcal {B}-minimal, where A_i(1\leq i\leq k) are hermitian matrices in M_n(\Bbb C). [ABSTRACT FROM AUTHOR]