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

Abstract:

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 $M_n(\Bbb C)$.