Constructions of minimal Hermitian matrices related to a C*-subalgebra of 𝑀_{𝑛}(ℂ)

Abstract

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