CONSTRUCTIONS OF MINIMAL HERMITIAN MATRICES RELATED TO A C*-SUBALGEBRA OF M_n(C)

This paper provides a constructive method using unitary diagonalizable elements to obtain all hermitian matrices A in Mn(C) such that ∥A∥ = _Bmin _∈B ∥A + B∥, where B is a C*-subalgebra of Mn(C), ∥ · ∥ denotes the operator norm. Such an A is called B-minimal. Moreover, for a C*-subalgebra B determined by a conditional expectation from Mn(C) onto it, this paper constructs ^∅k_i=1 Bminimal hermitian matrices in M_kn(C) through B-minimal hermitian matrices in Mn(C), and gets a dominated condition that the matrix  = diag(A1, A2, · · ·, A_k) is ^∅k_i=1 B-minimal if and only if ∥Â∥ ≤ ∥As∥ for some s ∈ (1, 2, · · ·, k) and As is B-minimal, where Ai(1 ≤ i ≤ k) are hermitian matrices in Mn(C).