Jian-Hua Zhang

Matrix-traces on C∗-algebra Mn(A)

Abstract In this note, for a C ∗ -algebra A , we define a matrix-trace on the C ∗ -algebra M n ( A ) to be a positive linear mapping τ : M n ( A )→ A such that τ(u ∗ au)=τ(a) (∀a∈M n (A), ∀u∈U(M n (A))) and τ(a 2 )⩽(τ(a)) 2 (∀a⩾0) . We give some basic properties of a matrix-trace and prove that if A is a unital Abelian C ∗ -algebra, then a map τ is an A -module matrix-trace on M n ( A ) if and only if there exists an element λ of A with 0⩽ λ ⩽ λ 2 such that τ(a)=λ· tr (a) (∀a=[a ij ]∈M n (A)) , where tr( a )=∑ i =1 n a ii .