<atl>Matrix-traces on <f>C&ast;</f>-algebra <f>Mn(A)</f>

In this note, for a <f>C&ast;</f>-algebra A, we define a matrix-trace on the <f>C&ast;</f>-algebra <f>Mn(A)</f> to be a positive linear mapping <f>τ:Mn(A)→A</f> such that <f>τ(u&ast;au)=τ(a) (∀a∈Mn(A), ∀u∈U(Mn(A)))</f> and <f>τ(a2)&les;(τ(a))2 (∀a&ges;0)</f>. We give some basic properties of a matrix-trace and prove that if A is a unital Abelian <f>C&ast;</f>-algebra, then a map <f>τ</f> is an A-module matrix-trace on <f>Mn(A)</f> if and only if there exists an element <f>λ</f> of A with <f>0&les;λ&les;λ2</f> such that <f>τ(a)=λ·tr(a) (∀a=[aij]∈Mn(A))</f>, where <f>tr(a)=∑i=1naii</f>. [Copyright &y& Elsevier]