Matrix-traces on C*-algebra Mn(A) Original Research Article

In this note, for a C*-algebra A, we define a matrix-trace on the C*-algebra Mn(A) to be a positive linear mapping τ:Mn(A)→A such that τ(u*au)=τ(a) (for allaset membership, variantMn(A), for alluset membership, variantU(Mn(A))) and τ(a2)less-than-or-equals, slant(τ(a))2 (for allagreater-or-equal, slanted0). 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 Mn(A) if and only if there exists an element λ of A with 0less-than-or-equals, slantλless-than-or-equals, slantλ2 such that τ(a)=λ·tr(a) (for alla=[aij]set membership, variantMn(A)), where tr(a)=∑i=1naii.