Faithfulness Of Conditional Expectation In Reduced Amalgamated Free Product

Reduced amalgamated free products of C*-algebras is generalization of free product of C*-algebra. we investigate the problem of reduced amalgamated free products of C~*-algebras specified conditional expectation.Let a unital C~*-algebras B, for a set I,has at least two elements,and every i ∈I ,unital C~*-algebras A_i is a containing a copy of B as a unital C~*-subalgebras and having a conditional expectation whose GNS representation is faithful.The paper proved that if every conditional expectation is left-module and right-module, B is center of every Ai,then in reduced amalgamated free products of C~*-algebras over B specified conditional expectation is faithful.