This thesis is an extension of the work done by Filali , Neufang , and Monfared . In this disseration , Suppose that A is a Banach algebra and Y is a Banach space and ?:A? L ( Y) is a representation . If ? is continuous with respect to the norm topologies of A and L (Y) then we ay that ? is a continuous representation . If both A and Y are dual spaces , and ?:A? L (Y), is a continuous representation such that ? is also continuous in the w*-topologies of A and L (Y) then we say that ? is a normal representation. By a coordinate function of a representation ?:A? L (Y) corresponding to y Y and ? Y* , we mean the linear functional on A defined by (a)= ?(a)y,? , (a A) .
