Let A be an algebra. A centralizer on A is a linear operator w:A?A satisfying w(ab)=w(a)b=aw(b) for any a,b?A. Let A and B be Banach algebras. As usual, homomorphism from A into B is a linear map ?:A?B such that ?(ab)=?(a)?(b) for any a,b?A .We call a map ?:A?A is a weighted homomorphism if there exist a homomorphism ?:A? B and invertible centralizer w:B?B such that ?=wo?. A linear map ? from a Banach algebra A into another B preserves zero products if ?(a)?(b)=? whenewer a,b?A are such that ab=?. In this thesis, we consider several types of orthogonality conditions on group algebrasL^? (G) of a locally campact group G such as: (i) one-sided orthogonality conditions A? : f*g=? A? : f*(g^*)=?; A? : (f^*)*g=?. (ii) two- sided orthogonality conditions A? : f*g=g*f=?; A? :f*(g^*)=(g^*)*f=?.