Let $ \\mathcal{A} $ be a complex commutative Banach algebra and let $ \\Delta ( \\mathcal{A}) $ be the maximal ideal space of $ \\mathcal{A} $ . We say that $ \\mathcal{A} $ has the bouunded separating property if there exists a constant $ C 0 $ such that for every two distinct points $ \\phi _{1}, \\phi _{2} \\in \\Delta ( \\mathcal{A}) $ , there is an element $ a \\in \\mathcal{A} $ for which \\begin{center} $ \\widehat{a}( \\phi _{1})=1 ,\\quad \\widehat{a}( \\phi _{2})=0,\\quad \\Vert a \\Vert \\leq C.$ \\end{center} where $ \\widehat{a} $ is the Gelfand transform of $ a \\in \\mathcal{A} $ .\\\\ In chapter 2, we show that compact homomorphisms are of finite dimensional range for a large dir=ltr In chapter 3, we show that if $ \\mathcal{A} $ is a uniformly regular Ditkin algebra,then every weakly compact homomorphism of $ \\mathcal{A} $ into a Banach algebra is finite dimensional range. \\\\ In chapter 4, we give some applications of previous chapter on Arense regular and weakly sequentially compact Banach algrbras. For example as a main result, we show that if $ \\mathcal{A} $ is an Arens regular Banach algebra with b.a.i.p., and $ \\mathcal{B} $ is an weakly sequentially complete Banach algebra and one of the algebras is commutative, then for every continuous, homomorphism $ h:\\mathcal{A} \\longrightarrow \\mathcal{B} $ , the algebra $ h( \\mathcal{A}) $ is semisimple and finite dimensional