N-Homomorphisms, Mappings Preserving N-Zero Products And Local N-Isomorphisms

This thesis consists of three chapters which concerns the mapping theory such as nhomomorphism, n-zero product preserving and n-isomorphism between operator algebras.In chapter 1,we study that automatic continuity of n-homomorphisms on unital(*) Banach algebras,characterize the structure of the n-homomorphism and discuss the relations among n-homomorphism,homomorphism,mapping preserving zero product and associative mapping.Using the characterization of the n-homomorphism on a unital Banach algebras,we show that each surjective n-homomorphism and * n-homomorphism from a unital Banach algebra and *-Banach algebra,respectively,into a semi-simple abelian Banach algebra and a C^*-algebra,is automatically continuous.We also show that each * n-homomorphism on C^*-algebras is isometric,preserving spectral radius and preserving two-side self-adjoint and show that n-homomorphism preserving idempotents on unital algebras and the mapping which are both n-homomorphism and n+1-homomorphism on factorizable algebra are homomorphisms.In addition,it is devoted to examine the relations between n-homomorphism and other mappings.For example,it is shown that bounded n-homomorphism on unital Banach algebras is associative,surjective n-homomorphism from an algebra onto unital algebra and factorizable and semi-prime algebra,respectively, are zero product preserving.In chapter 2,we introduce the notion of thc mappings preserving n-zero products, compared with the mappings preserving zero product,characterize the structure relations between surjective mapping preserving two-side n-zero product and mapping preserving zero product.In this chapter,it is shown that a surjective n-zero product preserving from an algebra linearly generated by its idempotents onto unital algebra is associative. From this,we conclude that surjective n-zero product preserving mapping on B(H)has thc form:φ(T)=λATA^-1,whcrcλ∈C,A is invertible in B(H).it is also shown that a bounded surjective n-zero product preserving on unital C^*-algebra is a product of an algebra homomorphism and a central element. In chapter 3,we introduce the notion of local n-isomorphism.We show that each surjective local n-isomorphism on B(X),the operator algebra of all bounded lincar operators on an infinite dimensional Banach space X,and surjective local n-isomorphism from unital Banach algebra onto semi-simple abelian Banach algebra are n-isomorphisms. We also show that each continuous surjective local n-isomorphism from von Neumann algebra onto unital Banach algebra is a product of a Jordan homomorphism and a central clement.