Some Properties Of Matrix-Valued Lipschitz Mappings And Derivations In A Banach Algebra

In this paper, we discuss non-commutatitive Lipschitz spaces, non-commutatitive Lipschitz algebras and derivations in a Banach algebra. We obtain some properties of them and some Lipschitz properties of analytic functions.In Chapter 1, we define Lipschitz-a operators from a compact metric space (K, d) to the matrix space Mm,n(F) and obtain some properties of them. We prove that both L (K, Mm,n(F)) and l (K,Mm,n(F)) are Banach spaces in the norm || || (||f|| = ||f|| + L (f)). We show that l (K, Mm,n(F)) is a closed subspace of L (K,Mm,n(F)) and that if 0 < < < 1 then L (K,Mm,n(F)) is a closed subspace of L (K,Mm,n(F)).In Chapter 2, we discuss the Lipschitz algebras L (K,Mn(F)) and l (K,Mn(F)) and prove that they are all unital regular and *-subalgebras of the C*-algebra C(K, Mn(F)). We obtain some properties of the Lipschitz algebras and limit Lipschitz algebras. Finally, we prove that if (K, d)and ( , ) are compact metric spaces which have more than one elements then Ll(K, C( , C)) =In Chapter 3, we study the properties of Lipschitz space L ((K,d},R), l ((K,d),R) andK), l1((K,d),R), and give some examples to demonstrate them.In Chapter 4, we generalize K. B. Sinha's results about derivations (A) on B(H) to general Banach algebras. Let A be a unital complex Banach algebra and a € A. For every analytic function f on a region in the complex plane, we introduce a bounded linear operator Df(a) and give an integral representation of Df(a). We also obtain estimations of its spectral radius and its norm. Secondly, we study the relation between Df(a) and a and prove that f(a) = Df(a) a = aDf(a), for all f H( ), a € A. We obtain some properties of the map and prove that is a bounded linear operator from H( ) to B(A). Finally, we get some Lipschitz properties of analytic functions and prove that Df(a) is a Lipschitz-1 operator on A. For each and is a closed curve in such that (a) , we definefor all , where f(a) represents the Riesz functional calculus of a by /. Then for all a, b ,there exists an M with 0 < M < + such that ||f(a) - f(b)|| M||a - b||.