Banach代数上同态的自动连续性

Automatic Continuity of Homomorphisms on Banach Algebras

本文引入了具有性质OUDP的Banach代数A,证明了从A到任一Banach代数的任一同态自动连续.对有单位的Banach代数A,证明了从A到任一Banach代数的满同态自动连续当且仅当从矩阵代数M_n(A)到任一Banach代数的满同态自动连续.同时还证明:若从A(M_n(A))的任一闭双理想出发的同态自动连续,则从M_n(A)(A)的任一闭双理想出发的满同态的分离空间由拟幂零元组成.

In this paper, the Banach algebra with the property OUDP is defined and it is proved that every homomorphism from such algebra into a Banach algebra is continuous. For an unit Banach algebra A, it is shown that every homomorphism from A is continuous if and only if so does it from M_n(A) and that if each homomorphism from every closed bi-ideal of A(resp. M_n(A)) is continuous, then the separating space of a homomorphism from a closed bi-ideal of M_n(A)(resp.A ) onto a Banach algebra B consists of qusi-nilpotent elements of B.