Without the successful work of Professor Kakutani on representing a unit vector space as a dense vector sub-lattice of in 1941, where X is a compact Hausdorff space and C(X) is the space of real continuous funct...Without the successful work of Professor Kakutani on representing a unit vector space as a dense vector sub-lattice of in 1941, where X is a compact Hausdorff space and C(X) is the space of real continuous functions on X. Professor M. H. Stone would not begin to work on “The generalized Weierstrass approximation theorem” and published the paper in 1948. Latter, we call this theorem as “Stone-Weierstrass theorem” which provided the sufficient and necessary conditions for a vector sub-lattice V to be dense in . From the theorem, it is not clear and easy to see whether 1) “the vector sub-lattice V of C(X) contains constant functions” is or is not a necessary condition;2) Is there any clear example of a vector sub-lattice V which is dense in , but V does not contain constant functions. This implies that we do need some different version of “Stone-Weierstrass theorem” so that we will be able to understand the “Stone-Weierstrass theorem” clearly and apply it to more places where they need this wonderful theorem.展开更多
In this paper, we shall give an elementary proof of a Bishop–Stone–Weierstrass theorem for （M2（C））n with respect to its pure states. To be more precise, we shall show that the pure-state Bishop hull of a unital ...In this paper, we shall give an elementary proof of a Bishop–Stone–Weierstrass theorem for （M2（C））n with respect to its pure states. To be more precise, we shall show that the pure-state Bishop hull of a unital subalgebra （not necessarily self-adjoint） of （M2（C））n is equal to itself.展开更多
文摘Without the successful work of Professor Kakutani on representing a unit vector space as a dense vector sub-lattice of in 1941, where X is a compact Hausdorff space and C(X) is the space of real continuous functions on X. Professor M. H. Stone would not begin to work on “The generalized Weierstrass approximation theorem” and published the paper in 1948. Latter, we call this theorem as “Stone-Weierstrass theorem” which provided the sufficient and necessary conditions for a vector sub-lattice V to be dense in . From the theorem, it is not clear and easy to see whether 1) “the vector sub-lattice V of C(X) contains constant functions” is or is not a necessary condition;2) Is there any clear example of a vector sub-lattice V which is dense in , but V does not contain constant functions. This implies that we do need some different version of “Stone-Weierstrass theorem” so that we will be able to understand the “Stone-Weierstrass theorem” clearly and apply it to more places where they need this wonderful theorem.
基金supported by National Natural Science Foundation of China (Grant No.11201146)Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry
文摘In this paper, we shall give an elementary proof of a Bishop–Stone–Weierstrass theorem for （M2（C））n with respect to its pure states. To be more precise, we shall show that the pure-state Bishop hull of a unital subalgebra （not necessarily self-adjoint） of （M2（C））n is equal to itself.
