Seminorms Associated with Subadditive Weights on C~*-Algebras

摘要

Let Φ be a subadditive weight on a C~*-algebra A, and let M~+_Φ be the set of all elements x in A~+ with Φ(ⅹ) ＜ +∞. A seminorm ‖·‖ _Φ introduced on the lineal M~sa~Φ = lin_R M~+_Φ and a sufficient condition for the seminorm to be a norm is given. Let / be the unit of the algebra A, and let Φ(Ⅰ) = 1. Then, for every element x of A~sa, the limit ρΦ(ⅹ) = lim_t→0+(Φ(Ⅰ + tx) - 1)/t exists and is finite. Properties of ρΦ, are investigated, and examples of subadditive weights on C~*-algebras are considered. On the basis of Lozinskii's 1958 results, specific subadditive weights on M_n(C) are considered. An estimate for the difference of Cayley transforms of Hermitian elements of a von Neumann algebra is obtained.