Seminorms Associated with Subadditive Weights on C*-Algebras

Let ϕ be a subadditive weight on a C* -algebra A, and let Mϕ+ be the set of all elements x in A+ with ϕ(x) < +00. A seminorm ‖ • ‖ is introduced on the lineal Mϕsa = linRMϕ+, and a sufficient condition for the seminorm to be a norm is given. Let I be the unit of the algebra A, and let ϕ(I) = 1. Then, for every element x of Asa, the limit ρϕ(x) = limt→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 Mn(C) are considered. An estimate for the difference of Cayley transforms of Hermitian elements of a von Neumann algebra is obtained.