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Seminorms Associated with Subadditive Weights on C*-Algebras

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Abstract

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.

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Funding

The research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (project No 1.13556.2019/13.1).

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  1. Kazan (Volga Region) Federal University, Kazan, 420008, Russia

    A. M. Bikchentaev

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  1. A. M. Bikchentaev
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Correspondence to A. M. Bikchentaev.

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Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 3, pp. 341-350.

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Bikchentaev, A.M. Seminorms Associated with Subadditive Weights on C*-Algebras. Math Notes 107, 383–391 (2020). https://doi.org/10.1134/S0001434620030025

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