From Matrix to Operator Inequalities

Canadian mathematical bulletin

Terry A. Loring¹ 

[1] Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, U.S.A.

关键词: $C*$-algebras;    matrices;    bounded operators;    relations;    operator norm;    order;    commutator;    exponential;    residually finite dimensional;   

DOI  :  10.4153/CMB-2011-063-8

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We generalize Löwner's method for proving that matrix monotonefunctions are operator monotone. The relation $xleq y$ on boundedoperators is our model for a definition of $C^{*}$-relationsbeing residually finite dimensional.Our main result is a meta-theorem about theorems involving relationson bounded operators. If we can show there are residually finite dimensionalrelations involved and verify a technical condition, then such atheorem will follow from its restriction to matrices.Applications are shown regarding norms of exponentials, the normsof commutators, and "positive" noncommutative $*$-polynomials.

【 授权许可】

Unknown