Positive linear maps on C^⁎-algebras and rigid functions

详细信息    查看全文

• 作者：Mitsuru Uchiyama¹ ; ^{uchiyama@riko.shimane-u.ac.jp}
• 关键词：Positive linear map ; C⁎C⁎-homomorphism ; Davis and Choi's inequality ; Operator convex function ; Rigid function ; Distribution
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 May 2017
• 年：2017
• 卷：449
• 期：2
• 页码：1472-1478
• 全文大小：282 K
• 卷排序：449

文摘

Let a linear map Φ between two unital C⁎C⁎-algebras be positive and unital. Kadison showed that if f(t)=|t|f(t)=|t| and Φ(f(X))=f(Φ(X))Φ(f(X))=f(Φ(X)) for all selfadjoint operators X  , then Φ(X2)=Φ(X)2Φ(X2)=Φ(X)2 for all selfadjoint operators X  , that is, Φ is a C⁎C⁎-homomorphism. Choi proved this fact for an operator convex function f, and then conjectured that this fact would hold for a non-affine continuous function f  . We shall prove a refinement of his conjecture. Petz has further proved that if f(Φ(A))=Φ(f(A))f(Φ(A))=Φ(f(A)) for a non-affine operator convex function f and a fixed A  , then Φ(A2)=Φ(A)2Φ(A2)=Φ(A)2. Arveson called such a function f a rigid function  . We shall directly show power functions trtr are rigid functions on (0,∞)(0,∞) if r≠0,r≠1.