Isometries between matrix algebras

Journal of the Australian Mathematical Society

Wai-Shun Cheung¹ 

[1] Chi-Kwong Li

关键词: primary 15A04;    15A60;   

DOI  :  10.1017/S1446788700010119

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

As an attempt to understand linear isometries between C*-algebras without the surjectivity assumption, we study linear isometries between matrix algebras. Denote by Mm the algebra of m × m complex matrices. If k ≥ n and φ: Mn → Mk has the form X ↦ U[X ⊕ f(X)] V or X ↦ U[X1 ⊕ f(X)]V for some unitary U, V ∈ Mk and contractive linear map f: Mn → Mk, then ║φ(X)║ = ║X║ for all X ∈ Mn. We prove that the converse is true if k ≤ 2n - 1, and the converse may fail if k ≥ 2n. Related results and questions involving positive linear maps and the numerical range are discussed.

【 授权许可】

Unknown