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Zbl 1224.46107
Zhang, Min; Jin, Shuilin
(Jin, Shui-lin)
Characterization of quantum independence by joint numerical range. (Chinese. English summary)
[J] J. Jilin Univ., Sci. 48, No. 2, 234-236 (2010). ISSN 1671-5489

Summary: A necessary and sufficient condition for $C^*$ algebras to be independent is given by use of joint numerical range, that is, sub-$C^*$ algebras $\cal{A}$ and $\cal{B}$ of $\cal{C}$ are quantum independent, if and only if for all $\boldsymbol{A}\in \cal{A}^+, \boldsymbol{B}\in \cal{B}^+$, $$W(\boldsymbol{A}, \boldsymbol{B})=W(\boldsymbol{A})\times W(\boldsymbol{B}),$$ where $W(\boldsymbol{A}, \boldsymbol{B})$ denotes the joint numerical ranges of the operator tuple $(\boldsymbol{A}, \boldsymbol{B})$.

MSC 2000:: *46L05 General theory of C*-algebras
47A12 Numerical range of linear operators
81P40

Keywords: joint spectrum; joint numerical range; quantum independence

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