Conditions equivalent to $C^{*}$ independence

Shuilin Jin¹ ; Li Xu² ; Qinghua Jiang³ ; Li Li⁴

¹ Department of Mathematics Harbin Institute of Technology Harbin, China 150001
² College of Computer Science and Technology Harbin Engineering University Harbin, China 150001
³ Academy of Fundamental and Interdisciplinary Sciences Harbin Institute of Technology Harbin, China 150001
⁴ Natural Science Research Center Harbin Institute of Technology Harbin, China 150001

Studia Mathematica, Tome 211 (2012) no. 3, pp. 191-197

Résumé

Let $\mathcal{A}$ and $\mathcal{B}$ be mutually commuting unital $C^{*}$ subalgebras of $\mathcal{B}(\mathcal{H})$. It is shown that $\mathcal{A}$ and $\mathcal{B}$ are $C^{*}$ independent if and only if for all natural numbers $n, m$, for all $n$-tuples $A=(A_1, \ldots, A_n)$ of doubly commuting nonzero operators of $\mathcal{A}$ and $m$-tuples $B=(B_1, \ldots, B_m)$ of doubly commuting nonzero operators of $\mathcal{B}$, $$ \mathrm{Sp}(A, B)= \mathrm{Sp}(A) \times \mathrm{Sp}(B), $$ where $\mathrm{Sp}$ denotes the joint Taylor spectrum.

Détail

DOI : 10.4064/sm211-3-1

Keywords: mathcal mathcal mutually commuting unital * subalgebras mathcal mathcal shown mathcal mathcal * independent only natural numbers n tuples ldots doubly commuting nonzero operators mathcal m tuples ldots doubly commuting nonzero operators mathcal mathrm mathrm times mathrm where mathrm denotes joint taylor spectrum