Conditions equivalent to $C^{*}$ independence
Volume 211 / 2012
Studia Mathematica 211 (2012), 191-197
MSC: Primary 47A13; Secondary 46L06, 81Q10.
DOI: 10.4064/sm211-3-1
Abstract
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.
