A joint spectral characterization of \(C^*\) independence (Q2909262)

The paper is devoted to a spectral characterization of \(C^{*}\)-independence of two commuting unital \(C^{*}\)-subalgebras of \(B(H)\). Recall that a couple of \(C^{*}\)-subalgebras \(A\) and \(B\) in \(B(H)\) is said to be a statistical independent if any states \(\varphi_1\) and \(\varphi_2\) over \(A\) and \(B\), respectively, admit an extension up to a state \(\varphi\) on \(C^{*}(A,B)\) in the sense that \(\varphi|_{A}=\varphi_1\) and \(\varphi|_{B}=\varphi_2\). Roos's characterization of independence of commuting \(C^{*}\)-subalgebras \(A\) and \(B\) asserts that actually we deal with the property \(a\in A\backslash {0}\), \(b\in B\backslash {0} \Rightarrow ab\neq 0\), which in turn is equivalent to \(a\in A^+\backslash {0}\), \(b\in B^+\backslash {0} \Rightarrow ab>0\). The main result of the present paper asserts that independence of commuting \(C^{*}\)-subalgebras \(A\) and \(B\) is equivalent to the property \(\sigma(a,b)=\sigma(a)\times\sigma(b)\) for all \(a\in A^+, b\in B^+\), or just for all \(a\in A\), \(b\in B\), where \(\sigma (a,b)\) is joint Taylor spectrum of commuting operators. The first key point in the proof is to possibility of using Gelfand-Naimark's theorem for the couple \((a,b)\) of commuting positive operators in \(B(H)\) which allows to have continuous functions \(f(a)\in A\), \(g(b)\in B\) and \((f\otimes g)(a,b)\in B(H)\). Another moment is to apply Curto's characterization of the joint Taylor spectrum \(\sigma (a,b)\) in terms of \(a, a^{*}, b\) and \(b^{*}\). Namely, \((0,0)\not\in \sigma (a,b)\) iff \(aa^{*}+bb^{*}\), \(aa^{*}+b^{*}b\), \(a^{*}a+bb^{*}\) and \(a^{*}a+b^{*}b\) are invertible in \(B(H)\).