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On the left and right joint spectra in Banach algebras. (English) Zbl 0737.46039

Let \(A\) be a complex unital Banach algebra with (Jacobson) radical \(J\). The left joint spectrum of \((a_ 1,\dots,a_ n)\in A^ n\) is the set \(\sigma_{\ell}:=\{\lambda_ 1,\dots,\lambda_ n)\in \mathbb{C}^ n: \sum_{j=1}^ n A(a_ j-\lambda_ j)\neq A\}\). Its left approximate spectrum is the set \(\tau_{\ell}(a_ 1,\dots,a_ n):=\{(\lambda_ 1,\dots,\lambda_ n)\in \mathbb{C}^ n: \inf_{\| a\| =1}\sum_{j=1}^ n\| (a_ j-\lambda_ j)a\| = 0\}\). Right joint and approximate spectra are defined similarly. Since \(\sigma_{\ell}(a_ 1,\dots,a_ n)\) coincides with the left approximate spectrum of \((a_ 1+J,\dots,a_ n+J)\) in \(A/J\) and similarly for \(\sigma_ r\), we have \(\sigma_{\ell}(a_ 1,\dots,a_ n)=\sigma_ r(a_ 1,\dots,a_ n)\) for all \(a_ j\in A\) if \(A/J\) is commutative.
The authors prove the converse: if \(\sigma_{\ell}(a_ 1,\dots,a_ n)\subset\sigma_ r(a_ 1,\dots,a_ n)\) for every finite set of \(a_ 1,\dots,a_ n\) in \(A\), then \(A/J\) is commutative. They pose the question whether there could be an integer \(N=N(A)\) such that the validity of the inclusion for all \(n\leq N\) would still imply \(A/J\) commutative. They answer this affirmatively, with \(N=2\), for all von Neumann algebras \(A\). They also examine connections between joint and approximate spectra, showing for example that in any unital Banach algebra if \(\tau_{\ell}(a)\subset\tau_ r(a)\) for every \(a\), then \(\sigma_{\ell}(a)=\sigma_ r(a)\) for every \(a\) and offer a nontrivial example to show that the converse of this is false. They ask whether \(\tau_{\ell}(a_ 1,\dots,a_ n)\subset\tau_ r(a_ 1,\dots,a_ n)\) for all \(a_ j\in A\) implies the commutativity of \(A/J\) and show that the converse [trivially true for joint spectra, as noted above] is false. The example showing this is modified to yield an \(A\) with \(A/J\) commutative and \(\tau_{\ell}(a_ 1,\dots,a_ n)\subset \tau_ r(a_ 1,\dots,a_ n)\) for all \(a_ j\in A\) yet for some pair \(\tau_{\ell}(a_ 1,a_ 2)\neq \tau_ r(a_ 1,a_ 2)\).

MSC:

46H05 General theory of topological algebras
46L10 General theory of von Neumann algebras
46J45 Radical Banach algebras
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