Operator algebras with contractive approximate identities: a large operator algebra in \(c_0\) (Q2790590)

The paper under review is an important contribution to the theory of non-self-adjoint operator algebras (that is, norm-closed subalgebras of the algebra of all bounded, linear operator on a Hilbert space). The authors construct a singly generated, semisimple, commutative operator algebra \(A\) with the following properties: {\parindent=0.8cm \begin{itemize}\item[(1)]The spectrum of the generator \(g\) is a null sequence and 0, but \(A\) properly contains the closure of the socle. \item[(2)]The multiplication operator \(T_ga=ga\) (\(a\in A\)) is not weakly compact (which is the same as saying that \(A\) is not an ideal of the bidual \(A^{**}\) endowed with the Arens product; note that operator algebras are automatically Arens-regular). \item[(3)]The algebra \(A\) has a contractive approximate identity. NEWLINENEWLINE\end{itemize}} Moreover, one can choose \(A\) to contain properly the closure of the linear span of minimal spectral idempotents associated to the elements of the spectrum of \(g\) (spectral projections are projections obtained by applying the holomorphic functional calculus to the isolated points of the spectrum) or make this linear span dense in the strong operator topology.NEWLINENEWLINEThe algebra \(A\) is the completion of a renorming of the subalgebra of \(c_0\) generated by the canonical unit vectors \(e_k\) (\(k\in \mathbb {N}\)) and the sequence \(g = \sum_{i=1}^\infty \tfrac{1}{2^i}\cdot e_i\) with the following properties: {\parindent=0.8cm \begin{itemize} \item[(1)]\(A\) has a contractive approximate identity and is topologically generated by \(g\); \item[(2)]\(\sigma(g)=\{\tfrac{1}{n}: n\in\mathbb N\}\); \item[(3)]\(g\) is not in the closed linear span of \(\{e_k: k\in \mathbb{N}\}\); \item[(4)]\(A\) is semi-simple. NEWLINENEWLINE\end{itemize}} (The word `renorming' is crucial here -- \(A\) is a subalgebra of \(c_0\) but, of course, the norm in \(A\) is not the supremum norm.) It is noteworthy that \(A\) is also a natural Banach sequence algebra, that is, it is a sequence algebra which (clearly) contains all finitely supported functions and whose characters are coordinate-evaluations.NEWLINENEWLINEThe algebra \(A\) appears to be the first example of a natural Banach sequence algebra with a bounded approximate identity (bai) whose socle is not dense (that is, \(A\) is a natural Banach algebra with bai which is not Tauberian) or, even more generally, the first example of a natural Banach algebra with a non-weakly compact multiplication operator which has a bai at the same time. (Feinstein constructed a natural Banach sequence algebra whose socle is not dense, yet his example does not have a bai; see Section 4.1 of [\textit{H. G. Dales}, Banach algebras and automatic continuity. Oxford: Clarendon Press (2000; Zbl 0981.46043)]).NEWLINENEWLINEThe authors also prove that in semi-simple, (topologically) singly generated operator algebras whose socle has a bai, the socle is dense if and only if all multiplication operators are compact.NEWLINENEWLINEIt is also noted (Remark 1.7) that the last assertion of Theorem 5.10 (4) in [\textit{M. Almus} et al., Stud. Math. 212, No. 1, 65--93 (2012; Zbl 1271.46038)] is false.NEWLINENEWLINEThe paper is well written, however, the statement of the main theorem is probably too densely packed. From the mathematical side, it is to be noted that the growth estimates which lie at the heart of the construction are particularly impressive.