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On some problems in the space \(C^{(n)}[0,1]\). (English) Zbl 1399.46073

Summary: We consider the so-called \(\alpha\)-Duhamel product, denoted by \(\underset\alpha \otimes\), on the space \(C^{(n)}[0,1]\) and prove that, with this product, this space has the structure of a unital Banach algebra, and then show that its maximal ideal space consists of the homomorphisms \(\varphi_{\alpha}\) defined by \(\varphi_{\alpha}(f) = f(\alpha)\). Moreover, we consider the usual convolution product \(\underset\alpha \ast\) and study the \(\underset\alpha \ast\)-generators of the Banach algebra \(\left (C^{(n)}[0,1], \underset\alpha \ast \right)\). Some other related questions are also discussed. Our results improve the work of M. T. Karaev [Sib. Mat. Zh. 46, No. 3, 553–566 (2005; Zbl 1224.46100); translation in Sib. Math. J. 46, No. 3, 431–442 (2005)], M. T. Karaev and H. Tuna [Complex Variables, Theory Appl. 49, No. 6, 449–457 (2004; Zbl 1069.47034)], M. Karaev et al. [Math. Nachr. 284, No. 13, 1678–1689 (2011; Zbl 1232.46049)], and R. Tapdigoglu [Houston J. Math. 39, No. 1, 169–176 (2013; Zbl 1285.47012)] where the case \(\alpha = 0\) was considered.

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46J20 Ideals, maximal ideals, boundaries
46J45 Radical Banach algebras
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