A Perron–Frobenius type result in Banach algebras via asymptotic closeness to a cone

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For an element a of a Banach algebra (scaled to spectral radius 1) we prove that the spectral radius is contained in the spectrum, if the sequence of powers (ak)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(a^k)$$\\end{document} is asymptotically not too far from a normal cone.