Geodesic
Convexity around the Unit of a Banach Algebra
Serdica Mathematical Journal, Tome 34 (2008) no. 3, pp. 619-628.

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We estimate the (midpoint) modulus of convexity at the unit 1 of a Banach algebra A showing that inf {max±||1 ± x|| − 1 : x ∈ A, ||x||=ε} ≥ (π/4e)ε²+o(ε²) as ε → 0. We also give a characterization of two-dimensional subspaces of Banach algebras containing the identity in terms of polynomial inequalities.
Keywords: Unital Banach Algebra, Strongly Extreme Point, Midpoint Modulus of Local Convexity 
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     title = {Convexity around the {Unit} of a {Banach} {Algebra}},
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Kadets, Vladimir; Katkova, Olga; Martín, Miguel; Vishnyakova, Anna. Convexity around the Unit of a Banach Algebra. Serdica Mathematical Journal, Tome 34 (2008) no. 3, pp. 619-628. http://geodesic.mathdoc.fr/item/SMJ2_2008_34_3_a6/