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2000 Mathematics Subject Classification: Primary: 46B20. Secondary: 46H99, 47A12.
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.
Let be the cone of real univariate polynomials of degree ≤ 2n which are nonnegative on the real axis and have nonnegative coefficients. We describe the extremal rays of this convex cone and the class of linear operators, acting diagonally in the standard monomial basis, preserving this cone.
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