A commutative Banach algebra whose unit

A commutative Banach algebra whose unit A commutative Banach algebra whose unit ball has no extreme point (and so it isn't the dual space of any Banach space by the Krein-Milman theorem (cf. [Con, Theorem 7.4])).

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The unit ball of c₀ has no extreme point. For see this, let (x_n) belongs to the ball of c₀. lim_n x_n = 0, so there exists a number N such that for all n > N, |x_n| < [ 1/ 2]. Let y_n = z_n = x_n for n Ł N, and let y_n = x_n+2^-n and z_n = x_n-2^-n for n > N, then (y_n) and (z_n) belong to the unit ball of c₀ and (x_n) = [ 1/ 2](y_n)+[ 1/ 2](z_n). So (x_n) isn't is not an extreme point.

Ref.

[Con]J.B. Conway, A course in functional analysis, New York, Springer-Verlag, 1990.

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On 22 Feb 2001, 00:14.