Order and Commutativity in Banach Algebras
نویسنده
چکیده
S. Sherman has shown [4] that if the self adjoint elements of a C* algebra form a lattice under their natural ordering the algebra is necessarily commutative. In this note we extend this result to real Banach algebras with an identity and arbitrary Banach * algebras with an identity. The central fact for a real Banach algebra A is that if the positive cone is defined to be the uniform closure of the set of finite sums of squares of elements of A, and if A is a lattice under the ordering induced by this cone, then extreme points of the unit sphere of the dual cone are multiplicative linear functionals. A similar situation holds for * algebras.
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