The positive cone in Banach algebras

TLDR: In this paper, the authors show that the real radical of a real commutative algebra is the extreme point of the intersection of the dual cone and the unit sphere in the adjoint of the algebra, and the "real radical" consists of elements x such that x 2 is approximately a sum of squares.

Abstract: This paper concerns Banach algebras which are real or * algebras and possess a unit. The principal method of at.tack is via an ordering of the algebra, the positive cone being the closure of the set of sums of squares (sums of elements xx*) in contrast to the positive open cone used by Raikov [9] (3) and others. An important role is played by an identity on norms, which together with a few preliminary lemmas is proved in ?1. In ?2 the real homomorphisms of a real commutative algebra are found to be the extreme points of the intersection of the dual cone and the unit sphere in the adjoint of the algebra, and the "real radical" is shown to consist of elements x such that -x2 is approximately a sum of squares. The theorem of Arens [1] characterizing real function algebras is derived. In ??3 and 4 these results are applied to * algebras. The new norm of an element x, which Gelfand and Naimark [3] introduced by means of positive functionals, is proved to be the square root of the distance from -xx* to the positive cone. Some results relating general * algebras to operator algebras, including the representation theorem of Gelfand and Naimark [2], are derived. In ?5, a refinement of the basic identity is established for the Fourier transform of a measure (discrete + absolutely continuous) on a locally compact Abelian group. R. V. Kadison [5 ] has recently investigated Banach algebras by means of an order relation. The positive cone he uses is identical with that used here only when 1 +xx* always has an inverse. The principal overlap with Kadison's work, outside of the deduction of certain known theorems by order methods, seems to be the geometric characterization of the real homomorphisms of a real algebra (see 2.1). Like Kadison's work, this paper is essentially self-contained. (Some notable exceptions occur in ?5.) 1. Preliminaries. 1.1. DEFINITIONS. A set C is a cone in a real Banach space R if it is closed, nonvoid, the sum of two members of C is a member of C, and non-negative scalar multiples of members of C are members of C. If C is a cone in R, then C', the dual cone, is the set of bounded linear functionals which are non