Linear maps on real C* - algebras and related structures

Abstract

In this thesis we obtain new results on the structures of real C*-algebras and nonsurjective isometries between them. Some of the results have been published in [1]. We prove a spectral inequality for real Banach*-algebras and give characterisations of real C*-algebras among Banach*-algebras. We study the ideal and facial structures in real C*-algebras and show that there is a bijection from the class of norm-closed left ideals I of a real C*-algebra A to the class of weak*-closed faces F of the state space S(A). The bijection is given by I 7! F = f 2 S(A) : (a a) = 0 for all a 2 Ig, with inverse F 7! I = fa 2 A : (a a) = 0 for all 2 Fg. As an application, we use the structures of faces to show an algebraic property of linear maps on real C*-algebras. We prove that if T : A ! B is a linear contraction between real C*-algebras A and B, then there is a projection p in the second dual B00 of B such that T(aa a)p = T(a)T(a) T(a)p (a 2 A). If T is an isometry, not necessarily surjective, we obtain a stronger result which also extends a celebrated result of Kadison on surjective isometries between complex C*-algebras. We prove the following theorem. Let T be a linear isometry between two real C*-algebras A and B, which can be non-surjective. Then for each a 2 A there exists a partial isometry u 2 B00 and a projection p 2 B00 such that (i) fu; T(ff; g; hg); ug = fu; fT(f); T(g); T(h)g; ug; (ii) T(ff; g; hg)p = fT(f); T(g); T(h)gp, for all f; g; h in the real JB*-triple A(a) generated by a 2 A, where ff; g; hg is the triple product defined by 2ff; g; hg = fg h + hg f. Moreover, fu; T( ):ug : A(a) ! B00 and T( )p : A(a) ! B00 are isometries. This theorem cannot be proved by simple complexification. We give an example of a real linear isometry which cannot be complexified to a complex isometry. We conclude by proving a theorem which states that a Jordan*-homomorphism T : A ! B between real C*-algebras A and B is a sum of a C*-homomorphism and a C*-antihomomorphism, extending a well-known result for complex C*- algebras.