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dc.contributor.authorApazoglou, Maria
dc.description.abstractIn 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.en_US
dc.titleLinear maps on real C* - algebras and related structuresen_US
dc.rights.holderThe copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author

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  • Theses [3928]
    Theses Awarded by Queen Mary University of London

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