## Linear maps on real C* - algebras and related structures

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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.

##### Authors

Apazoglou, Maria##### Collections

- Theses [2220]