Universiteit Leiden

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Dissertation

Positive representations of algebras of continuous functions

If X is a locally compact Hausdorff space, then a representation of the complex C* algebra C_0(X) on a Hilbert space $H$ is given by a spectral measure that takes its values in the orthogonal projections on $H$.

Author
X. Jiang
Date
25 September 2018
Links
Thesis in Leiden Repository

If X is a locally compact Hausdorff space, then a representation of the complex C* algebra C_0(X) on a Hilbert space $H$ is given by a spectral measure that takes its values in the orthogonal projections on $H$. It is natural to ask whether something similar is true for a positive representation of the ordered Banach algebra C_0(X) on a Banach lattice E. If E is a KB-space or if E is reflexive, then the answer is affirmative: the representation is given by a spectral measure that takes its values in the positive projections on X. The proofs of above results make use of the fact that E is a Banach space, but there should be a purely order-theoretic approach. In chapter 1 of this thesis, we shall explain that this is indeed the case. In chpater 2, we are talking about simultaneous power factorization in Banach algebras. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity.

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