2021/09/21 by Marcel de Jeu, de Jeu, Marcel, Xingni Jiang +1 · 1 citation
Mathematics · #Advanced Banach Space Theory #Functional Equations Stability Results #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2109.10690
Let X be a locally compact Hausdorff space, let A be a partially ordered algebra, and let π\colon C\mathrm c(X)→ A be a positive algebra homomorphism. Under conditions on A that are satisfied in a good number of cases of practical interest, it is shown that π is represented by a unique regular spectral measure μ on the Borel σ-algebra of X, taking its values in the positive idempotents in A. The measure μ, which is σ-additive in an ordered sense, represents π via the order integral (a generalisation of the Lebesgue integral) that goes back to J.D.M. Wright and which was investigated earlier by the authors. The positive algebra homomorphism π can be extended from C\mathrm c(X) to a positive linear map from the accompanying L1-space of μ into A. It is shown that, quite often, this L1-space is closed under multiplication, so that it is a vector lattice algebra, and that the extended map from L1 into A is not only an algebra homomorphism but, even when A is not a vector lattice, also a vector lattice homomorphism in a sense that is explained in the paper. When A has the countable sup property, the image of L1 (or of its positive cone) is described in terms of consecutive ups and downs of the image of \mathrm C\mathrm c(X) (or of its positive cone). The general results are applied in three different contexts, showing how various spectral theorems have a common order-theoretical root: representations on Banach lattices, on Hilbert spaces, and (the algebra need not consist of operators) spectral theory for JBW-algebras.