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Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Chapter (peer-reviewed) › peer-review
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TY - CHAP
T1 - Lattice homomorphisms in harmonic analysis
AU - Dales, Harold Garth
AU - de Jeu, Marcel
N1 - The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-030-10850-2_6
PY - 2019/8/10
Y1 - 2019/8/10
N2 - Let S be a non-empty, closed subspace of a locally compact group G that is a subsemigroup of G. Suppose that X,Y , and Z are Banach lattices that are vector sublattices of the order dual Cc(S,R)∼ of the real-valued, continuous functions with compact support on S, and where Z is Dedekind complete. Suppose that ∗ : X ×Y → Z is a positive bilinear map such that supp(x ∗ y) ⊆ suppx · suppy for all x ∈ X+ and y ∈ Y + with compact support. We show that, under mild conditions, the canonically associated map from X into the vector lattice of regular operators from Y into Z is then a lattice homomorphism. Applications of this result are given in the context of convolutions, answering questions previously posed in the literature. As a preparation, we show that the order dual of the continuous, compactly supported functions on a closed subspace of a locally compact space can be canonically viewed as an order ideal of the order dual of the continuous, compactly supported functions on the larger space. As another preparation, we show that Lp-spaces and Banach lattices of measures on a locally compact space can be embedded as vector sublattices of the order dual of the continuous, compactly supported functions on that space
AB - Let S be a non-empty, closed subspace of a locally compact group G that is a subsemigroup of G. Suppose that X,Y , and Z are Banach lattices that are vector sublattices of the order dual Cc(S,R)∼ of the real-valued, continuous functions with compact support on S, and where Z is Dedekind complete. Suppose that ∗ : X ×Y → Z is a positive bilinear map such that supp(x ∗ y) ⊆ suppx · suppy for all x ∈ X+ and y ∈ Y + with compact support. We show that, under mild conditions, the canonically associated map from X into the vector lattice of regular operators from Y into Z is then a lattice homomorphism. Applications of this result are given in the context of convolutions, answering questions previously posed in the literature. As a preparation, we show that the order dual of the continuous, compactly supported functions on a closed subspace of a locally compact space can be canonically viewed as an order ideal of the order dual of the continuous, compactly supported functions on the larger space. As another preparation, we show that Lp-spaces and Banach lattices of measures on a locally compact space can be embedded as vector sublattices of the order dual of the continuous, compactly supported functions on that space
U2 - 10.1007/978-3-030-10850-2_6
DO - 10.1007/978-3-030-10850-2_6
M3 - Chapter (peer-reviewed)
SN - 9783030108496
SP - 79
EP - 129
BT - Positivity and Noncommutative Analysis
A2 - Buskes, Gerard
A2 - de Jeu, Marcel
A2 - Dodds, Peter
A2 - Schep, Anton
A2 - Sukochev, Fedor
A2 - van Neerven, Jan
A2 - Wickstead, Anthony
PB - Springer Birkhäuser
ER -