Abstract
Let A be a Banach algebra with a bounded left approximate identity {e_lambda }_{lambda in Lambda }, let pi be a continuous representation of A on a Banach space X, and let S be a non-empty subset of X such that lim _{lambda }pi (e_lambda )s=s uniformly on S. If S is bounded, or if {e_lambda }_{lambda in Lambda } is commutative, then we show that there exist ain A and maps x_n: Srightarrow X for nge 1 such that s=pi (a^n)x_n(s) for all nge 1 and sin S. The properties of ain A and the maps x_n, as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair’s power factorization theorem. 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. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of mathrm {C}_0({mathbb R}) and unbounded S is included.
Highlights
Introduction and overviewLet A be a real or complex Banach algebra with a bounded left approximate identity {eλ}λ∈, and let π be a continuous representation of A on a Banach space X
Restricting ourselves to positive pointwise non-power factorization: if A is an ordered Banach algebra with a positive bounded left approximate identity {eλ}λ∈, if π is a positive representation of A on an ordered Banach space X, and if s is an element of the positive cone X + of X such that limλ π(eλ)s = s, do there exist a ∈ A+ and x1(s) ∈ A+ such that s = π(a)x1(s)? As we shall see, such a ∈ A+ exists whenever the positive cone A+ of A is closed, but it may be impossible to arrange that x1(s) ∈ X +
The results obtained assert the existence of a positive simultaneous pointwise power factorization for sets of bounded maps with values in a subset S of the positive cone of an ordered Banach space, where S is such that limλ eλs − s = 0 uniformly on S for some positive bounded left approximate identity {eλ}λ∈ of A, and where S is bounded or {eλ}λ∈ is commutative
Summary
Let A be a real or complex Banach algebra with a bounded left approximate identity {eλ}λ∈ , and let π be a continuous representation of A on a Banach space X. One wants to establish the existence of a ∈ A and a map x1 : S → X such that s = π(a)x1(s) for all s ∈ S, together with some additional properties of a and x1 This is possible, for example, if S is bounded and such that limλ π(eλ)s = s uniformly on S; see e.g. Restricting ourselves to positive pointwise non-power factorization: if A is an ordered Banach algebra with a positive bounded left approximate identity {eλ}λ∈ , if π is a positive representation of A on an ordered Banach space X , and if s is an element of the positive cone X + of X such that limλ π(eλ)s = s, do there exist a ∈ A+ and x1(s) ∈ A+ such that s = π(a)x1(s)? A positive simultaneous power factorization result for ordered Banach algebras of bounded functions, Theorem 5.7, can be established precisely because of this freedom.
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