Orthogonally additive polynomials on non-commutative $$L^p$$ L p -spaces
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Orthogonally additive polynomials on non-commutative Lp -spaces Jerónimo Alaminos1 · María L. C. Godoy1 · Armando R. Villena1 Received: 15 March 2019 / Accepted: 12 October 2019 © Universidad Complutense de Madrid 2019
Abstract Let M be a von Neumann algebra with a normal semifinite faithful trace τ . We prove that every continuous m-homogeneous polynomial P from L p (M , τ ), with 0 < p < ∞, into each topological linear space X with the property that P(x +y) = P(x)+ P(y) whenever x and y are mutually orthogonal positive elements of L p (M , τ ) can be represented in the form P(x) = Φ(x m ) (x ∈ L p (M , τ )) for some continuous linear map Φ : L p/m (M , τ ) → X . Keywords Non-commutative L p -space · Schatten classes · Orthogonally additive polynomial Mathematics Subject Classification 46L10 · 46L52 · 47H60
1 Introduction In [16], the author succeeded in providing a useful representation of the orthogonally additive homogeneous polynomials on the spaces L p ([0, 1]) and p with 1 ≤ p < ∞. In [12] (see also [6]), the authors obtained a similar representation for the space C(K ), for a compact Hausdorff space K . These results were generalized to Banach
The authors were supported by MINECO Grant PGC2018–093794–B–I00. and Junta de Andalucía Grant FQM–185. The second named author was supported by Contrato Predoctoral FPU, Plan propio de Investigación y Transferencia 2018, University of Granada.
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Armando R. Villena [email protected] Jerónimo Alaminos [email protected] María L. C. Godoy [email protected]
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Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
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lattices [4] and Riesz spaces [9]. Further, the problem of representing the orthogonally additive homogeneous polynomials has been also considered in the context of Banach function algebras [1,19] and non-commutative Banach algebras [2,3,11]. Notably, [11] can be thought of as the natural non-commutative analogue of the representation of orthogonally additive polynomials on C(K )-spaces, and the purpose to this paper is to extend the results of [16] on the representation of orthogonally additive homogeneous polynomials on L p -spaces to the non-commutative L p -spaces. The non-commutative L p -spaces that we consider are those associated with a von Neumann algebra M equipped with a normal semifinite faithful trace τ . From now on, S(M ,τ ) stands for the linear span of the positive elements x of M such that τ supp(x) < ∞; here supp(x) stands for the support of x. Then S(M , τ ) is a ∗subalgebra of M with the property that |x| p ∈ S(M , τ ) for each x ∈ S(M , τ ) and each 0 < p < ∞. For 0 < p < ∞, we define · p : S(M , τ ) → R by x p = τ (|x| p )1/ p (x ∈ S(M , τ )). Then · p is a norm or a p-norm according to 1 ≤ p < ∞ or 0 < p < 1, and the space L p (M , τ ) can be defined as the completion of S(M , τ ) with respect to · p . Nevertheless, for our purposes here, it is important to realize the elements of L p (M , τ ) as measurable operators. Specifically, the set L 0 (M ,
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