Factorizing multilinear operators on Banach spaces, C<sup>*</sup>-algebras and JB<sup>*</sup>-triples

Palazuelos, Carlos; Peralta, Antonio M.; Villanueva, Ignacio

doi:10.4064/sm192-2-3

Cited by 1 publication

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“…Jointly w-right-norm continuous multilinear operators have been studied in [22,30] and [37]. A multilinear operator T : The polarization formula (4.1) guarantees that an m-homogeneous polynomial P is w-right-norm continuous if and only if its generating multilinear operator is jointly w-right-norm continuous (at 0) if and only if P is w-right-norm continuous at 0.…”

Section: W-right-norm Continuous Holomorphic Mappingsmentioning

confidence: 99%

Sequential w‐right continuity and summing operators

Diestel

Peralta

Puglisi

2011

Mathematische Nachrichten

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We continue the study of the w-right and strong * topologies in general Banach spaces started in [36,37] and [35]. We show that in L1 (μ)-spaces the w-right convergence of sequences admits a simpler control. Some considerations about these topologies will be contemplated in the particular cases of C*-algebras and JB*-triples in connection with summing operators. We also study (sequential) w-right-norm and strong*-norm continuity for holomorphic mappings. and the second author of the present note introduced in [36] two interesting topologies: the strong * and the w-right topology in the following way: let X and Y be two Banach spaces, for every bounded linear operator T : X −→ Y , we can consider a seminorm on X defined by x T := T (x) . The strong * -topology is the topology generated by the family of seminorms · T , where T : X −→ H is a bounded linear operator from X to some Hilbert space H (such a topology is denoted by S * (X, X * )). Similarly, the w-right-topology is the topology generated by the family of seminorms · T where T runs in the set of all bounded linear operators from X to a reflexive space [36].In Section 2, we establish new methods for controlling w-right convergent sequences in L 1 (μ) spaces. Section 3 is devoted to a more detailed study of strong*-norm continuous operators between Banach spaces. In the particular cases of operators whose domain is a C*-algebra or a JB*-triple, we explore the connections with p-C*-summing and p-JB*-triple-summing operators. We prove an extension property for 2-C*-summing and 2-JB*-triple-summing operators (see Theorems 3.6 and 3.9). In this section we shall also introduce and develop p-JB*-triple-summing operators on JB*-triples as suitable generalizations of p-C*-summing operators on C*-algebras in the sense introduced by Pisier in [39].The last section of the paper is devoted to the study of those holomorphic mappings of bounded type which are sequentially w-right-norm continuous. The main result in [35] establishes that a bounded linear operator T : X −→ Y is weakly compact if and only if T is w-right-norm continuous. We shall provide examples showing that none of these implications holds for continuous polynomials in general Banach spaces. In the linear case, T is weakly compact if and only if T * * is Y -valued. In the setting of multilinear operators, this equivalence has been recently studied in [37]. One of the main results in the just quoted paper proves that when X 1 , . . . , X k are non zero sequentially right Banach spaces and T : X 1 × · · · × X k −→ X is a multilinear operator, then T is RQCC (i.e., T is jointly sequentially w-right-norm continuous) if and only if all of the Aron-Berner extensions of T are X-valued if and only if T has an X-valued Aron-Berner extension. We shall consider here holomorphic mappings of bounded type f between two Banach spaces X and Y with X being a sequentially right Banach

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Section: W-right-norm Continuous Holomorphic Mappingsmentioning

confidence: 99%