Banach Lattice Structures and Concavifications in Banach Spaces

[EN] Let (Omega,sigma,mu) be a finite measure space and consider a Banach function space Y(mu). We say that a Banach space E is representable by Y(mu) if there is a continuous bijection I:Y(mu)-> E. In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.The authors would like to thank the referees for their valuable comments, which helped to improve the manuscript. The work of the second author was supported by the Ministerio de Ciencia e Innovacion, Agencia Estatal del Investigacion (Spain) and FEDER under project #PGC2018-095366-B-100. The work of the fourth author was supported by the Ministerio de Ciencia e Innovacion, Agencia Estatal del Investigacion (Spain) and FEDER under project #MTM2016 77054-C2-1-P. We did not receive any funds for covering the costs of publishing in open access.Agud Albesa, L.; Calabuig, JM.; Juan, MA.; Sánchez Pérez, EA. (2020). Banach Lattice Structures and Concavifications in Banach Spaces. Mathematics. 8(1):1-20. https://doi.org/10.3390/math801012712081Bu, Q., Buskes, G., Popov, A. I., Tcaciuc, A., & Troitsky, V. G. (2012). The 2-concavification of a Banach lattice equals the diagonal of the Fremlin tensor square. Positivity, 17(2), 283-298. doi:10.1007/s11117-012-0166-8Buskes, G., & van Rooij, A. (2001). Squares of Riesz Spaces. 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