Descriptive complexity of some isomorphism classes of Banach spaces
From the day that Cantor emerged with a proof of the existence of transcendental numbers without even mentioning one, but merely thinking about the complexity of the process, the door is open to make deep mathematics describing the complexity of a process or a class of objects. That is the topic of this paper placing as central objects different classes of Banach spaces. There are two ways to tackle the problem: How complex is a given family of Banach spaces? How complex is a relation between Banach spaces? In both cases, the first step to be given is to present a frame for the problem. For instance, to consider a class of separable Banach spaces, one can choose as frame the subspaces of \(C[0,1]\). Such is the first difficulty overcome in this paper: set admissible Polish topologies on the set of closed subspaces of \(C[0,1]\) to study then the complexity of classes of Banach spaces, mainly \(l_2\) and asymptotically Hilbert spaces. The results are technically difficult to describe and the notation is terrifying, which is why we opted for a descriptive (non-complex) review. There are two recent papers somewhat connected with the open problems listed at the end of the paper: One is [\textit{R. Anisca} et al., J. Funct. Anal. 272, No. 9, 3845--3868 (2017; Zbl 1370.46008)] in which the authors study the complexity of the relation of equivalence between positions of the subspaces of a Banach space, particularly in \(L_ p\) spaces; and the other is [\textit{J. Suárez de la Fuente}, ``A weak Hilbert space that is a twisted Hilbert space'', J. Inst. Math. Jussieu (to appear; \url{doi:10.1017/S1474748018000221})] which complements the works of \textit{R. Anisca} [Proc. Am. Math. Soc. 138, No.~4, 1405--1413 (2010; Zbl 1196.46012)] and \textit{W. Cuellar} [J. Math. Anal. Appl. 440, No.~2, 624--635 (2016; Zbl 1351.46009)] showing an ergodic twisted Hilbert space (that is not Hilbert).
scientific article; zbMATH DE number 6879851
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