Unconditional and bimonotone structures in high density Banach spaces

2016/04/15 by Talponen, Jarno
#03E02 #03E55 #03E75 #05C55 #46A50 #46B15 #46B26 (Primary) #54A25 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO)

paper · doi:10.48550/arxiv.1604.04408

Abstract

It is shown that every normalized weakly null sequence of length κ_λ in a Banach space has a subsequence of length λ which is an unconditional basic sequence; here κ_λ is a large cardinal depending on a given infinite cardinal λ. Transfinite topological games on Banach spaces are analyzed which determine the existence of a long unconditional basic sequence. Then 'asymptotic disentanglement' condition in a transfinite setting is studied which ensures a winning strategy for the unconditional basic sequence builder in the above game. The following problem is investigated: When does a Markushevich basic sequence with length uncountable regular cardinal κ admit a subsequence of the same length which is a bimonotone basic sequence? Stabilizations of projectional resolutions of the identity (PRI) are performed under a density contravariance principle to gain some additional strong regularity properties, such as bimonotonicity.

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