Test-space characterizations of some classes of Banach spaces

Let P be a class of Banach spaces and let T = {Tα}α∈A be a set of metric spaces. We say that T is a set of test-spaces for P if the following two conditions are equivalent: (1) X / ∈ P; (2) The spaces {Tα}α∈A admit uniformly bilipschitz embeddings into X. The first part of the paper is devoted to a simplification of the proof of the following test-space characterization obtained in M. I. Ostrovskii [Different forms of metric characterizations of classes of Banach spaces, Houston J. Math., to appear]: For each sequence {Xm}m=1 of finite-dimensional Banach spaces there is a sequence {Hn}n=1 of finite connected unweighted graphs with maximum degree 3 such that the following conditions on a Banach space Y are equivalent: (A) Y admits uniformly isomorphic embeddings of {Xm}m=1; (B) Y admits uniformly bilipschitz embeddings of {Hn}n=1. The second part of the paper is devoted to the case when {Xm}m=1 is an increasing sequence of spaces. It is shown that in this case the class of spaces given by (A) can be characterized using one test-space, which can be chosen to be an infinite graph with maximum degree 3. 2010 Mathematics Subject Classification: Primary: 46B07; Secondary: 05C12, 46B85, 54E35