On the embeddability of the family of countably branching trees into quasi-reflexive Banach spaces

In this note we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk index from the dual are both equal to the first infinite ordinal omega. In particular we show that the family of countably branching trees does neither embed into the James space J(p) nor into its dual space J(p)* for p is an element of (1, infinity). (C) 2020 Elsevier Inc. All rights reserved.