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Lipschitz-free Spaces on Finite Metric Spaces

Published online by Cambridge University Press:  13 February 2019

Stephen J. Dilworth
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA Email:
Denka Kutzarova
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, USA Institute of Mathematics and Informatics, Bulgarian Academy of Sciences Email:
Mikhail I. Ostrovskii
Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, NY 11439, USA Email:


Main results of the paper are as follows:

(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.

(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.

Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

© Canadian Mathematical Society 2019

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Author S. D. was supported by the National Science Foundation under Grant Number DMS–1361461. Authors S. D. and D. K. were supported by the Workshop in Analysis and Probability at Texas A&M University in 2017. Author M. O. was supported by the National Science Foundation under Grant Number DMS–1700176.


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