Completely coarse maps are ℝ-linear

Braga, Bruno; Chávez-Domínguez, Javier

doi:10.1090/proc/15289

Completely coarse maps are ${\mathbb {R}}$-linear
HTML articles powered by AMS MathViewer

by Bruno M. Braga and Javier Alejandro Chávez-Domínguez

Proc. Amer. Math. Soc. 149 (2021), 1139-1149

DOI: https://doi.org/10.1090/proc/15289

Published electronically: January 21, 2021

PDF | Request permission

Abstract:

A map between operator spaces is called completely coarse if the sequence of its amplifications is equi-coarse. We prove that all completely coarse maps must be ${\mathbb {R}}$-linear. On the opposite direction of this result, we introduce a notion of embeddability between operator spaces and show that this notion is strictly weaker than complete ${\mathbb {R}}$-isomorphic embeddability (in particular, weaker than complete ${\mathbb {C}}$-isomorphic embeddability). Although weaker, this notion is strong enough for some applications. For instance, we show that if an infinite dimensional operator space $X$ embeds in this weaker sense into Pisier’s operator space $\mathrm {OH}$, then $X$ must be completely isomorphic to $\mathrm {OH}$.

References

David P. Blecher and Christian Le Merdy, Operator algebras and their modules—an operator space approach, London Mathematical Society Monographs. New Series, vol. 30, The Clarendon Press, Oxford University Press, Oxford, 2004. Oxford Science Publications. MR 2111973, DOI 10.1093/acprof:oso/9780198526599.001.0001
J. Bourgain, Real isomorphic complex Banach spaces need not be complex isomorphic, Proc. Amer. Math. Soc. 96 (1986), no. 2, 221–226. MR 818448, DOI 10.1090/S0002-9939-1986-0818448-2
Bruno M. Braga and Thomas Sinclair, A Ribe theorem for noncommutative ${L}_1$-spaces and Lipschitz free operator spaces, arXiv:1911.05505 [math.OA] (2019).
Arthur Engel, Problem-solving strategies, Problem Books in Mathematics, Springer-Verlag, New York, 1998. MR 1485512
Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
Valentin Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (2007), no. 1, 462–488. MR 2331251, DOI 10.1016/j.aim.2007.01.003
G. Godefroy, G. Lancien, and V. Zizler, The non-linear geometry of Banach spaces after Nigel Kalton, Rocky Mountain J. Math. 44 (2014), no. 5, 1529–1583. MR 3295641, DOI 10.1216/RMJ-2014-44-5-1529
Nigel J. Kalton, The nonlinear geometry of Banach spaces, Rev. Mat. Complut. 21 (2008), no. 1, 7–60. MR 2408035, DOI 10.5209/rev_{R}EMA.2008.v21.n1.16426
Timur Oikhberg and Éric Ricard, Operator spaces with few completely bounded maps, Math. Ann. 328 (2004), no. 1-2, 229–259. MR 2030376, DOI 10.1007/s00208-003-0481-2
Gilles Pisier, The operator Hilbert space $\textrm {OH}$, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103. MR 1342022, DOI 10.1090/memo/0585
Gilles Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. MR 2006539, DOI 10.1017/CBO9781107360235
Zhong-Jin Ruan, Complexifications of real operator spaces, Illinois J. Math. 47 (2003), no. 4, 1047–1062. MR 2036989
Zhong Jin Ruan, On real operator spaces, Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 3, 485–496. International Workshop on Operator Algebra and Operator Theory (Linfen, 2001). MR 2014029, DOI 10.1007/s10114-003-0278-y
Walter Rudin, Positive definite sequences and absolutely monotonic functions, Duke Math. J. 26 (1959), 617–622. MR 109204
I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J. 9 (1942), 96–108. MR 5922, DOI 10.1215/S0012-7094-42-00908-6
Sonia Sharma, Real operator algebras and real completely isometric theory, Positivity 18 (2014), no. 1, 95–118. MR 3167068, DOI 10.1007/s11117-013-0233-9