Approximately multiplicative maps between algebras of bounded operators on Banach spaces

Choi, Yemon; Horváth, Bence; Laustsen, Niels Jakob

doi:10.1090/tran/8687

Approximately multiplicative maps between algebras of bounded operators on Banach spaces
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by Yemon Choi, Bence Horváth and Niels Jakob Laustsen PDF

Trans. Amer. Math. Soc. 375 (2022), 7121-7147 Request permission

Abstract:

We show that for any separable reflexive Banach space $X$ and a large class of Banach spaces $E$, including those with a subsymmetric shrinking basis but also all spaces $L_p[0,1]$ for $1\le p \le \infty$, every bounded linear map $\mathcal {B}(E)\to \mathcal {B}(X)$ which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism $\mathcal {B}(E)\to \mathcal {B}(X)$. That is, the pair $(\mathcal {B}(E),\mathcal {B}(X))$ has the AMNM property in the sense of Johnson [J. London Math. Soc. (2) 37 (1988), pp. 294–316]. Previously this was only known for $E=X=\ell _p$ with $1<p<\infty$; even for those cases, we improve on the previous methods and obtain better constants in various estimates. A crucial role in our approach is played by a new result, motivated by cohomological techniques, which establishes AMNM properties relative to an amenable subalgebra; this generalizes a theorem of Johnson (op cit.).

References

Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
Kevin Beanland, Tomasz Kania, and Niels Jakob Laustsen, Closed ideals of operators on the Tsirelson and Schreier spaces, J. Funct. Anal. 279 (2020), no. 8, 108668, 28. MR 4111752, DOI 10.1016/j.jfa.2020.108668
Earl Berkson and Horacio Porta, Representations of ${\mathfrak {B}}(X)$, J. Functional Analysis 3 (1969), 1–34. MR 0235419, DOI 10.1016/0022-1236(69)90048-2
A. Blanco and N. Grønbæk, Amenability of algebras of approximable operators, Israel J. Math. 171 (2009), 127–156. MR 2520105, DOI 10.1007/s11856-009-0044-7
M. Burger, N. Ozawa, and A. Thom, On Ulam stability, Israel J. Math. 193 (2013), no. 1, 109–129. MR 3038548, DOI 10.1007/s11856-012-0050-z
P. G. Casazza, W. B. Johnson, and L. Tzafriri, On Tsirelson’s space, Israel J. Math. 47 (1984), no. 2-3, 81–98. MR 738160, DOI 10.1007/BF02760508
Peter G. Casazza and Thaddeus J. Shura, Tsirel′son’s space, Lecture Notes in Mathematics, vol. 1363, Springer-Verlag, Berlin, 1989. With an appendix by J. Baker, O. Slotterbeck and R. Aron. MR 981801, DOI 10.1007/BFb0085267
Yemon Choi, Approximately multiplicative maps from weighted semilattice algebras, J. Aust. Math. Soc. 95 (2013), no. 1, 36–67. MR 3123743, DOI 10.1017/S1446788713000189
Matthew Daws, Dual Banach algebras: representations and injectivity, Studia Math. 178 (2007), no. 3, 231–275. MR 2289356, DOI 10.4064/sm178-3-3
Matthew Daws, Hung Le Pham, and Stuart White, Conditions implying the uniqueness of the weak${}^*$-topology on certain group algebras, Houston J. Math. 35 (2009), no. 1, 253–276. MR 2491881, DOI 10.1007/s00233-009-9186-5
T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no $l_{p}$, Compositio Math. 29 (1974), 179–190. MR 355537
N. Grønbæk, B. E. Johnson, and G. A. Willis, Amenability of Banach algebras of compact operators, Israel J. Math. 87 (1994), no. 1-3, 289–324. MR 1286832, DOI 10.1007/BF02773000
B. Horváth, Algebras of operators on Banach spaces, and homomorphisms thereof, Ph.D. thesis, Lancaster University, 2019.
Bence Horváth and Zsigmond Tarcsay, Perturbations of surjective homomorphisms between algebras of operators on Banach spaces, Proc. Amer. Math. Soc. 150 (2022), no. 2, 747–761. MR 4356184, DOI 10.1090/proc/15666
R. A. J. Howey, Approximately multiplicative maps between some Banach algebras, Ph.D. thesis, University of Newcastle upon Tyne, 2000.
B. E. Johnson, Approximately multiplicative maps between Banach algebras, J. London Math. Soc. (2) 37 (1988), no. 2, 294–316. MR 928525, DOI 10.1112/jlms/s2-37.2.294
W. B. Johnson, N. C. Phillips, and G. Schechtman, The SHAI property for the operators on $L^p$, J. Funct. Anal. 282 (2022), no. 4, Paper No. 109333, 12. MR 4349334, DOI 10.1016/j.jfa.2021.109333
D. Kazhdan, On $\varepsilon$-representations, Israel J. Math. 43 (1982), no. 4, 315–323. MR 693352, DOI 10.1007/BF02761236
Tomasz Kochanek, Approximately order zero maps between $\rm C^\ast$-algebras, J. Funct. Anal. 281 (2021), no. 2, Paper No. 109025, 49. MR 4242962, DOI 10.1016/j.jfa.2021.109025
Niels Jakob Laustsen, On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces, Glasg. Math. J. 45 (2003), no. 1, 11–19. MR 1972689, DOI 10.1017/S0017089502008947
J. Lindenstrauss and H. P. Rosenthal, The ${\cal L}_{p}$ spaces, Israel J. Math. 7 (1969), 325–349. MR 270119, DOI 10.1007/BF02788865
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
Paul McKenney and Alessandro Vignati, Ulam stability for some classes of $C$*-algebras, Proc. Roy. Soc. Edinburgh Sect. A 149 (2019), no. 1, 45–59. MR 3922807, DOI 10.1017/S0308210517000397
Haskell P. Rosenthal, On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273–303. MR 271721, DOI 10.1007/BF02771562
Volker Runde, Amenable Banach algebras, Springer Monographs in Mathematics, Springer-Verlag, New York, [2020] ©2020. A panorama. MR 4179584, DOI 10.1007/978-1-0716-0351-2
Raymond A. Ryan, Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2002. MR 1888309, DOI 10.1007/978-1-4471-3903-4
H. H. Schaefer and M. P. Wolff, Topological vector spaces, 2nd ed., Graduate Texts in Mathematics, vol. 3, Springer-Verlag, New York, 1999. MR 1741419, DOI 10.1007/978-1-4612-1468-7
Ivan Singer, Bases in Banach spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 154, Springer-Verlag, New York-Berlin, 1970. MR 0298399, DOI 10.1007/978-3-642-51633-7