## Approximately multiplicative maps between algebras of bounded operators on Banach spaces

HTML articles powered by AMS MathViewer

- 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

## Additional Information

**Yemon Choi**- Affiliation: Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster, LA1 4YF, United Kingdom
- MR Author ID: 795462
- ORCID: 0000-0001-7447-248X
- Email: y.choi1@lancaster.ac.uk
**Bence Horváth**- Affiliation: Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic
- MR Author ID: 1377121
- Email: horvath@math.cas.cz, hotvath@gmail.com
**Niels Jakob Laustsen**- Affiliation: Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster, LA1 4YF, United Kingdom
- MR Author ID: 640805
- ORCID: 0000-0003-1658-2415
- Email: n.laustsen@lancaster.ac.uk
- Received by editor(s): October 7, 2021
- Received by editor(s) in revised form: February 3, 2022, and February 4, 2022
- Published electronically: July 29, 2022
- Additional Notes: The second author acknowledges support from the Czech Science Foundation (GAČR project 19-07129Y; RVO 67985840).
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 7121-7147 - MSC (2020): Primary 39B82, 47L10; Secondary 46B03, 46M18, 47B49
- DOI: https://doi.org/10.1090/tran/8687