Abstract
We now focus on (associative) Banach algebras. Our approach to zpd Banach algebras is based on the related notion of a Banach algebra with property \(\mathbb B\). In Banach algebras having bounded (left) approximate identities, property \(\mathbb B\) is just equivalent to the zpd property. However, in many ways it is technically more suitable. Using it, we will be able to show, in particular, that all C ∗-algebras and all group algebras L 1(G), where G is a locally compact group, are zpd Banach algebras. Also, we will rely on property \(\mathbb B\) in the study of the stability of the zpd property under various constructions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J. Alaminos, M. Brešar, J. Extremera, A.R. Villena, Maps preserving zero products. Studia Math. 193, 131–159 (2009)
G. Brown, W. Moran, Point derivations on M(G). Bull. London Math. Soc. 8, 57–64 (1976)
H.G. Dales, Banach Algebras and Automatic Continuity. London Mathematical Society Monographs, New Series, vol. 24 (Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2000)
E. Erdogan, Ö. Gök, Convolution factorability of bilinear maps and integral representations. Indag. Math. 29, 1334–1349 (2018)
E. Erdogan, Ö. Gök, E.A. Sánchez Pérez, Product factorability of integral bilinear operators on Banach function spaces. Positivity 23, 671–696 (2019)
E. Erdogan, E.A. Sánchez Pérez, Integral representation of product factorable bilinear operators and summability of bilinear maps on C(K)-spaces. J. Math. Anal. Appl. 483, 123629, 25 (2020)
E. Kaniuth, A Course in Commutative Banach Algebras. Graduate Texts in Mathematics, vol. 246 (Springer, New York, 2009)
E. Kaniuth, A.T.-M. Lau, Fourier and Fourier-Stieltjes Algebras on Locally Compact Groups. Mathematical Surveys and Monographs, vol. 231 (American Mathematical Society, Providence, 2018)
C.J. Read, Discontinuous derivations on the algebra of bounded operators on a Banach space. J. London Math. Soc. 40, 305–326 (1989)
S. Roch, P.A. Santos, B. Silbermann, Non-commutative Gelfand Theories. A Tool-kit for Operator Theorists and Numerical Analysts. Universitext (Springer, Berlin, 2011)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Brešar, M. (2021). Zero Product Determined Banach Algebras. In: Zero Product Determined Algebras. Frontiers in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-80242-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-80242-4_5
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-80241-7
Online ISBN: 978-3-030-80242-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)