Abstract
The primary objective of this paper is to establish the property of zero product determinacy for the algebra \(\textrm{Alg} \mathcal {L}\), where \(\mathcal {L}\) is either a completely distributive commutative subspace lattice or a subspace lattice with two atoms. This objective is achieved by employing a technical approach that involves demonstrating the isomorphism between the multiplier algebra of the algebra consisting of compact operators belonging to \(\textrm{Alg} \mathcal {L}\) and \(\textrm{Alg} \mathcal {L}\). Furthermore, we investigate the properties of derivations and homomorphisms on these algebras as an application of our main result. Additionally, we prove that in the finite-dimensional case, a unital algebra generated by a single operator is zero product determined if and only if every local derivation from the algebra to any of its Banach bimodules is a derivation.
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The research was supported by the National Natural Science Foundation of China (Grant No. 11871021).
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Communicated by Mohammad Sal Moslehian.
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Li, J., Pan, S. & Su, S. A Class of Zero Product Determined Banach Algebras. Bull. Malays. Math. Sci. Soc. 47, 24 (2024). https://doi.org/10.1007/s40840-023-01625-9
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DOI: https://doi.org/10.1007/s40840-023-01625-9