Zero product determined Banach algebras

TL;DR

This paper investigates zero product determined properties of certain Banach algebras, including subspace lattice algebras, matrix algebras, and UHF algebras, providing new characterizations and structural insights.

Contribution

It establishes that specific classes of Banach algebras are zero product determined and offers criteria linking local derivations to derivations.

Findings

Algebras associated with certain subspace lattices are zero product determined.

Characterization of zero product determined property for matrix algebras.

Countable dimensional locally matrix and UHF algebras are zero Lie product determined.

Abstract

Let $\mathcal{L}$ be a completely distributive commutative subspace lattice or a subspace lattice with two atoms, we use a unified approach to study the derivations, homomorphisms on $\mathrm{Alg} \mathcal{L}$. We verify that the multiplier algebra of $\mathrm{Alg} \mathcal{L}\cap \mathcal{K}(\mathcal{H})$ is isomorphic to $\mathrm{Alg} \mathcal{L}$ and $\mathrm{Alg} \mathcal{L}$ is zero product determined. For $T$ in $M_{n}(\mathbb{C})$, $n\geq 2$, we show that $\mathcal{A}_{T}$ is zero product determined if and only if every local derivation from $\mathcal{A}_{T}$ into any Banach $\mathcal{A}_{T}$-bimodule is a derivation. In addition, we establish some equivalent conditions for an algebra to be zero product determined. For countable dimensional locally matrix algebras and triangular UHF algebras, we also show that they are zero Lie product determined.