Zero product determined Banach algebras

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 $M_n(\mathbb{C})$, $n\ge 2$, 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.