ScalingStacks

Definition 2.18. Let $\mathcal{A}_{1},\mathcal{A}_{2}$ be two Banach $k$-algebras, and $\phi:A_{1}\to A_{2}$ be a homomorphism of $k$-algebras. We say that $\phi$ is a homomorphism of Banach $k$-algebras if it is bounded as a $k$-linear map. A homomorphism of Banach $k$-algebra $\phi$ is often denoted by $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$. A homomorphism of Banach $k$-algebra $\phi$ is called an isomorphism of Banach $k$-algebras if there exists a homomorphism of Banach $k$-algebras $\psi:\mathcal{A}_{2}\to\mathcal{A}_{1}$ such that $\phi\circ\psi=\mathrm{Id}_{\mathcal{A}_{2}}$ and $\psi\circ\phi=\mathrm{Id}_{\mathcal{A}_{1}}$.