Definition:Spectrum of Banach Algebra

Definition

Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.

Let $\Phi_A$ be the set of characters on $A$.

Let $w^\ast$ be the weak-$\ast$ topology on $A^\ast$.

Let $w^\ast_{\Phi_A}$ be the subspace topology on $\Phi_A$ inherited from $\struct {A^\ast, w^\ast}$.

We call $\struct {\Phi_A, w^\ast_{\Phi_A} }$ the spectrum of $A$.

Sources

• 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $4.10$: The continuity of characters