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Let A be a commutative Banach algebra and let ${\Sigma }_A$ be its structure space. The norm spectrum σ(f) of the functional f ∈ A* is defined by $\sigma \left(f\right)=\overline{f·a:a\in A}\cap {\Sigma }_A$, where f·a is the functional on A defined by ⟨f·a,b⟩ = ⟨f,ab⟩, b ∈ A. We investigate basic properties of the norm spectrum in certain classes of commutative Banach algebras and present some applications.

Let A be a complex, commutative Banach algebra and let $M_A$ be the structure space of A. Assume that there exists a continuous homomorphism h:L¹(G) → A with dense range, where L¹(G) is a group algebra of the locally compact abelian group G. The main results of this note can be summarized as follows: (a) If every weakly almost periodic functional on A with compact spectra is almost periodic, then the space $M_A$ is scattered (i.e., $M_A$ has no nonempty perfect subset). (b) Weakly almost periodic functionals...