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A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras

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Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function $φ_a(t):=φ(α_t a)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum $σ_w*(φ_a)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define $Ʌ_φ^a$ to be the union of all sets $σ_w*(φ_a)$ where a ∈ A, and $Λ_α$ to be the closure of the union of all sets $Ʌ_φ^a$ where φ ∈ ∆(A), and call $Λ_α$ the unitary spectrum of α. Starting by showing that the closure of $Ʌ_φ^a$ (for fixed φ ∈ ∆(A)) is a subsemigroup of Ĝ we characterize the structure properties of the group representation α such as norm continuity, growth and existence of non-trivial invariant subspaces through its unitary spectrum $Λ_α.$ For an automorphism T of a semisimple commutative Banach algebra A we consider the group representation T: ℤ → Aut (A) defined by $T_n:=T^n$ for all n ∈ ℤ. It is shown that $Λ_T=σ(T)∩𝕋$, where σ(T) is the spectrum of T and 𝕋 is the unit circle. From this fact we give an easy proof of the Kamowitz-Scheinberg theorem which asserts that the spectrum σ(T) either contains 𝕋 or is a finite union of finite subgroups of 𝕋.
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