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

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 ${\displaystyle {\phi }_a\left(t\right):=\phi \left({\alpha }_{t}a\right)}$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum ${\displaystyle {\sigma }_w\cdot \left({\phi }_a\right)}$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define ${\displaystyle {Ʌ}_{\phi }^a}$ to be the union of all sets ${\displaystyle {\sigma }_w\cdot \left({\phi }_a\right)}$ where a ∈ A, and ${\displaystyle {\Lambda }_{\alpha }}$ to be the closure of the union of all sets ${\displaystyle {Ʌ}_{\phi }^a}$ where φ ∈ ∆(A), and call ${\displaystyle {\Lambda }_{\alpha }}$ the unitary spectrum of α. Starting by showing that the closure of ${\displaystyle {Ʌ}_{\phi }^a}$ (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 ${\displaystyle {\Lambda }_{\alpha }.}$ For an automorphism T of a semisimple commutative Banach algebra A we consider the group representation T: ℤ → Aut (A) defined by ${\displaystyle T_n:=T^n}$ for all n ∈ ℤ. It is shown that ${\displaystyle {\Lambda }_T=\sigma \left(T\right)\cap }$, where σ(T) is the spectrum of T and 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 or is a finite union of finite subgroups of .