### A remark on functional continuity of certain Fréchet 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 ${\phi}_{a}\left(t\right):=\phi \left({\alpha}_{t}a\right)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum ${\sigma}_{w}*\left({\phi}_{a}\right)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define ${\u0245}_{\phi}^{a}$ to be the union of all...

It is shown that reducing bands of measures yield decompositions not only of an operator representation itself, but also of its commutant. This has many consequences for commuting Hilbert space representations and for commuting operators on Hilbert spaces. Among other things, it enables one to construct a Lebesgue-type decomposition of several commuting contractions without assuming any von Neumann-type inequality.

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