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A spectral mapping theorem for Banach modules

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Seferoğlu H.

Studia Mathematica

2003

tom 156

nr 2

99-103

Let G be a locally compact abelian group, M(G) the convolution measure algebra, and X a Banach M(G)-module under the module multiplication μ ∘ x, μ ∈ M(G), x ∈ X. We show that if X is an essential L¹(G)-module, then $σ(T_{μ}) = \overline {μ̂(sp(X))}$ for each measure μ in reg(M(G)), where $T_{μ}$ denotes the operator in B(X) defined by $T_{μ}x = μ ∘ x$, σ(·) the usual spectrum in B(X), sp(X) the hull in L¹(G) of the ideal $I_{X} = {f ∈ L¹(G) | T_{f} = 0}$, μ̂ the Fourier-Stieltjes transform of μ, and reg(M(G)) the largest closed regular subalgebra of M(G); reg(M(G)) contains all the absolutely continuous measures and discrete measures.

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1 Studia Mathematica

1 Seferoğlu H.

1 2003

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A spectral mapping theorem for Banach modules