A spectral mapping theorem for Banach modules

H. Seferoğlu

Studia Mathematica, Tome 157 (2003), p. 99-103 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2003-01-01

Zbl 1028.46073

EUDML-ID : urn:eudml:doc:284829