A spectral mapping theorem for Banach modules

H. Seferoğlu. "A spectral mapping theorem for Banach modules." Studia Mathematica 156.2 (2003): 99-103. <http://eudml.org/doc/284829>.

@article{H2003,
abstract = {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.},
author = {H. Seferoğlu},
journal = {Studia Mathematica},
keywords = {Banach module; spectrum; Banach algebra; Fourier-Stieltjes transform},
language = {eng},
number = {2},
pages = {99-103},
title = {A spectral mapping theorem for Banach modules},
url = {http://eudml.org/doc/284829},
volume = {156},
year = {2003},
}

TY - JOUR
AU - H. Seferoğlu
TI - A spectral mapping theorem for Banach modules
JO - Studia Mathematica
PY - 2003
VL - 156
IS - 2
SP - 99
EP - 103
AB - 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.
LA - eng
KW - Banach module; spectrum; Banach algebra; Fourier-Stieltjes transform
UR - http://eudml.org/doc/284829
ER -