2023/05/09 by Jyunji Inoue, Sin‐Ei Takahasi, Inoue, Jyunji +1
Mathematics · #43A20 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2305.05617
openalex publication_date 2023/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a non-discrete LCA group with the dual group Γ. We define generalized group algebra, \mathcal L1(G), and generalized measure algebra, \mathcal M(G), on G as generalizations of the group algebra L1(G) and the measure algebra M(G), respectively. Generalized Fourier transforms of elements of \mathcal L1(G) and generalized Fourier-Stieltjes transforms of elements of \mathcal M(G) are also defined as generalizations of the Fourier transforms and the Fourier-Stieltjes transforms, respectively. The image \mathcal A(Γ) of \mathcal L1(G) by the generalized Fourier transform becomes a function algebra on Γ with norm inherited from \mathcal L1(G) through this transform. It is shown that \mathcal A(Γ) is a natural Banach function algebra on Γ which is BSE and BED. It turns out that \mathcal L1(G) contains all Rajchman measures. Segal algebras in \mathcal L1(G) are defined and investigated. It is shown that there exists the smallest isometrically character invariant Segal algebra in \mathcal L1(G), which (eventually) coincides with the smallest isometrically character invariant Segal algebra in L1(G), the Feichtinger algebra of G. A notion of locally bounded elements of \mathcal M(G) is introduced and investigated. It is shown that for each locally bounded element μ of \mathcal M(G) there corresponds a unique Radon measure ιμ on G which characterizes μ. We investigate the multiplier algebra \mathbbM(\mathcal L1(G)) of \mathcal L1(G), and obtain a result that there is a natural continuous isomorphism from \mathbbM (\mathcal L1(G)) into A(G)^*, the algebra of pseudomeasures on G. When G is compact, this map becomes surjective and isometric.