Small-2 Sets Are Riesz Sets
Abstract
Let $ G $ be a compact metrizable Abelian group, $ L^{1}(G) $ its group algebra and $ M(G) $ its measure algebra. For each proper subset $ E $ of the dual group $ \hat{G} $, let $ L^{1}_{E}(G)=\{f\in L^{1}(G):\hat{f}=0 \text{ on } \hat{G}\setminus E \}$ and $M_{E}=\{μ\in M(G):\hatμ=0 \text{ on }\hat{G}\setminus E\} $. If $ M_{E}(G)=L^{1}_{E}(G) $ then the set $ E $ is said to be a Riesz sets. If $ M_{E}(G)*M_{E}(G)\subseteq L_{E}^{1}(G) $ then $ E $ is said to be a small-2 set. The main results of this paper are the following: 1. Every small-2 set is a Riesz set. 2. The ideal $ L^{1}_{E}(G) $ is Arens regular iff $ E $ is a Riesz set. Let $ A=L_{E}(G) $ and equip $ A^{**} $ with the first Arens product. 3. The centre of $ A^{**} $ is $ Z(A^{**})=A+N(A^{**}) $, where $ N(A^{**})=\{r\in A^{**}:rA^{**}=\{0\}\} $. These results settle three long-standing open problems in this area.
Related
BibTeX
@misc{lau2026small2,
title = {{Small-2 Sets Are Riesz Sets}},
author = {A. To-Ming Lau and A. Ülger},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.30570},
eprint = {2606.30570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30570},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Small-2 Sets Are Riesz Sets
AU - A. To-Ming Lau
AU - A. Ülger
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2606.30570
UR - https://arxiv.org/abs/2606.30570
AB - Let $ G $ be a compact metrizable Abelian group, $ L^{1}(G) $ its group algebra and $ M(G) $ its measure algebra. For each proper subset $ E $ of the dual group $ \hat{G} $, let $ L^{1}_{E}(G)=\{f\in L^{1}(G):\hat{f}=0 \text{ on } \hat{G}\setminus E \}$ and $M_{E}=\{μ\in M(G):\hatμ=0 \text{ on }\hat{G}\setminus E\} $. If $ M_{E}(G)=L^{1}_{E}(G) $ then the set $ E $ is said to be a Riesz sets. If $ M_{E}(G)*M_{E}(G)\subseteq L_{E}^{1}(G) $ then $ E $ is said to be a small-2 set. The main results of this paper are the following: 1. Every small-2 set is a Riesz set. 2. The ideal $ L^{1}_{E}(G) $ is Arens regular iff $ E $ is a Riesz set. Let $ A=L_{E}(G) $ and equip $ A^{**} $ with the first Arens product. 3. The centre of $ A^{**} $ is $ Z(A^{**})=A+N(A^{**}) $, where $ N(A^{**})=\{r\in A^{**}:rA^{**}=\{0\}\} $. These results settle three long-standing open problems in this area.
ER -