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| Main Authors: | , , |
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| Format: | Preprint |
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2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2008.06031 |
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| _version_ | 1866914225278943232 |
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| author | Gupta, Ved Prakash Jain, Ranjana Talwar, Bharat |
| author_facet | Gupta, Ved Prakash Jain, Ranjana Talwar, Bharat |
| contents | For any locally compact group $G$ and any Banach algebra $A$, a characterization of the closed Lie ideals of the generalized group algebra $L^1(G,A)$ is obtained in terms of left and right actions by $G$ and $A$. In addition, when $A$ is unital and $G$ is an ${\bf [SIN]}$ group, we show that the center of $L^1(G,A)$ is precisely the collection of all center valued functions which are constant on the conjugacy classes of $G$. As an application, we establish that $\mathcal{Z}(L^1(G) \otimes^γ A)= \mathcal{Z}(L^1(G)) \otimes^γ \mathcal{Z}(A)$, for a class of groups and Banach algebras. And, prior to these, for any finite group $G$, the Lie ideals of the group algebra $\mathbb{C}[G]$ are identified in terms of some canonical spaces determined by the irreducible characters of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_06031 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On closed Lie ideals and center of generalized group algebras Gupta, Ved Prakash Jain, Ranjana Talwar, Bharat Functional Analysis 22D15, 43A20, 46M05 For any locally compact group $G$ and any Banach algebra $A$, a characterization of the closed Lie ideals of the generalized group algebra $L^1(G,A)$ is obtained in terms of left and right actions by $G$ and $A$. In addition, when $A$ is unital and $G$ is an ${\bf [SIN]}$ group, we show that the center of $L^1(G,A)$ is precisely the collection of all center valued functions which are constant on the conjugacy classes of $G$. As an application, we establish that $\mathcal{Z}(L^1(G) \otimes^γ A)= \mathcal{Z}(L^1(G)) \otimes^γ \mathcal{Z}(A)$, for a class of groups and Banach algebras. And, prior to these, for any finite group $G$, the Lie ideals of the group algebra $\mathbb{C}[G]$ are identified in terms of some canonical spaces determined by the irreducible characters of $G$. |
| title | On closed Lie ideals and center of generalized group algebras |
| topic | Functional Analysis 22D15, 43A20, 46M05 |
| url | https://arxiv.org/abs/2008.06031 |