On strong Arens irregularity of projective tensor product of Hilbert-Schmidt space
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866914225367023616 |
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| author | Gupta, Ved Prakash Singh, Lav Kumar |
| author_facet | Gupta, Ved Prakash Singh, Lav Kumar |
| contents | It was shown in [16] that the Banach algebra $A:=S_2(\ell^2)\otimes^γ S_2(\ell^2)$ is not Arens regular, where $S_2(\ell^2)$ denotes the Banach algebra of the Hilbert-Schmidt operators on $\ell^2$. In this article, employing the notion of limits along ultrafilters, we prove that the irregularity of $S_2(\ell^2)\otimes^γ S_2(\ell^2)$ is not strong. Along the way, we provide a class of functionals in $A^{**}$ which lie in the topological center but are not in $A$; and, as a consequence, we deduce that $A^{**}$ is not an annihilator Banach algebra with respect to any of the two Arens products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_13946 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On strong Arens irregularity of projective tensor product of Hilbert-Schmidt space Gupta, Ved Prakash Singh, Lav Kumar Functional Analysis 47B10, 46B28, 46M05 It was shown in [16] that the Banach algebra $A:=S_2(\ell^2)\otimes^γ S_2(\ell^2)$ is not Arens regular, where $S_2(\ell^2)$ denotes the Banach algebra of the Hilbert-Schmidt operators on $\ell^2$. In this article, employing the notion of limits along ultrafilters, we prove that the irregularity of $S_2(\ell^2)\otimes^γ S_2(\ell^2)$ is not strong. Along the way, we provide a class of functionals in $A^{**}$ which lie in the topological center but are not in $A$; and, as a consequence, we deduce that $A^{**}$ is not an annihilator Banach algebra with respect to any of the two Arens products. |
| title | On strong Arens irregularity of projective tensor product of Hilbert-Schmidt space |
| topic | Functional Analysis 47B10, 46B28, 46M05 |
| url | https://arxiv.org/abs/2108.13946 |