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Türkiye'deki Matematiksel Etkinlikler

06 Mayıs 2016, 15:40

### İstanbul Analiz Seminerleri

Anar Dosi
Middle East Technical University, Northern Cyprus Campus, Türkiye

The present talk is devoted to classification of multinormed (or locally convex) von Neumann algebras of type I. A multinormed von Neumann algebra is defined as an inverse limit of von Neumann algebras whose connecting maps are $w^{\ast }$-continuous $\ast$-homomorphisms. It is known that the bounded part of a multinormed von Neumann algebra is a von Neumann algebra, and every multinormed von Neumann algebra is a central completion $\mathcal{M% }_{\mathcal{E}}$ of a von Neumann algebra $\mathcal{M}$ equipped with a domain $\mathcal{E}$ of its central projections. Moreover, $\mathcal{M}_{% \mathcal{E}}$ admits the predual $\left( \mathcal{M}_{\mathcal{E}}\right) _{\ast }$ which is an $\ell ^{1}$-normed space equipped with the canonical bornology $\left\{ \text{ball}\, \mathcal{M}_{\ast }e:e\in \mathcal{E}\right\}$. We prove that $\left( \mathcal{M}_{\mathcal{E}}\right) _{\ast }=\mathcal{M}% _{\ast \mathcal{E}}=\sum_{e\in \mathcal{E}}\mathcal{M}_{\ast }e=\mathcal{M}% _{\ast }\otimes _{\mathcal{E}}\mathcal{E}_{\ast }$ and the bornological dual $\left( \mathcal{M}_{\ast \mathcal{E}}\right) ^{\prime }$ is identified with $\mathcal{M}_{\mathcal{E}}$ up to an isometric isomorphism of polynormed spaces. In the case of $L^{\infty }\left( \mathcal{T}\right)$, the domain $% \mathcal{E}$ corresponds to a measurable covering of a locally compact space equipped with a positive Radon integral $\int :C_{c}\left( \mathcal{T}% \right) \rightarrow \mathbb{C}$, and the algebra $L^{\infty }\left( \mathcal{% T}\right) _{\mathcal{E}}$ is represented by means of $\mathcal{E}$-locally essentially bounded functions, that is, those functions $f\in \mathfrak{L}% \left( \mathcal{T}\right)$ such that $\text{esssup}\, \left\vert f|E\right\vert <\infty$ for all $E\in \mathcal{E}$. The bornological predual $L^{\infty }\left( \mathcal{T}\right) _{\ast \mathcal{E}}$ of the multinormed von Neumann algebra $L^{\infty }\left( \mathcal{T}\right) _{% \mathcal{E}}$ is reduced to the $\ell ^{1}$-normed space $L^{1}\left( \mathcal{T}\right) _{\mathcal{E}}$ which consists of those $g\in L^{1}\left( \mathcal{T}\right)$ such that $\left\Vert g\right\Vert _{1}=\int_{E}\left\vert g\right\vert$ for some $E\in \mathcal{E}$. If the $% \sigma$-covering is reduced to the trivial one $\mathcal{E}=\left( \mathcal{% T}\right)$ then $L^{\infty }\left( \mathcal{T}\right) _{\mathcal{E}% }=L^{\infty }\left( \mathcal{T}\right)$, $L^{1}\left( \mathcal{T}\right) _{% \mathcal{E}}=L^{1}\left( \mathcal{T}\right)$ and we obtain the classical result $L^{1}\left( \mathcal{T}\right) ^{\ast }=L^{\infty }\left( \mathcal{T}% \right)$. In the case of $L^{\infty }\left( \mathcal{T}\right) \overline{% \otimes }\mathcal{M}$ with a measurable covering $\mathcal{E}$ of $\mathcal{T% }$ and a (multiplicity) von Neumann algebra $\mathcal{M}\subseteq \mathcal{B}% \left( H\right)$, we obtain the multinormed von Neumann algebra $L^{\infty }\left( \mathcal{T}\right) _{\mathcal{E}}\overline{\otimes }\mathcal{M}$, which consists of unbounded decomposable operators $\int^{\oplus }x\left( t\right)$ on their common domain $\mathcal{O=\cup }_{E\in \mathcal{E}% }\left( \left[ E\right] \otimes 1\right) \left( L_{H}^{2}\left( \mathcal{T}% \right) \right)$ defined by means of (unbounded) measurable functions $% x\left( \cdot \right) :\mathcal{T\rightarrow M}$ which are $\mathcal{E}$% -locally bounded in the sense that all functions $\left( \left[ E\right] x\right) \left( \cdot \right) :\mathcal{T\rightarrow M}$, $e\in \mathcal{E}$ are bounded. Moreover, $\left( L^{\infty }\left( \mathcal{T}\right) _{% \mathcal{E}}\overline{\otimes }\mathcal{M}\right) _{\ast \mathcal{E\otimes }% 1}=L^{1}\left( \mathcal{T}\right) _{\mathcal{E}}\otimes _{\ell ^{1}}\mathcal{% M}_{\ast }$, which consists of those $y\in L_{\mathcal{M}_{\ast }}^{1}\left( \mathcal{T}\right)$ such that $\left\Vert y\right\Vert _{1}=\int_{E}\left\Vert y\left( t\right) \right\Vert$ for some $E\in \mathcal{E}$. Finally, every multinormed von Neumann algebra of type I can be obtained by means of $L^{\infty }\left( \mathcal{T}\right) _{\mathcal{E}}% \overline{\otimes }\mathcal{M}$ for various multiplicities $\mathcal{M}$.
Analiz İngilizce
Sabancı University, Karaköy Communication Center, Bankalar Caddesi 2, Karaköy

admin 20.03.2020

## İLETİŞİM

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## DESTEK VERENLER

31. Journees Arithmetiques Konferansı Organizasyon Komitesi

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