turkmath.org

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 $\mathcal{M}$.