İstanbul Analysis Seminars
Multinormed von Neumann algebras of Type I
Anar Dosi
Middle East Technical University, Northern Cyprus Campus, Turkey
Özet :
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 $\mathcal{M}$.
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Tarih |
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06.05.2016 |
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Saat |
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15:40 |
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Yer |
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Sabancı University, Karaköy Communication Center, Bankalar Caddesi 2, Karaköy |
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Dil |
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English |