Comparison of topologies on ⁎-algebras of locally measurable operators

Comparison of topologies on ⁎-algebras of locally measurable operators We consider the local measure topology $${t(\mathcal{M})}$$ on the ⁎-algebra $${LS(\mathcal{M})}$$ of all locally measurable operators and on the ⁎-algebra $${S(\mathcal{M},\tau)}$$ of all τ-measurable operators affiliated with a von Neumann algebra $${\mathcal{M}}$$ . If τ is a semifinite but not a finite trace on $${\mathcal{M},}$$ then one can consider the τ-local measure topology t τ l and the weak τ-local measure topology t w τ l . We study relationships between the topology $${t(\mathcal{M})}$$ and the topologies t τ l , t w τ l , and the (o)-topology $${t_o(\mathcal{M})}$$ on $${LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^\ast=T\}}$$ . We find that the topologies $${t(\mathcal{M})}$$ and t τ l (resp. $${t(\mathcal{M})}$$ and t w τ l ) coincide on $${S(\mathcal{M},\tau)}$$ if and only if $${\mathcal{M}}$$ is finite, and $${t(\mathcal{M})=t_o(\mathcal{M})}$$ on $${LS_h(\mathcal{M})}$$ holds if and only if $${\mathcal{M}}$$ is a σ-finite and finite. Moreover, it turns out that the topology t τ l (resp. t w τ l ) coincides with the (o)-topology on $${S_h(\mathcal{M},\tau)}$$ only for finite traces. We give necessary and sufficient conditions for the topology $${t(\mathcal{M})}$$ to be locally convex (resp., normable). We show that (o)-convergence of sequences in $${LS_h(\mathcal{M})}$$ and convergence in the topology $${t(\mathcal{M})}$$ coincide if and only if the algebra $${\mathcal{M}}$$ is an atomic and finite algebra. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Comparison of topologies on ⁎-algebras of locally measurable operators

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Publisher
Springer Journals
Copyright
Copyright © 2011 by Springer Basel AG
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0152-6
Publisher site
See Article on Publisher Site

Abstract

We consider the local measure topology $${t(\mathcal{M})}$$ on the ⁎-algebra $${LS(\mathcal{M})}$$ of all locally measurable operators and on the ⁎-algebra $${S(\mathcal{M},\tau)}$$ of all τ-measurable operators affiliated with a von Neumann algebra $${\mathcal{M}}$$ . If τ is a semifinite but not a finite trace on $${\mathcal{M},}$$ then one can consider the τ-local measure topology t τ l and the weak τ-local measure topology t w τ l . We study relationships between the topology $${t(\mathcal{M})}$$ and the topologies t τ l , t w τ l , and the (o)-topology $${t_o(\mathcal{M})}$$ on $${LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^\ast=T\}}$$ . We find that the topologies $${t(\mathcal{M})}$$ and t τ l (resp. $${t(\mathcal{M})}$$ and t w τ l ) coincide on $${S(\mathcal{M},\tau)}$$ if and only if $${\mathcal{M}}$$ is finite, and $${t(\mathcal{M})=t_o(\mathcal{M})}$$ on $${LS_h(\mathcal{M})}$$ holds if and only if $${\mathcal{M}}$$ is a σ-finite and finite. Moreover, it turns out that the topology t τ l (resp. t w τ l ) coincides with the (o)-topology on $${S_h(\mathcal{M},\tau)}$$ only for finite traces. We give necessary and sufficient conditions for the topology $${t(\mathcal{M})}$$ to be locally convex (resp., normable). We show that (o)-convergence of sequences in $${LS_h(\mathcal{M})}$$ and convergence in the topology $${t(\mathcal{M})}$$ coincide if and only if the algebra $${\mathcal{M}}$$ is an atomic and finite algebra.

Journal

PositivitySpringer Journals

Published: Dec 16, 2011

References

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