Pseudotopologies with applications to one-parameter groups, von Neumann algebras, and Lie algebra representations

Jan Rusinek

Studia Mathematica (1993)

  • Volume: 107, Issue: 3, page 273-286
  • ISSN: 0039-3223

Abstract

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For any pair E,F of pseudotopological vector spaces, we endow the space L(E,F) of all continuous linear operators from E into F with a pseudotopology such that, if G is a pseudotopological space, then the mapping L(E,F) × L(F,G) ∋ (f,g) → gf ∈ L(E,G) is continuous. We use this pseudotopology to establish a result about differentiability of certain operator-valued functions related with strongly continuous one-parameter semigroups in Banach spaces, to characterize von Neumann algebras, and to establish a result about integration of Lie algebra representations.

How to cite

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Rusinek, Jan. "Pseudotopologies with applications to one-parameter groups, von Neumann algebras, and Lie algebra representations." Studia Mathematica 107.3 (1993): 273-286. <http://eudml.org/doc/216033>.

@article{Rusinek1993,
abstract = {For any pair E,F of pseudotopological vector spaces, we endow the space L(E,F) of all continuous linear operators from E into F with a pseudotopology such that, if G is a pseudotopological space, then the mapping L(E,F) × L(F,G) ∋ (f,g) → gf ∈ L(E,G) is continuous. We use this pseudotopology to establish a result about differentiability of certain operator-valued functions related with strongly continuous one-parameter semigroups in Banach spaces, to characterize von Neumann algebras, and to establish a result about integration of Lie algebra representations.},
author = {Rusinek, Jan},
journal = {Studia Mathematica},
keywords = {pseudotopology; continuity; composition of operators; differentiability; one-parameter semigroup; von Neumann algebra; integration; Lie algebra representation; pseudotopological vector spaces; differentiability of certain operator- valued functions; strongly continuous one-parameter semigroups in Banach spaces; von Neumann algebras; integration of Lie algebra representations},
language = {eng},
number = {3},
pages = {273-286},
title = {Pseudotopologies with applications to one-parameter groups, von Neumann algebras, and Lie algebra representations},
url = {http://eudml.org/doc/216033},
volume = {107},
year = {1993},
}

TY - JOUR
AU - Rusinek, Jan
TI - Pseudotopologies with applications to one-parameter groups, von Neumann algebras, and Lie algebra representations
JO - Studia Mathematica
PY - 1993
VL - 107
IS - 3
SP - 273
EP - 286
AB - For any pair E,F of pseudotopological vector spaces, we endow the space L(E,F) of all continuous linear operators from E into F with a pseudotopology such that, if G is a pseudotopological space, then the mapping L(E,F) × L(F,G) ∋ (f,g) → gf ∈ L(E,G) is continuous. We use this pseudotopology to establish a result about differentiability of certain operator-valued functions related with strongly continuous one-parameter semigroups in Banach spaces, to characterize von Neumann algebras, and to establish a result about integration of Lie algebra representations.
LA - eng
KW - pseudotopology; continuity; composition of operators; differentiability; one-parameter semigroup; von Neumann algebra; integration; Lie algebra representation; pseudotopological vector spaces; differentiability of certain operator- valued functions; strongly continuous one-parameter semigroups in Banach spaces; von Neumann algebras; integration of Lie algebra representations
UR - http://eudml.org/doc/216033
ER -

References

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  1. [B] A. Bastiani, Applications différentiables et variétés différentiables de dimension infinie, J. Anal. Math. 13 (1964), 1-114. Zbl0196.44103
  2. [B-R] O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer, New York, 1979. 
  3. [D] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980. Zbl0457.47030
  4. [F-B] A. Frölicher and W. Bucher, Calculus in Vector Spaces without Norm, Lecture Notes in Math. 30, Springer, New York, 1966. Zbl0156.38303
  5. [J-M] P. E. T. Jørgensen and T. Moore, Operator Commutation Relations, Reidel, Dordrecht, 1984. 
  6. [Ke] H. H. Keller, Differenzierbarkeit in topologischen Vektorräumen, Comment. Math. Helv. 38 (1964), 308-320. Zbl0124.06402
  7. [K] J. Kisyński, On the integration of a Lie algebra representation in a Banach space, internal report, International Centre for Theoretical Physics, Triest, 1974. 
  8. [R] J. Rusinek, Analytic vectors and integrability of Lie algebra representations, J. Funct. Anal. 74 (1987), 10-23. Zbl0643.22007
  9. [T-W] J. Tits and L. Waelbroeck, The integration of a Lie algebra representation, Pacific J. Math. 26 (1968), 595-600. Zbl0172.18602

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