Korovkin theory in normed algebras

Ferdinand Beckhoff

Studia Mathematica (1991)

  • Volume: 100, Issue: 3, page 219-228
  • ISSN: 0039-3223

Abstract

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If A is a normed power-associative complex algebra such that the selfadjoint part is normally ordered with respect to some order, then the Korovkin closure (see the introduction for definitions) of T ∪ {t* ∘ t| t ∈ T} contains J*(T) for any subset T of A. This can be applied to C*-algebras, minimal norm ideals on a Hilbert space, and to H*-algebras. For bounded H*-algebras and dual C*-algebras there is even equality. This answers a question posed in [1].

How to cite

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Beckhoff, Ferdinand. "Korovkin theory in normed algebras." Studia Mathematica 100.3 (1991): 219-228. <http://eudml.org/doc/215884>.

@article{Beckhoff1991,
abstract = {If A is a normed power-associative complex algebra such that the selfadjoint part is normally ordered with respect to some order, then the Korovkin closure (see the introduction for definitions) of T ∪ \{t* ∘ t| t ∈ T\} contains J*(T) for any subset T of A. This can be applied to C*-algebras, minimal norm ideals on a Hilbert space, and to H*-algebras. For bounded H*-algebras and dual C*-algebras there is even equality. This answers a question posed in [1].},
author = {Beckhoff, Ferdinand},
journal = {Studia Mathematica},
keywords = {normed power-associative complex algebra; selfadjoint part; Korovkin closure; -algebras; minimal norm ideals; -algebras; bounded -algebras; dual -algebras},
language = {eng},
number = {3},
pages = {219-228},
title = {Korovkin theory in normed algebras},
url = {http://eudml.org/doc/215884},
volume = {100},
year = {1991},
}

TY - JOUR
AU - Beckhoff, Ferdinand
TI - Korovkin theory in normed algebras
JO - Studia Mathematica
PY - 1991
VL - 100
IS - 3
SP - 219
EP - 228
AB - If A is a normed power-associative complex algebra such that the selfadjoint part is normally ordered with respect to some order, then the Korovkin closure (see the introduction for definitions) of T ∪ {t* ∘ t| t ∈ T} contains J*(T) for any subset T of A. This can be applied to C*-algebras, minimal norm ideals on a Hilbert space, and to H*-algebras. For bounded H*-algebras and dual C*-algebras there is even equality. This answers a question posed in [1].
LA - eng
KW - normed power-associative complex algebra; selfadjoint part; Korovkin closure; -algebras; minimal norm ideals; -algebras; bounded -algebras; dual -algebras
UR - http://eudml.org/doc/215884
ER -

References

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  1. [1] F. Altomare, Korovkin closures in Banach algebras, in: Advances in Invariant Subspaces and Other Results of Operator Theory, Proc. 9th Internat. Conf. on Operator Theory, Timișoara and Herculane 1984, Oper. Theory: Adv. Appl. 17, Birkhäuser, Basel 1986, 35-42. 
  2. [2] W. Ambrose, Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc. 57 (1945), 364-386. Zbl0060.26906
  3. [3] F. Beckhoff, Korovkin-Theory in Algebren, Schriftenreihe Math. Inst. Univ. Münster, Ser. 2, Heft 45, 1987. Zbl0626.46049
  4. [4] F. Beckhoff, A counterexample in Korovkin theory, Rend. Circ. Mat. Palermo (2) 37 (1988), 469-473. Zbl0674.46031
  5. [5] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer, Berlin 1979. Zbl0421.46048
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  7. [7] N. Dunford and J. T. Schwartz, Linear Operators II, Interscience Publ., 1963. Zbl0128.34803
  8. [8] R. V. Kadison, A generalized Schwarz inequality and algebraic invariants of operator algebras, Ann. of Math. 56 (1952), 494-503. Zbl0047.35703
  9. [9] I. Kaplansky, Groups with representations of bounded degree, Canad. J. Math. 1 (1949), 105-112. Zbl0037.01603
  10. [10] B. V. Limaye and M. N. N. Namboodiri, Korovkin approximation on C*-algebras, J. Approx. Theory 34 (1982), 237-246. Zbl0501.46049
  11. [11] B. V. Limaye and M. N. N. Namboodiri, Weak Korovkin approximation by completely positive linear maps on β(H), ibid. 42 (1984), 201-211. Zbl0552.41028
  12. [12] B. V. Limaye and M. N. N. Namboodiri, Weak approximation by positive maps on C*-algebras, to appear. Zbl0593.41020
  13. [13] L. H. Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand, New York 1953. 
  14. [14] M. Pannenberg, Korovkin approximation in Waelbroeck algebras, Math. Ann. 274 (1986), 423-437. Zbl0576.41014
  15. [15] W. M. Priestley, A noncommutative Korovkin theorem, J. Approx. Theory 16 (1976), 251-260. 
  16. [16] A. G. Robertson, A Korovkin theorem for Schwarz maps on C*-algebras, Math. Z. 56 (1977), 205-207. Zbl0344.46116
  17. [17] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, Berlin 1960. 
  18. [18] M. Takesaki, Theory of Operator Algebras I, Springer, New York 1979. 
  19. [19] B. Yood, Hilbert algebras as topological algebras, Ark. Mat. 12 (1974), 131-151. Zbl0286.46054

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