# Korovkin theory in normed algebras

Studia Mathematica (1991)

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

## Access Full Article

top## Abstract

top## How to cite

topBeckhoff, 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

top- [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] W. Ambrose, Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc. 57 (1945), 364-386. Zbl0060.26906
- [3] F. Beckhoff, Korovkin-Theory in Algebren, Schriftenreihe Math. Inst. Univ. Münster, Ser. 2, Heft 45, 1987. Zbl0626.46049
- [4] F. Beckhoff, A counterexample in Korovkin theory, Rend. Circ. Mat. Palermo (2) 37 (1988), 469-473. Zbl0674.46031
- [5] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer, Berlin 1979. Zbl0421.46048
- [6] J. Dixmier, Von Neumann Algebras, North-Holland Math. Library 27, 1981.
- [7] N. Dunford and J. T. Schwartz, Linear Operators II, Interscience Publ., 1963. Zbl0128.34803
- [8] R. V. Kadison, A generalized Schwarz inequality and algebraic invariants of operator algebras, Ann. of Math. 56 (1952), 494-503. Zbl0047.35703
- [9] I. Kaplansky, Groups with representations of bounded degree, Canad. J. Math. 1 (1949), 105-112. Zbl0037.01603
- [10] B. V. Limaye and M. N. N. Namboodiri, Korovkin approximation on C*-algebras, J. Approx. Theory 34 (1982), 237-246. Zbl0501.46049
- [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] B. V. Limaye and M. N. N. Namboodiri, Weak approximation by positive maps on C*-algebras, to appear. Zbl0593.41020
- [13] L. H. Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand, New York 1953.
- [14] M. Pannenberg, Korovkin approximation in Waelbroeck algebras, Math. Ann. 274 (1986), 423-437. Zbl0576.41014
- [15] W. M. Priestley, A noncommutative Korovkin theorem, J. Approx. Theory 16 (1976), 251-260.
- [16] A. G. Robertson, A Korovkin theorem for Schwarz maps on C*-algebras, Math. Z. 56 (1977), 205-207. Zbl0344.46116
- [17] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, Berlin 1960.
- [18] M. Takesaki, Theory of Operator Algebras I, Springer, New York 1979.
- [19] B. Yood, Hilbert algebras as topological algebras, Ark. Mat. 12 (1974), 131-151. Zbl0286.46054

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.