Totally convex algebras

Dieter Pumplün; Helmut Röhrl

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 2, page 205-235
  • ISSN: 0010-2628

Abstract

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By definition a totally convex algebra A is a totally convex space | A | equipped with an associative multiplication, i.eȧ morphism μ : | A | ⊗ | A | ⟶ | A | of totally convex spaces. In this paper we introduce, for such algebras, the notions of ideal, tensor product, unitization, inverses, weak inverses, quasi-inverses, weak quasi-inverses and the spectrum of an element and investigate them in detail. This leads to a considerable generalization of the corresponding notions and results in the theory of Banach spaces.

How to cite

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Pumplün, Dieter, and Röhrl, Helmut. "Totally convex algebras." Commentationes Mathematicae Universitatis Carolinae 33.2 (1992): 205-235. <http://eudml.org/doc/247413>.

@article{Pumplün1992,
abstract = {By definition a totally convex algebra $A$ is a totally convex space $|A|$ equipped with an associative multiplication, i.eȧ morphism $\mu :|A|\otimes |A|\longrightarrow |A|$ of totally convex spaces. In this paper we introduce, for such algebras, the notions of ideal, tensor product, unitization, inverses, weak inverses, quasi-inverses, weak quasi-inverses and the spectrum of an element and investigate them in detail. This leads to a considerable generalization of the corresponding notions and results in the theory of Banach spaces.},
author = {Pumplün, Dieter, Röhrl, Helmut},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {totally convex algebra; Eilenberg-Moore algebra; Banach algebra; ideal; (weak) inverse; spectrum; totally convex spaces; totally convex algebras; category of Eilenberg- Moore algebras of the unit-ball functor; category of Banach algebras and contractive homomorphisms; ideal; tensor product; inverses; spectrum; weakly invertible; spectral mapping theorem},
language = {eng},
number = {2},
pages = {205-235},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Totally convex algebras},
url = {http://eudml.org/doc/247413},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Pumplün, Dieter
AU - Röhrl, Helmut
TI - Totally convex algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 2
SP - 205
EP - 235
AB - By definition a totally convex algebra $A$ is a totally convex space $|A|$ equipped with an associative multiplication, i.eȧ morphism $\mu :|A|\otimes |A|\longrightarrow |A|$ of totally convex spaces. In this paper we introduce, for such algebras, the notions of ideal, tensor product, unitization, inverses, weak inverses, quasi-inverses, weak quasi-inverses and the spectrum of an element and investigate them in detail. This leads to a considerable generalization of the corresponding notions and results in the theory of Banach spaces.
LA - eng
KW - totally convex algebra; Eilenberg-Moore algebra; Banach algebra; ideal; (weak) inverse; spectrum; totally convex spaces; totally convex algebras; category of Eilenberg- Moore algebras of the unit-ball functor; category of Banach algebras and contractive homomorphisms; ideal; tensor product; inverses; spectrum; weakly invertible; spectral mapping theorem
UR - http://eudml.org/doc/247413
ER -

References

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  1. Bonsall F.F., Duncan J., Complete normed algebras, Springer, Erg. Math., Berlin-Heidelberg-New York, 1973. Zbl0271.46039MR0423029
  2. Bourbaki N., Eléments de mathématiques, Algèbre, chap. III, Hermann, Paris, 1970. Zbl0455.18010MR0274237
  3. Cohn P.M., Universal algebra, Harper & Row, New York-Evanston-London, 1965. Zbl0461.08001MR0175948
  4. Pelletier J.W., Rosický J., Generating the monadic theory of C * -algebras and related categories, Categorical topology and its relation to analysis, algebra and combinatorics, Conf. Proc. Prague 1988, World Scientif. Publ. Singapore, New Jersey, London, Hongkong (1989), 163-180. MR1047899
  5. Pierce R.S., Introduction to the theory of abstract algebras, Holt, Rinehart and Winston, New York, 1965. MR0227070
  6. Pumplün D., Röhrl H., Banach spaces and totally convex spaces I, Comm. Alg. 12 (1984), 953-1019. (1984) MR0735910
  7. Pumplün D., Röhrl H., Banach spaces and totally convex spaces II, Comm. Alg. 13 (1985), 1047-1113. (1985) MR0780637
  8. Pumplün D., Röhrl H., Separated totally convex spaces, man. math. 50 (1985), 145-183. (1985) MR0784142
  9. Pumplün D., Röhrl H., The coproduct of totally convex spaces, Beitr. Alg. u. Geom. 24 (1987), 249-278. (1987) MR0888218
  10. Pumplün D., Röhrl H., Congruence relations on totally convex spaces, Comm. Alg. 18 (1990), 1469-1496. (1990) MR1059742
  11. Pumplün D., Regularly ordered Banach spaces and positively convex spaces, Results Math. 7 (1984), 85-112. (1984) MR0758773
  12. Pumplün D., The Hahn-Banach Theorem for totally convex spaces, Dem. Math. XVIII (1985), 567-588. (1985) MR0819335
  13. Pumplün D., Eilenberg-Moore algebras revisited, Seminarberichte, FB Mathematik u. Inf., Fernuniversität, 29 (1988), 97-144. (1988) 
  14. Rickart Ch.E., General theory of Banach algebras, R.E. Krieger Publ. Co. Huntington, N.Y., 1974. Zbl0095.09702
  15. Tholen W., Relative Bildverzerlegungen und algebraische Kategorien, Ph.D. thesis, U. Münster, 1974. 

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