# Totally convex algebras

Commentationes Mathematicae Universitatis Carolinae (1992)

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

## Access Full Article

top## Abstract

top## How to cite

topPumplün, 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

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

## NotesEmbed ?

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