Topological algebra

2020 Mathematics Subject Classification: Primary: 46H [MSN][ZBL]

A universal algebra that is a topological space such that the algebraic operations are continuous.

An algebra (in the sense of a "ring with operators" ) $A$ over a topological field or commutative ring $R$ that is a topological space in which the operations of addition and multiplication, as well as the mapping $R\times A\to A$ ($(r,a)\to ra$), are continuous. A Banach algebra is an example of a topological algebra over the field of complex numbers.

The branch of algebra which studies topological algebraic structures, i.e. groups, semi-groups, rings, lattices, vector spaces, modules, and others, equipped with topologies in which the algebraic operations considered are continuous.

The notion of a topological group arose in connection with the study of groups of continuous transformations. Thus, in the second half of the 19th century, S. Lie and his school developed the theory of an important class of topological groups (groups of differentiable transformations from a manifold into itself), which were subsequently called Lie groups (cf. Lie group). The study of general topological groups began in the 1920's (cf. [Sc], [Le]). By the early 1930's, topological groups, rings and fields were being systematically investigated.

A.N. Kolmogorov [Ko] developed an axiomatic approach to the study of topological projective geometries. Their classification depends essentially on the description of locally compact skew-fields (cf. Locally compact skew-field). A complete description of connected locally compact skew-fields was given in 1932 by L.S. Pontryagin (cf. [Po], Chapt. 4).

Numerous problems in analysis led to the general definition of a Banach space (cf. also [Bo]), which motivated the systematic study of topological modules over topological rings and Banach algebras (cf. [GeRaSh], [Na]).

The basic branches of topological algebra at present are: topological groups and their generalizations (in particular, topological semi-groups and quasi-groups, cf. Topological group); topological rings (cf. Topological ring) (in particular, topological fields and skew-fields) and topological modules (cf. Topological module) over them (in particular, topological vector spaces, cf. Topological vector space); topological lattices (in particular, topological projective planes); and topological universal algebras (cf. [Bo], [ArVoMi], [Sk], [Ma]).

The following lines of research in topological algebra can be distinguished: the existence of topologies in algebraic systems (groups, rings and others), to transform them into topological algebraic systems with special properties; questions about extending topologies to extensions of algebraic systems, and the possibility of imbedding them into topological algebraic systems of specific classes; properties of the topology of a topological algebraic system, in particular the possibility of specifying the topology by a metric or norm; the construction of different classes of topological algebraic systems (including the theory of radicals of a topological algebraic system); free topological algebraic systems; and questions of duality of topological algebraic systems.

Comments

The concept of a topological semi-group has been studied in functional analysis and topological algebra (in the sense of 3) above) since about 1950. Cf. [HiPh]–[HoLaPy].

An important aspect of topological algebra is the interplay between topological and algebraic structures on the same set; in particular, the way in which otherwise inequivalent topological properties are forced to become equivalent by the presence of a compatible algebraic structure, and vice versa. For example, if the underlying topological space of a topological group satisfies the $T_0$ separation axiom, then it is automatically a Hausdorff space (and even completely regular). Similarly, if the underlying space of a topological ring (with $1$) is compact and Hausdorff, then it is also zero-dimensional. In the other direction, if a lattice admits a compact Hausdorff topology compatible with its lattice structure, then it is necessarily complete (i.e. has suprema and infima of arbitrary subsets). In this case (and even for topological semi-lattices), the topology is uniquely determined by the algebraic structure (plus the requirement that it should be compact and Hausdorff). For more information on topological lattices, see [GiHoKeLaMiSc], [Jo].

References

[ArVoMi]	V.I. Arnautov, M.I. Vodinchar, A.V. Mikhalev, "Introduction to the theory of topological rings and modules", Kishinev (1981) (In Russian) MR0633148 Zbl 0518.16013
[Bo]	N. Bourbaki, "Topological vector spaces", Springer (1987) (Translated from French) MR0910295 Zbl 0622.46001
[GeRaSh]	I.M. Gel'fand, D.A. Raikov, G.E. Shilov, "Commutative normed rings", Chelsea, reprint (1964) (Translated from Russian) Zbl 0201.45703
[GiHoKeLaMiSc]	G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.V. Mislove, D.S. Scott, "A compendium of continuous lattices", Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001
[Gl]	V.M. Glushkov, "Structure of locally bicompact groups and Hilbert's fifth problem" Uspekhi Mat. Nauk, 12 : 2 (1957) pp. 3–41 (In Russian)
[Go]	J.A. Goldstein, "Semigroups of linear operators and applications", Oxford Univ. Press (1985) MR0790497 Zbl 0592.47034
[Gu]	A. Guichardet, "Special topics in topological algebras", Gordon & Breach (1968) MR0243351
[HiPh]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups", Amer. Math. Soc. (1957) MR0089373 Zbl 0392.46001 Zbl 0033.06501
[Ho]	K.H. Hofmann, "Topological semigroups: history, theory, applications" Jahresber. Deutsch. Math.-Verein., 78 (1976) pp. 9–59 MR0424997 Zbl 0361.22001
[HoLaPy]	K.H. Hofmann (ed.) J.D. Lawson (ed.) J.S. Pym (ed.), The analytical and topological theory of semigroups, de Gruyter (1990) MR1072781 Zbl 0702.00013
[HoMo]	K.H. Hofmann, P.S. Mostert, "Elements of compact semigroups", C.E. Merrill (1966) MR0209387 Zbl 0161.01901
[Jo]	P.T. Johnstone, "Stone spaces", Cambridge Univ. Press (1982) MR0698074 Zbl 0499.54001
[Ko]	A.N. Kolmogorov, "Zur Begründung der projektiven Geometrie" Ann. of Math., 33 (1932) pp. 175–176 MR1503044
[Le]	F. Leja, "Sur la notion du groupe topologique" Fund. Math., 9 (1927) pp. 37–44 JFM Zbl 53.0110.03
[Ma]	A.I. [A.I. Mal'tsev] Mal'cev, "The metamathematics of algebraic systems, Collected papers 1936–1967", North-Holland (1971) (Translated from Russian) Zbl 0231.02002
[Na]	M.A. Naimark, "Normed rings", Reidel (1984) (Translated from Russian) MR1292007 MR0355601 MR0355602 MR0205093 MR0110956 MR0090786 MR0026763 Zbl 0218.46042 Zbl 0137.31703 Zbl 0089.10102 Zbl 0073.08902
[Po]	L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0882.01025 Zbl 0534.22001 Zbl 0079.03903 Zbl 0058.26003 Zbl 0022.17104 Zbl 0016.20305 Zbl 0016.20304 Zbl 0015.24901
[Sc]	O. Schreier, "Abstrakte kontinuierliche Gruppen" Abh. Math. Sem. Univ. Hamburg, 4 (1926) pp. 15–32 JFM Zbl 51.0112.04
[Sk]	L.A. Skornyakov, "Topological projective planes" Trudy Moskov. Mat. Obshch., 3 (1954) pp. 347–373 (In Russian) MR0063026 Zbl 0057.36201 Zbl 0055.16902