UDC 519.54

MATHEMATICS

I. V. KARKLINISH

INDUCTIVE AND PROJECTIVE LIMITS OF COMMUTATIVE TOPOLOGICAL GROUPS

(Presented by Academician P. S. Novikov on 6 V 1966)

We shall consider three categories of commutative groups in which there exist limits of direct and inverse spectra, as well as sums and products. A topology \(t\) on a group \(X\) will be called a group topology if \((X,t)\) is a topological group. \(R\) denotes the additive group of real numbers; \(Z\) the subgroup of integers; \(T=R/Z\); \(\vartheta:R\to T\) the canonical representation, and \(W=\vartheta([-1/4,1/4])\).

1. Let \(X'\) be the group of all characters of the topological group \((X,t)\) (i.e. of its continuous representations in \(T\)); \(\langle x,x'\rangle\) is the value of the character \(x'\in X'\) at the point \(x\in X\); \(M^0=\{x'\in X':\langle M,x'\rangle\subset W\}\) for any set \(M\subset X\), and \(M'^0=\{x\in X:\langle x,M'\rangle\subset W\}\) for any set \(M'\subset X'\). A set \(M\subset X\) is called, following N. Ya. Vilenkin \((^1)\), quasi-convex if \((M^0)^0=M\). If \(\varphi\) is a continuous representation of \((X,t)\) in \((Y,v)\), then \(\varphi(M^{00})\subset(\varphi M)^{00}\) for any set \(M\subset X\), and \(\varphi^{-1}N\) is quasi-convex for any quasi-convex set \(N\subset Y\). The group \((X,t)\) is called locally quasi-convex* (an \(lc\)-group) \((^1)\) if in \((X,t)\) there exists a fundamental system of neighborhoods of zero consisting of quasi-convex sets. The topology of an \(lc\)-group is called an \(lc\)-topology. If \((X,t)\) is a separated \(lc\)-group, then for every \(x\in X\setminus\{0\}\) there exists an \(x'\in X'\) such that \(\langle x,x'\rangle\ne0\).

Theorem 1. Let \((X_\alpha,t_\alpha)_{\alpha\in A}\) be an arbitrary family of topological groups, \(X\) a group, and for every \(\alpha\in A\) let a representation \(\varphi_\alpha:X_\alpha\to X\) be defined.

Then among those group (respectively \(lc\)-) topologies on \(X\) for which all \(\varphi_\alpha\) are continuous, there exists a strongest topology \(t\) (respectively an \(lc\)-topology \(c\)), and
\[ (X,c)'=(X,t)'. \]
If \(\mathfrak B\) is a fundamental system of neighborhoods of zero in \((X,t)\), then \((U^{00})_{U\in\mathfrak B}\) is a fundamental system of neighborhoods of zero in \((X,c)\). If \((Y,v)\) is an \((lc\)-)group, then the representation \(\varphi:(X,t)\to(Y,v)\) \(((X,c)\to(Y,v))\) is continuous if and only if \(\varphi\varphi_\alpha\) is continuous for every \(\alpha\in A\).

Theorem 2. Let \((X^\alpha,t^\alpha)_{\alpha\in A}\) be an arbitrary family of \(lc\)-groups, \(X\) a group, and for every \(\alpha\in A\) let a representation \(\varphi^\alpha:X\to X^\alpha\) be defined.

Then the weakest topology \(t\) on \(X\) for which all \(\varphi^\alpha\) are continuous is an \(lc\)-topology.

A boundedness in a group \(X\) \((^1)\) is a set \(\mathfrak B X\) of subsets of \(X\) satisfying the following axioms: \(M\cup N\in\mathfrak B X\) and \(M+N\in\mathfrak B X\) for any \(M,N\in\mathfrak B X\); \(\bigcup_{x} M=X\); if \(M\subset N\) and \(N\in\mathfrak B X\), then \(M\in\mathfrak B X\); if \(M\in\mathfrak B X\), then \(-M\in\mathfrak B X\). A set \(\mathfrak B_1X\subset\mathfrak B X\) is called a base of the boundedness \(\mathfrak B X\) if for every \(M\in\mathfrak B X\) there exists an \(N\in\mathfrak B_1X\) such that \(M\subset N\). Examples of boundednesses are the sets \(b(X,t)\) of all bounded sets \((^2)\) and \(pc(X,t)\) of all precompact sets of the topological group \((X,t)\); the set \(c(X,t)\) of all bicompact sets is a base of boundedness

* Here, as everywhere below, by a “group” is meant a commutative group.

in \(X\). The group \(X\), endowed with the boundedness \(\mathfrak{B}X\), will be denoted by \([X,\mathfrak{B}X]\). A representation
\[ \varphi:[X,\mathfrak{B}X]\to [Y,\mathfrak{B}Y] \]
is called bounded if \(\varphi M\in \mathfrak{B}Y\) for every \(M\in \mathfrak{B}X\). The groups \([X,\mathfrak{B}X]\) and \([Y,\mathfrak{B}Y]\) are called isomorphic if there exists such an algebraic isomorphism \(\varphi\) of \(X\) onto \(Y\) that \(\varphi\) and \(\varphi^{-1}\) are bounded representations. On the set of all boundednesses of the group \(X\) an order relation is introduced by inclusion. A boundedness \(\mathfrak{B}X\) in a topological group \((X,t)\) is called quasi-convex if for \(\mathfrak{B}X\) there exists a base consisting of quasi-convex sets. If \(\mathfrak{B}X\) is a boundedness in \((X,t)\), then the family \((M^{00})_{M\in \mathfrak{B}X}\) is a base of a quasi-convex boundedness in \(X\). We shall denote it by \(\mathfrak{B}^{00}X\). An \(lc\)-group endowed with a quasi-convex boundedness is called an \(lcb\)-group.

Theorem 3. Let \([X_\alpha,t_\alpha,\mathfrak{B}X_\alpha]_{\alpha\in A}\) be a family of \(lcb\)-groups; let \(X\) be a group; for every \(\alpha\in A\) a representation \(\varphi_\alpha:X_\alpha\to X\) is defined, and let \(t\) be the strongest \(lc\)-topology in \(X\) for which all \(\varphi_\alpha\) are continuous.

Then in \(X\) there exists the weakest boundedness \(\mathfrak{B}X\) for which all \(\varphi_\alpha\) are bounded representations, and \((M^{00})_{M\in \mathfrak{B}X}\) is a base of the weakest quasi-convex boundedness in \(X\) for which all \(\varphi_\alpha\) are bounded representations. If \([Y,v,\mathfrak{B}Y]\) is an \(lcb\)-group, then
\[ \varphi:X\to Y \]
is a continuous bounded representation if and only if \(\varphi\varphi_\alpha\) is a continuous bounded representation for every \(\alpha\in A\).

Theorem 4. Let \([X^\alpha,t^\alpha,\mathfrak{B}X^\alpha]_{\alpha\in A}\) be a family of \(lcb\)-groups; let \(X\) be a group; for every \(\alpha\in A\) a representation
\[ \varphi^\alpha:X\to X^\alpha \]
is defined, and let \(t\) be the weakest \(lc\)-topology in \(X\) for which all \(\varphi_\alpha\) are continuous.

Then in \(X\) there exists the strongest boundedness \(\mathfrak{B}X\) for which all \(\varphi^\alpha\) are bounded representations, and moreover \(\mathfrak{B}X\) is a quasi-convex boundedness. If \([Y,v,\mathfrak{B}Y]\) is an \(lcb\)-group, then
\[ \varphi:Y\to X \]
is a continuous bounded representation if and only if \(\varphi^\alpha\varphi\) is a continuous bounded representation for every \(\alpha\in A\).

Let \(\mathcal{K}_1,\mathcal{K}_2\), and \(\mathcal{K}_3\) be, respectively, the categories of all (commutative) topological groups, \(lc\)-groups, and \(lcb\)-groups, whose morphisms are all possible continuous representations in the case of \(\mathcal{K}_1\) and \(\mathcal{K}_2\), or continuous bounded representations in the case of \(\mathcal{K}_3\). In each of these categories there exist limits of direct spectra and sums (realized on the basis of Theorem 1 or 3), and also limits of inverse spectra and products (realized on the basis of Theorem 2 or 4). Thus, the limit of the direct spectrum
\[ \{[X_\alpha,t_\alpha,\mathfrak{B}X_\alpha];\pi^\alpha_\beta\}_A \]
in \(\mathcal{K}_3\) is its limit \(X\) in the category of groups, endowed with the strongest \(lc\)-topology \(t\) and the weakest quasi-convex boundedness \(\mathfrak{B}X\), for which all canonical representations
\[ \pi_\alpha:X_\alpha\to X \]
are continuous and bounded. The sum of an arbitrary family of \(lcb\)-groups
\[ [X_\alpha,t_\alpha,\mathfrak{B}X_\alpha]\quad(\alpha\in A) \]
in \(\mathcal{K}_3\) is the sum
\[ \sum_{\alpha\in A} X_\alpha \]
of this family in the category of groups, endowed with the strongest \(lc\)-topology and the weakest quasi-convex boundedness for which all canonical representations
\[ \iota_\alpha:X_\alpha\to \sum_{\alpha\in A} X_\alpha \]
are continuous and bounded. In \(\mathcal{K}_3\) the quotient group of an \(lcb\)-group \([X,t,\mathfrak{B}X]\) by a subgroup \(Y\) is defined as the quotient group \(X/Y\), endowed with the strongest \(lc\)-topology and the weakest quasi-convex boundedness for which the canonical representation
\[ \omega:X\to X/Y \]
is continuous and bounded. Similarly, in \(\mathcal{K}_3\) the limit of an inverse spectrum and the product, which is determined by a subgroup, are realized. Many results known for limits of direct and inverse spectra in the category of groups \((^3)\) are also valid in \(\mathcal{K}_1,\mathcal{K}_2\), and \(\mathcal{K}_3\). Moreover, for some classes of spectra \((^{4,5})\) the limits have such important properties as separability, completeness, and others.

1. In this part we consider the category \(\mathcal{K}_3\). Let \([X,t,\mathfrak{B}X]\) be an \(lcb\)-group; its character group \(X'\), endowed with the \(lc\)-topology in which

a fundamental system of neighborhoods of zero consists of sets of the form \(M^0\), where \(M \in {\mathfrak B}X\), and a basis of the quasi-convex boundedness consists of sets of the form \(U^0\), where \(U\) is a neighborhood of zero in \((X,t)\), is called the group conjugate to \([X,t,{\mathfrak B}X]\), and is denoted by \([X,t,{\mathfrak B}X]'\) (1). \([X,t,{\mathfrak B}X]'\) is a separable \(lcb\)-group. If
\[ \varphi : [X,t,{\mathfrak B}X] \to [Y,v,{\mathfrak B}Y] \]
is a continuous bounded representation, then also \(\varphi' : Y' \to X'\), defined by the formula
\[ \langle x,\varphi'y'\rangle=\langle \varphi x,y'\rangle \quad (x\in X,\ y'\in Y'), \]
is a continuous bounded representation. \(\varphi'\) is called conjugate to \(\varphi\). A separable \(lcb\)-group \([X,t,{\mathfrak B}X]\) is called reflexive if the representation \(x\mapsto \langle x,\cdot\rangle\) is an isomorphism of the group \([X,t,{\mathfrak B}X]\) onto \([X,t,{\mathfrak B}X]''\).

Theorem 5. Let \(\{[X_\alpha,t_\alpha,{\mathfrak B}X_\alpha];\pi_\beta^\alpha\}_A\) be a direct spectrum in \({\mathfrak K}_3\). If \(X_\alpha' \ne \{0\}\) \((\alpha\in A)\), then
\[ \bigl(\lim_{\longrightarrow}\{[X_\alpha,t_\alpha,{\mathfrak B}X_\alpha];\pi_\beta^\alpha\}_A\bigr)' \]
is isomorphic to
\[ \lim_{\longleftarrow}\{[X_\alpha,t_\alpha,{\mathfrak B}X_\alpha]';(\pi_\alpha^\beta)'\}_A . \]

Theorem 6. Let \(\{[X^\alpha,t^\alpha,{\mathfrak B}X^\alpha];\rho_\alpha^\beta\}_A\) be an inverse spectrum in \({\mathfrak K}_3\) and
\[ \lim_{\longleftarrow}\{[X^\alpha,t^\alpha,{\mathfrak B}X^\alpha];\rho_\alpha^\beta\}_A=[X,t,{\mathfrak B}X]. \]
If the projection \(\rho^\alpha X\) is dense in \((X^\alpha,t^\alpha)\) \((\alpha\in A)\), then there exists a continuous \({\mathfrak B}\)-isomorphism of the group
\[ \lim_{\longrightarrow}\{[X^\alpha,t^\alpha,{\mathfrak B}X^\alpha)';(\rho_\alpha^\beta)'\}_A \]
onto \([X,t,{\mathfrak B}X]'\). If, moreover, all \([X^\alpha,t^\alpha,{\mathfrak B}X^\alpha]\) are reflexive, \({\mathfrak B}X^\alpha \subset pc^{00}(X^\alpha,t^\alpha)\) and \({\mathfrak B}X \supset c(X,t)\), then \([X,t,{\mathfrak B}X]'\) is isomorphic to
\[ \lim_{\longrightarrow}\{[X^\alpha,t^\alpha,{\mathfrak B}X^\alpha]';(\rho_\alpha^\beta)'\}_A . \]

1. A group of the I (II) kind (6) will mean a topological group isomorphic to the limit of a direct* (inverse) countable spectrum of separable locally bicompact groups (the formation of these limits in the categories of topological spaces \({\mathfrak K}_1\) or \({\mathfrak K}_2\) leads to isomorphic topological groups; in the case \({\mathfrak K}_3\) every topological group \((X,t)\) is endowed with the quasi-convex boundedness \(c^{00}(X,t)\)). The sum (product) of a countable family of real lines is not a locally bicompact group of the I (II) kind. This shows that there exist groups of the I (II) kind which are not groups of the II (I) kind. Denote by \(A\) and \(B\) the classes of all groups of the I and, respectively, II kind. \(A'=B\), \(B'=A\), and every group of these classes is \(c^{00}\)-reflexive. The classes \(A\) and \(B\) are closed with respect to passage to closed subgroups and to quotient groups by them. If \(X\) is a group of the I or II kind and \(Y\) is its closed subgroup, then
   \[ (X/Y)' \cong Y^0 \]
   and
   \[ Y' \cong X'/Y^0 \]
   (an isomorphism in \({\mathfrak K}_3\)). For any \(y'\in Y'\) and \(x\in X\setminus Y\) there exists an \(x'\in X'\) such that \(x'|_Y=y'\) and \(\langle x,x'\rangle\ne 0\). The last result (for groups of the I kind) is also contained in (7), but is proved there by another method (we used the fact that a quotient group of a group of the II kind is a group of the II kind).

I express my sincere gratitude to Prof. D. A. Raikov for valuable discussions.

Latvian State University
named after P. Stuchka

Received
6 V 1966

References

1. N. Ya. Vilenkin, Izv. Akad. Nauk SSSR, Ser. Mat., 15, 5, 439 (1951).
2. J. Hejcman, Czechoslovak Math. J., 9 (84), 4, 544 (1959).
3. N. Steenrod, S. Eilenberg, Foundations of Algebraic Topology, Moscow, 1958.
4. I. V. Karklin’sh, Latvian Math. Collection, 5, 1, 57 (1965).
5. I. V. Karklin’sh, Latvian Math. Yearbook, 347 (1966).
6. S. Kaplan, Duke Math. J., 17, 4, 419 (1950).
7. N. Th. Varopoulos, Proc. Cambridge Phil. Soc., 60, 3, 465 (1964).

* The direct spectrum \(\{X_\alpha;\pi_\beta^\alpha\}\) is considered here with monomorphisms \(\pi_\beta^\alpha\).