Unitary representability of free abelian topological groups

Long colimits of topological groups II: Free groups and vector spaces

We give a complete description of the topological spaces X such that the free abelian topological group A(X) embeds into the free abelian topological group A(I) on the closed unit interval. In particular, the free abelian topological group A(X) on any finite-dimensional compact metrizable space X embeds into A(I). To obtain our description, we study similar embeddings of the free locally convex spaces and continuous surjections between the spaces of continuous functions with the pointwise topology. Proofs are based on the classical Kolmogorov's Superposition Theorem.

Notes on countable tightness of the subspaces of free (Abelian) topological groups

The topological properties of q-spaces in free topological groups

Let n be a positive integer, Bn the closed unit ball in Euclidean n-space, and X any countable CW-complex of dimension at most n. It is shown that the free Abelian topological group on Bn, F(Bn), has F(X) as a closed subgroup. It is also shown that for every differentiable manifold Y of dimension at most n, F(Y) is a closed subgroup of F(Bn).

The kR-property of free Abelian topological groups and products of sequential fans

Weakly complete free topological groups

Free abelian topological groups and collapsing maps

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

We prove that the additive group (E*, τ k (E)) of an $${\mathcal {L}_\infty}$$ -Banach space E, with the topology τ k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ k (E)) is uniformly homeomorphic to a subset of $${\ell_2^\kappa}$$ for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group FA has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that FA is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and FA is a free A-groupp of Lyndon' type, then FA has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then FA has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, FA has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if FQ has a faithful Magnus representation in Q[[Y]].

Most of the results in this paper concern relationships between sequential properties of a pointed topological space (X, p) and sequential properties of the Graev free topological group on X. In particular, it is shown that the free group over a sequential ${k_\omega }$-space is sequential, and that a nondiscrete sequential free group has sequential order equal to ${\omega _1}$ (the first uncountable ordinal). The free topological group on a space X which includes a convergent sequence contains a closed subspace homeomorphic to ${S_\omega }$, a previously studied homogeneous, zero-dimensional sequential space. Finally, it is shown that there is no topological group homeomorphic to ${S_\omega }$.

We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to ℤ (known classically for ℵ1-free, recently for ℵn-free). We succeed to prove the existence of even $${\aleph _{{\omega _1} \cdot n}}$$ -free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every $${\aleph _{{\omega _1} \cdot \omega }}$$ -free Abelian group has non-trivial homomorphisms to ℤ.

Several authors have recently studied the problem of finding circumstances under which the amenability of a semigroup is inherited by a subsemigroup. It is known ([2] and [3]) that subgroups of an amenable group are amenable, but that in general subsemigroups of an amenable semigroup are not. We show here that subsemigroups of an amenbable group need not be amenable. Frey proved in [4] that subsemigroups of a group inherit amenability and the group contains no copy of the free semigroup S on two generators (S is not amenable). The problem is discussed in detail in [51. It is known [11 that a group generated by a copy of S contained in it need not be free; the question is whether it can be far enough from free to be amenable. Since solvable groups are amenable [6], our claim that subsemigroups of amenable groups need not be amenable will be established we can embed S in a solvable group. We proceed to do this. Granirer conjectured in his review [MR 35 #3423] of Wilde and Witz [7] that a group is amenable and it contains no copy of S. It will follow that the only if part of this conjecture is quite false. Let B be the free additive abelian group on the family of generators {b(m, n) }, m, n integers. Any permutation of these generators induces an automorphism of B. Let x and y be the automorphisms induced by b(m, n)F->b(m+1, n) and b(m, n)'->b(m, n+1), respectively. It is easy to see that the subgroup C of Aut B generated by x and y is the free abelian group on the generators x, y. We write C additively, and we indicate the action of an element c of C on an element b of B by be. Let be the normal product of B by C. consists of pairs [c, b] and the multiplication is given by -[c, b1i [c, b] [cl+c, b1+b]. c1->[c, 0], bF-*[O, b] are embeddings of C, B respectively into G, and B is thus a normal abelian subgroup with quotient G/B-C. Hence, is solvable and in fact the second derived group G of is trivial. The same holds for every subgroup of G. Now, let s= [x, b(O, 0)] and t= [y, b(0, 0)].

The topological fundamental group and free topological groups