On unitary representability of topological groups

The topological fundamental group and free topological groups

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

Weakly complete free topological groups

The topological properties of q-spaces in free topological groups

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

SynopsisThe compact weak topology (kw) on a Banach space is defined as the finest topology that agrees with the weak topology on weakly compact subsets. It appears in a natural manner in the study of certain classes of continuous and holomorphic maps between Banach spaces. In this paper we treat the kw topology and the finest locally convex topology contained in kw, which we call the ckw topology. We prove that kw = ckw if and only if the space is reflexive or Schur, and we derive characterisations of Banach spaces not containing l1, and of other classes of Banach spaces, in terms of these topologies. We also show that ckw is the topology of uniform convergence on (L)-subsets of the dual space. As a consequence, Banach spaces with the reciprocal Dunford–Pettis property are characterised.

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 ${S_\omega }$.

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

Minimal cusco maps have applications in functional analysis, in optimization, in the study of weak Asplund spaces, in the study of differentiability of functions, etc. It is important to know their topological properties. Let X be a Hausdorff topological space, MC(X) be the space of minimal cusco maps with values in R and ?UC be the topology of uniform convergence on compacta. We study completemetrizability and cardinal invariants of (MC(X), ?UC). We prove that for two nondiscrete locally compact second countable spaces X and Y, (MC(X), ?UC) and (MC(Y), ?UC) are homeomorphic and they are homeomorphic to the space C(Ic) of continuous real-valued functions on Ic with the topology of uniform convergence.

All spaces considered in this paper are assumed to be metrizable. A compactum is a compact space. A continuum is a connected compactum, and a mapping is a continuous function. For a space X we denote by C(X) the space of all real-valued mappings on X with the topology of uniform convergence. Then by Milutin's interesting work [8], we have known that for each pair of uncountable compacta X and Y, C{X) is linearly isomorphic to C(Y) (see [12] for the detailsand generalizations). On the other hand, for space X we denote by CP(X) the space of all real-valued mappings on X with the topology of pointwise convergence. Spaces X and Y are said to be l-equivalent[1] provided that CP(X) is linearly isomorphic to CP(Y), written CP(X) = CP(Y). Recently, Pavlovskii [11] showed the following.

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let G \mathcal {G} be a topological group, and A \mathcal {A} a unital symmetric C ∗ C^* -subalgebra of U C ( G ) \mathrm {UC}(\mathcal {G}) , the algebra of bounded uniformly continuous functions on G \mathcal {G} . Generalizing the notion of a stable metric, we study A \mathcal {A} -metrics δ \delta , i.e., the function δ ( e , ⋅ ) \delta (e, \cdot ) belongs to A \mathcal {A} ; the case A = W A P ( G ) \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}) , the algebra of weakly almost periodic functions on G \mathcal {G} , recovers stability. If the topology of G G is induced by a left invariant metric d d , we prove that A \mathcal {A} determines the topology of G \mathcal {G} if and only if d d is uniformly equivalent to a left invariant A \mathcal {A} -metric. As an application, we show that the additive group of C [ 0 , 1 ] C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [ Topological transformation groups: selected topics , Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now G \mathcal {G} be a metric group, and assume A ⊆ L U C ( G ) \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}) , the algebra of bounded left uniformly continuous functions on G \mathcal {G} , is a unital C ∗ C^* -algebra which is the uniform closure of coefficients of representations of G \mathcal {G} on members of F \mathscr {F} , where F \mathscr {F} is a class of Banach spaces closed under ℓ 2 \ell _2 -direct sums. We prove that A \mathcal {A} determines the topology of G \mathcal {G} if and only if G \mathcal {G} embeds into the isometry group of a member of F \mathscr {F} , equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability.

for t E T and s E S. This definition can be restated in equivalent ways in terms of particular flows. A flow (X, T) is a transformation group [8] where X is a topological space and Tis the additive group of the reals. We designate the action of t E T on x eX by x x,. A point xeX is said to be almost periodic if for each neighborhood U of x there is a relatively dense set S c T such that x. E U if s E S. We let B be the Banach space of uniformly continuous bounded real valued functions of the reals. The norm of an element of B shall be the supremum of its absolute values. We let (B, T) be the flow defined by x -) x, where x,(s) = x(t + s). Uniformly almost periodic functions may be defined in any one of the three following ways. (1) A function x in B is uniformly almost periodic if x is an almost periodic point of (B, T). (2) A function x in B is uniformly almost periodic if its orbit closure in (B,T) is compact. (3) A function x in B is uniformly almost periodic if it may be extended to a continuous function on the Bohr compactification of the reals. The fact that these three statements are equivalent may be found in [3; 12]. For many notions of functions being recursive under translation the compactopen topology is more natural than the topology of uniform convergence. With this in mind we let C be the set of continuous real valued functions of the reals with the compact open topology. This topology is equivalent to the topology induced by the metric

The topologies induced by two families of seminorms on a vector space of functions g: R + x R + x El -* El are compared. It is found that the continuous dependence of solutions of the Volterra equation x(t) f(t) + 5 , g(t, s, x(s)) ds does not hold for the weaker topology. This result corrects an error in the book of Miller, Benjamin, Menlo Park, Calif., 1971. The purpose of this paper is to give a counterexample to a proposition of R. K. Miller, concerning continuous dependence of solutions of Volterra integral equations. Consider the nonlinear Volterra equation x(t) = f(t) + f g(t, s, x(s)) ds, (E) where f E C(R +; E) (that is, f is a continuous function from R + = [0, oo) to the n-dimensional Euclidean space) and g is a function defined on R + X R + x En with values in En. In his book [2] Miller defines a vector space g of functions g: R + x R + x En--> En which satisfy certain hypotheses. Then he defines a family of seminorms on g (see (2) below) and proves, roughly speaking, that the solutions x of (E) depend continuously on the functionsf and g, where the functionsf belong to C(R +; En) topologized with the topology of uniform convergence on compact subsets and the functions g belong to 9 topologized with the locally convex vector topology induced by the seminorms mentioned above (cf. Theorem II.4.2 and its corollaries [2, pp. 108-116]). Miller gives an alternative characterization of the seminorms (see (4) below). In this note I will prove that this characterization is erroneous and that the seminorms on the right side of (4) lead to a topology on g for which the continuity theorems in [2] do not hold. The exact definition of g is given in [2, pp. 106-108]. For the purpose of this note it suffices to know that 9 contains all functions of the type g(t, s, x) = a(t s)h(s, x), (1) where h is a continuous function R + x En---En, and a is a locally integrable real valued function which vanishes on (oo, 0). The seminorms on 9 are defined in the following way (cf. [2, p. 107]): For every g E 9, every positive real number K and every bounded set B in Et, p(g, K, B) is defined by p(g, K, B) = sup{f Ig(t, s, p(s))I ds|t E[0, K], q E C([0, K]; B)}, (2) Received by the editors May 9, 1980. AMS (MOS) subject classifications (1970). Primary 45D05.

The study of the hypercyclicity of an operator is an old problem in mathematics; it goes back to a paper of Birkhoff in 1929 proving the hypercyclicity of the translation operators in the space of all entire functions with the topology of uniform convergence on compact subsets. This article studies the hypercyclicity of translation operators in some general reproducing kernel Hilbert spaces of entire functions. These spaces are obtained by duality in a complex separable Hilbert space H by means of an analytic H-valued kernel. A link with the theory of de Branges spaces is also established. An illustrative example taken from the Hamburger moment problem theory is included.

If H denotes the functions analytic in the open unit disk with the topology of uniform convergence on compact subsets, both the Tsuji functions in H and the functions in H with nonempty Tsuji sets comprise sets of first category in H.A question is posed about the category of a class of functions containing the Tsuji functions.