On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

Conditions for equality in Anderson’s theorem

Let G be a reflexive topological group, and Gits group of characters, endowed with the compact open topology. We prove that the evaluation mapping from Gx G into the torus T is continuous if and only if G is locally compact. This is an analogue of a well-known theorem of Arens on admissible topologies on C(X) . DEFINITIONS AND REMARKS Let X, Y be topological spaces, and let Z be a subset of yX . A topology on Z is said to be admissible if the evaluation mapping from the product Z x X into Y, defined by w(f, x) = f(x), is continuous. Let (S, V) = {f E Z; f(S) C V}. The family {(S, V)}, where S runs over the collection of all compact subsets of X and V runs over a basis of open sets in Y, is a subbase for the compact open topology on Z. An admissible topology on Z must be finer than the compact open topology [8]. A result of Arens states that the existence of a coarsest admissible topology for the class of real continuous functions on a completely regular space X is equivalent to X being locally compact [1]. In this paper we are interested in reflexive topological groups. Answering in the negative a question of Megrelishvili [7], we will see that the evaluation map for those groups need not be continuous. In fact, as specified in the theorem, the continuity of the evaluation characterizes locally compact Hausdorff abelian groups among reflexive groups. We prove this result by means of convergence spaces. For an account of theory of convergence spaces the reader is referred to [3, 4]. We only give here needed definitions. Let G be a Hausdorff topological abelian group, and let FG be the set of all continuous homomorphisms from G into the unidimensional torus T. If addition is defined pointwise in FG, it becomes an abelian group. The continuous Received by the editors October 5, 1993 and, in revised form, April 25, 1994; the contents of this paper were presented to the First Ibero-American Conference on Topology and its Applications in Benicassim, Spain, on March 28-30, 1995. 1991 Mathematics Subject Classification. Primary 22A05.

We give elementary proofs of two theorems concerning bounds on the maximum argument of the eigenvalues of a product of two unitary matrices--one by Childs et al. [J. Mod. Phys. 47, 155 (2000)] and the other one by Chau [Quant. Inf. Comp. 11, 721 (2011)]. Our proofs have the advantages that the necessary and sufficient conditions for equalities are apparent and that they can be readily generalized to the case of infinite-dimensional unitary operators. © 2011 Chau and Lam; licensee Springer.

In this note we obtain upper and lower bounds for the kth largest number in a set of real numbers in terms of their mean and standard deviation. For each inequality necessary and sufficient conditions for equality are given.

This paper presents an analogue of the spectral mapping theorem for the pseudo S-spectra of quaternionic operators. The theorem is formulated using two set inclusions, and necessary and sufficient conditions for equality are identified. The standard mapping theorem appears as a special case of this theorem. A weaker version is also derived with potential applications in various scenarios. The pseudo S-spectra of rank-one and rank-one nilpotent quaternionic operators are explicitly characterized. Finally, the mapping theorem is further demonstrated for specific classes of quaternionic operators, including rank-one nilpotent, normal, self-adjoint, orthogonal projection, and left multiplication operators.

The u-invariant of a field F, u = u(F), is defined to be the maximum of the dimensions of anisotropic quadratic forms over F. If F is a non-formally real field with a finite number q of square classes then it is known that u ≦ q. The purpose of this note is to give some necessary and sufficient conditions for equality in terms of the structure of the Witt ring of F.

Duality theory for convergence groups

Let G be a compact connected group and let H be a locally compact abelian group. Denote by C e ( G , H ) {C_e}(G,H) the space of all identity preserving continuous functions from G to H with the compact-open topology, and denote by Hom(G, H) the space of all homomorphisms in C e ( G , H ) {C_e}(G,H) . We prove that C e ( G , H ) {C_e}(G,H) is isomorphic to V × Hom ( G , H ) V \times {\text {Hom}}(G,H) , where V is a topological vector space. This is used to prove that every element of C e ( G , H ) {C_e}(G,H) is homotopic to precisely one element of Hom(G, H). We also prove that the fundamental group of H is isomorphic to Hom(K, H), K being the circle group, that π n ( H ) = 0 {\pi _n}(H) = 0 for n ≧ 2 n \geqq 2 , and that a compact connected abelian group admits essentially only one H-space structure.

A new topology on the universal path space

LetN be a right topological near-ring. A topologicalN-group is a pair (G, μ) whereG is a topological group and μ is a continuous function fromN ×G toG such that μ(a+b, x)=μ(a, x)+μ(b, x) and μ(ab, x)=μ(a, μ(b, x)) for alla, b∈N andx∈G. We show first that for locally compact HausdorffG, the problem of determining all such functions μ is equivalent to determining all homomorphisms from the topological near-ringN into the topological near-ringN(G), of all continuous selfmaps ofG, where addition of functions is pointwise, multiplication is composition and the topology onN(G) is the compact-open topology. This result is then applied to determine precisely what form certain functions μ must take in order that (G, μ) be a topologicalN-group when the near-ringN is a topological near-ring of continuous functions.

For a domain  in C, we denote by Aut() the group of holomorphic automorphisms of . It is obvious that Aut() is a topological group with respect to the law of composition and the compact-open topology. In particular, it is a theorem of H. Cartan that Aut() is in fact a Lie group if  is bounded. In light of the outstanding question “Which domains possess noncompact automorphism group?” there is much interest focused upon the existence and nonexistence of orbits of the automorphism group action accumulating at a given boundary point. The well-known Greene–Krantz conjecture belongs to such a line of research. In this paper, we discuss the finite-type boundary points that repel automorphism orbits. Denote by τ6(q) the D’Angelo type (see [10]) at q of the real hypersurface 6 in C. Now we present our main theorem.

Chapter 42 - Recent results and open questions relating Chu duality and Bohr compactifications of locally compact groups

This paper is devoted to an investigation of a topological ring of analytic functions. Specifically, this ring, denoted by R, is the set of functions analytic on the unit disc with the usual addition and scalar multiplication, the Hadamard product for its ring multiplication, and the compact-open topology. The ring R is identified algebrai cally with a subring RA of the ring of continuous functions on the non-negative integers X. The operations in fT are the usual pointwise operations, ana the structure of R is determined by considering its isomorph iT. In Chapter I we are concerned with the problems of identifying the maximal ideal space of R and describing the maximal ideals intrinsically. We first show, using theorems on general rings of continuous functions, that the maximal ideals are in one-to-one correspondence with the points of the Stone-Cech compactification px of X. We next give an intrinsic description of the maximal ideals, using the properties of the power series expansions for analytic functions. Using this description we strengthen the prev ious theorem appreciably and show that the maximal ideal space with the hul1-kernel topology is homeomorphic to pX. Finally, the Hadamard product is used to give a simple iv characterization of the dual space of the topological linear space of analytic functions on the unit disc. This dual space is isomorphic to the set of functions in R whose radius of convergence exceeds one, which is exactly the intersection of the maximal ideals corresponding to points of pX -X (the dense maximal ideals of R) . In Chapter II we continue the investigation of the maximal ideals by studying the structure of their associated residue class rings. The complex number field 0 is isomorphically embedded in R/M, where M is a maximal ideal of R. If M corresponds to a point of X, then R/M and the isomorph 0* of 0 are identical; whereas, if M corresponds to a point of ^X -X, then R/M is a transcendental extension of 0* having transcendence degree c, the cardinality of the continuum. Moreover, we show, in the second case, that R/M is algebraically closed. Using theorems on transcendental extensions and algebraically closed fields, we show that, in either case, R/M and 0 are isomorphic fields. The two classes of maximal ideals are distinguished by the fact that their residue class rings admit radically different types of complex-valued isomorphisms. In Chapter III we are concerned primarily with the structure of the closed ideals of R. The basic tool used is the rotational completeness theorem for analytic v functions, which we proved using the methods and results of harmonic analysis. We show that the closure of every principal ideal is principal, give a necessary and suffi cient condition that a principal ideal be closed, and show that every closed ideal is a principal ideal generated by an idempotent element of R. Using these theorems we indicate connections with the general theory of dual rings, of which R is an example, and raise several questions for further i investigation in the direction of releasing some of the restrictions with which most of the results so far have been obtained.

LetC(X)be the space of real-valued continuous functions on a Hausdorff completely regular topological spaceX. endowed with the compact-open topology. In this paper necessary and sufficient conditions are given for a subspace ofC(X)to be the range of a pointwise contractive projection inC(X).

Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$, denoted $\mathrm{PMap}(S)$, detects the homeomorphism type of $S$. As a corollary, every automorphism of $\mathrm{PMap}(S)$ is induced by a homeomorphism, which extends a theorem of Ivanov from the finite-type setting. In the process of proving these results, we show that $\mathrm{PMap}(S)$ is residually finite if and only if $S$ has finite genus, demonstrating that the algebraic structure of $\mathrm{PMap}(S)$ can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that $\mathrm{Map}(S)$ fails to be residually finite for any infinite-type surface $S$. In addition, we give a topological generating set for $\mathrm{PMap}(S)$ equipped with the compact-open topology. In particular, if $S$ has at most one end accumulated by genus, then $\mathrm{PMap}(S)$ is topologically generated by Dehn twists, otherwise the Dehn twists along with handle shifts topologically generate.