The objective of this thesis is to provide a detailed analysis of a new invariant for Grothendieck topoi in the special case of the topos of continuous G-sets and continuous G-equivariant maps. We use a well-known site to present the isotropy group in elementary terms, as systems of right cosets of open subgroups of G. We establish properties of the the isotropy group for an arbitrary topological group and use the developed theory to compute the isotropy group for the Schanuel topos.

The purpose of this paper is to develop an invariant integral for a compact topological group and, then to use that integral to prove the fundamental Peter-Weyl Theorem.

A topological group G is known as a B(𝑎) group if every continuous and almost open homomorphism from G onto a Hausdorff group is open. The permanence properties of the category of B(𝑎) groups are investigated and an internal characterization of such groups is established. Extensions of the closed graph and open mapping theorem are proved, employing this and related categories of groups. A similar concept is defined for topological semigroups, and further extensions of the open mapping and closed graph theorem are proved for them. / Thesis / Doctor of Philosophy (PhD)

For a topological space X, the fundamental group can be topologized as a quotient of the path space with the compact-open topology. For one-dimensional or planar Peano continua, the fundamental group with this topology is a topological group if and only if it is semilocally simply connected. In particular, we demonstrate that the group operation is not continuous in this setting.

É bem conhecido o fato de que todo grupo compacto tem sequências não triviais convergentes. A existência de grupos enumeravelmente compactos sem sequências não triviais convergentes, foi provada usando axiomas adicionais à axiomática usual ZFC: A. Hajnal e I. Juhász sob CH, E. K. van Douwen sob MA, A. H. Tomita sob MA(sigma-centrada) e R.E. Madariaga-Garcia e A. H. Tomita usando ultrafiltros seletivos. Neste trabalho, estudaremos algumas construções recentes relacionadas com as citadas acima, usando o Axioma de Martin, ultrafiltros seletivos e forcing. Essas construções estão relacionadas com algumas questões indicadas por A.D. Wallace, E. van Douwen, M. Tkachenko, D. Dikranjan e D. Shakhmatov / It is well known that every compact group has non-trivial convergent sequences. The existence of countably compact groups without non-trivial convergent sequences was proved using extra set-theoretical assumptions: A. Hajnal and I. Juhasz under CH, E. K. van Douwen under MA, A.H.Tomita under MA(centered) and R.E.Madariaga-Garcia and A.H. Tomita using a selective ultrafilter. I n this work, we study some recent constructions related to the ones given above using Martin Axiom, selective ultrafilters and forcing, related to questions raised by A.D. Wallace, E. van Douwen, M. Tkacenko, D. Dikranjan and D. Shakhmatov.

É bem conhecido o fato de que todo grupo compacto tem sequências não triviais convergentes. A existência de grupos enumeravelmente compactos sem sequências não triviais convergentes, foi provada usando axiomas adicionais à axiomática usual ZFC: A. Hajnal e I. Juhász sob CH, E. K. van Douwen sob MA, A. H. Tomita sob MA(sigma-centrada) e R.E. Madariaga-Garcia e A. H. Tomita usando ultrafiltros seletivos. Neste trabalho, estudaremos algumas construções recentes relacionadas com as citadas acima, usando o Axioma de Martin, ultrafiltros seletivos e forcing. Essas construções estão relacionadas com algumas questões indicadas por A.D. Wallace, E. van Douwen, M. Tkachenko, D. Dikranjan e D. Shakhmatov / It is well known that every compact group has non-trivial convergent sequences. The existence of countably compact groups without non-trivial convergent sequences was proved using extra set-theoretical assumptions: A. Hajnal and I. Juhasz under CH, E. K. van Douwen under MA, A.H.Tomita under MA(centered) and R.E.Madariaga-Garcia and A.H. Tomita using a selective ultrafilter. I n this work, we study some recent constructions related to the ones given above using Martin Axiom, selective ultrafilters and forcing, related to questions raised by A.D. Wallace, E. van Douwen, M. Tkacenko, D. Dikranjan and D. Shakhmatov.

The main theme of this thesis is to study the isometric algebra isomorphisms and the bipositive algebra isomorphisms between various Banach algebras associated with locally compact groups. Let $LUC(G)$ denote the $C^*$-algebra of left uniformly continuous functions with the uniform norm and let $C_0(G)^{\perp}$ denote the annihilator of $C_0(G)$ in $LUC(G)^*$. In Chapter 2 of this thesis, among other results, we show that if $G$ is a locally compact group and $H$ is a discrete group then whenever there exists a weak-star continuous isometric isomorphism between $C_0(G)^{\perp}$ and $C_0(H)^{\perp}$, $G$ is isomorphic to $H$ as a topological group. In particular, when $H$ is discrete $C_0(H)^{\perp}$ determines $H$ within the class of locally compact topological groups. In Chapter 3 of this thesis, we show that if $M(G,\omega_1)$ (the weighted measure algebra on $G$) is isometrically algebra isomorphic to $M(H,\omega_2)$, then the underlying weighted groups are isomorphic, i.e. there exists an isomorphism of topological groups $\phi:G\to H$ such that $\small{\displaystyle{\frac{\omega_1}{\omega_2\circ\phi}}}$ is multiplicative. Similarly, we show that any weighted locally compact group $(G,\omega)$ is completely determined by its Beurling group algebra $L^1(G,\omega)$, $LUC(G,\omega^{-1})^*$ and $L^1(G,\omega)^{**}$, when the two last algebras are equipped with an Arens product. Here, $LUC(G,\omega^{-1})$ is the weighted analogue of $LUC(G)$, for weighted locally compact groups. In Chapter 4 of this thesis, we show that the order structure combined with the algebra structure of each of the Banach algebras $L^1(G,\omega)$, $M(G,\omega)$, $LUC(G,\omega^{-1})^*$ and $L^1(G,\omega)^{**}$ completely determines the underlying topological group structure together with a constraint on the weight. In particular, we obtain new proofs for a previously known result of Kawada and results of Farhadi as special cases of our results. Finally, we provide an example of a bipositive algebra isomorphism between Beurling measure algebras that is not an isometry. We conclude this thesis with a selective list of open problems. / February 2016

Let H be a separable infinite dimensional complex Hilbert space, let U(H) be the Polish topological group of unitary operators on H, let G be a Polish topological group and φ:G→U(H) an algebraic isomorphism. Then φ is a topological isomorphism. The same theorem holds for the projective unitary group, for the group of *-automorphisms of L(H) and for the complex isometry group. If H is a separable real Hilbert space with dim(H)≥3, the theorem is also true for the orthogonal group O(H), for the projective orthogonal group and for the real isometry group. The theorem fails for U(H) if H is finite dimensional complex Hilbert space.

No description available.

We extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group G is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set G. As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.