Limits in Function Spaces and Compact Groups ∗
نویسنده
چکیده
If B is an infinite subset of ω and X is a topological group, let CX B be the set of all x ∈ X such that 〈xn : n ∈ B〉 converges to 1. If F is a filter of infinite sets, let DX F = ⋃{CX B : B ∈ F}. The CX B and DX F are subgroups of X when X is abelian. In the circle group T, it is known that CX B always has measure 0. We show that there is a filter F such that DT F has measure 0 but is not contained in any CX B . There is another filter G such that DX G = T. We also describe the relationship between DT F and the DX F for arbitrary compact groups X.
منابع مشابه
Shift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups
We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...
متن کاملAbstract structure of partial function $*$-algebras over semi-direct product of locally compact groups
This article presents a unified approach to the abstract notions of partial convolution and involution in $L^p$-function spaces over semi-direct product of locally compact groups. Let $H$ and $K$ be locally compact groups and $tau:Hto Aut(K)$ be a continuous homomorphism. Let $G_tau=Hltimes_tau K$ be the semi-direct product of $H$ and $K$ with respect to $tau$. We define left and right $tau$-c...
متن کاملCompact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions
We characterize compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.
متن کاملThe concentration function problem for $G$-spaces
In this note, we consider the concentration function problem for a continuous action of a locally compact group $G$ on a locally compact Hausdorff space $X$. We prove a necessary and sufficient condition for the concentration functions of a spread-out irreducible probability measure $mu$ on $G$ to converge to zero.
متن کاملOne-point extensions of locally compact paracompact spaces
A space $Y$ is called an {em extension} of a space $X$, if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {em equivalent}, if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Yleq Y'$, if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ ...
متن کاملInfinite-Dimensional Multiplicity-Free Spaces I: Limits of Compact Commutative Spaces
We study direct limits (G,K) = lim −→ (Gn,Kn) of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicityfree discrete direct sum of irreducible representations. Then we look at direct limits G/K = lim −→ Gn/Kn of compact riemannian symmetric spaces, where we combine our criterion with the Cartan–Helgason theorem to show in general that the reg...
متن کامل