arXiv subjects

M. R. Koushesh

Publications and source records attributed to M. R. Koushesh.

At least 19 recordsLinked to original sources

arXiv2018

On closed non-vanishing ideals in $C_B(X)$ II; compactness properties

For a completely regular space $X$, let $C_B(X)$ be the normed algebra of all bounded continuous scalar-valued mappings on $X$ equipped with pointwise addition and multiplication and the supremum norm and let $C_0(X)$ be its subalgebra consisting of mappings vanishing at infinity. For a non-vanishing closed ideal $H$ of $C_B(X)$ we study properties of its spectrum $\mathfrak{sp}(H)$ which may be characterized as the unique locally compact (Hausdorff) space $Y$ such that $H$ and $C_0(Y)$ are isometrically isomorphic. We concentrate on compactness properties of $\mathfrak{sp}(H)$ and find necessary and sufficient (algebraic) conditions on $H$ such that the spectrum $\mathfrak{sp}(H)$ satisfies (topological) properties such as the Lindelöf property, $σ$-compactness, countable compactness, pseudocompactness and paracompactness.

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arXiv2017

On closed subalgebras of CB(X)

For a completely regular space $X$ and a non-vanishing self-adjoint closed subalgebra $H$ of $C_B(X)$ which separates points from closed sets in $X$ we construct the Gelfand spectrum $\mathfrak{sp}(H)$ of $H$ as an open subspace of the compactification of $X$ generated by $H$. The simple construction of $\mathfrak{sp}(H)$ enables easier examination of its properties. We illustrate this by an example showing that the space $\mathfrak{sp}(H)$ is separable metrizable if and only if $H$ is countably generated.

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arXiv2017

On closed non-vanishing ideals in CB(X)

Let $X$ be a completely regular topological space. We study closed ideals $H$ of $C_B(X)$, the normed algebra of bounded continuous scalar-valued mappings on $X$ equipped with pointwise addition and multiplication and the supremum norm, which are non-vanishing, in the sense that, there is no point of $X$ at which every element of $H$ vanishes. This is done by studying the (unique) locally compact Hausdorff space $Y$ associated to $H$ in such a way that $H$ and $C_0(Y)$ are isometrically isomorphic. We are interested in various connectedness properties of $Y$. In particular, we present necessary and sufficient (algebraic) conditions for $H$ such that $Y$ satisfies (topological) properties such as locally connectedness, total disconnectedness, zero-dimensionality, strong zero-dimensionality, total separatedness or extremal disconnectedness.

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arXiv2017

The existence of one-point connectifications

P. Alexandroff proved that a locally compact $T_2$-space has a $T_2$ one-point compactification (obtained by adding a "point at infinity") if and only if it is non-compact. He also asked for characterizations of spaces which have one-point connectifications. Here, we study one-point connectifications, and in analogy with Alexandroff's theorem, we prove that in the realm of $T_i$-spaces ($i=3\frac{1}{2},4,5$) a locally connected space has a one-point connectification if and only if it has no compact component. We extend this theorem to the case $i=6$ by assuming some set-theoretic assumption, and to the case $i=2$ by slightly modifying its statement. We further extended the theorem by proving that a locally connected metrizable (resp. paracompact) space has a metrizable (resp. paracompact) one-point connectification if and only if it has no compact component. Contrary to the case of the one-point compactification, a one-point connectification, if exists, may not be unique. We consider the collection of all one-point connectifications of a locally connected locally compact space in the realm of $T_i$-spaces ($i=3\frac{1}{2},4,5$). We prove that this collection, naturally partially ordered, is a compact conditionally complete lattice whose order-structure determines the topology of all Stone-Cech remainders of components of the space.

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arXiv2017

A Gelfand-Naimark type theorem

Let $X$ be a completely regular space. For a non-vanishing self-adjoint Banach subalgebra $H$ of $C_B(X)$ which has local units we construct the spectrum $\mathfrak{sp}(H)$ of $H$ as an open subspace of the Stone-Cech compactification of $X$ which contains $X$ as a dense subspace. The construction of $\mathfrak{sp}(H)$ is simple. This enables us to study certain properties of $\mathfrak{sp}(H)$, among them are various compactness and connectedness properties. In particular, we find necessary and sufficient conditions in terms of either $H$ or $X$ under which $\mathfrak{sp}(H)$ is connected, locally connected and pseudocompact, strongly zero-dimensional, basically disconnected, extremally disconnected, or an $F$-space.

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arXiv2017

Embedding normed linear spaces into C(X)

It is well known that every (real or complex) normed linear space $L$ is isometrically embeddable into $C(X)$ for some compact Hausdorff space $X$. Here $X$ is the closed unit ball of $L^*$ (the set of all continuous scalar-valued linear mappings on $L$) endowed with the weak$^*$ topology, which is compact by the Banach-Alaoglu theorem. We prove that the compact Hausdorff space $X$ can indeed be chosen to be the Stone-Cech compactification of $L^*\setminus\{0\}$, where $L^*\setminus\{0\}$ is endowed with the supremum norm topology.

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arXiv2017

Connectifying a space by adding one point

It is a classical theorem of Alexandroff that a locally compact Hausdorff space has a one-point Hausdorff compactification if and only if it is non-compact. The one-point Hausdorff compactification is indeed obtained by adding the so called "point at infinity." Here we consider the analogous problem of existence of a one-point connectification, and keeping analogy, we prove that a locally connected normal space has a one-point normal connectification if and only if it has no compact component.

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arXiv2016

Connectedness modulo an ideal

For a topological space $X$ and an ideal $\mathscr{H}$ of subsets of $X$ we introduce the notion of connectedness modulo $\mathscr{H}$. This notion of connectedness naturally generalizes the notion of connectedness in its usual sense. In the case when $X$ is completely regular, we introduce a subspace $γ_{\mathscr H} X$ of the Stone--Čech compactification $βX$ of $X$, such that connectedness modulo ${\mathscr H}$ is equivalent to connectedness of $βX\setminusγ_{\mathscr H} X$. In particular, we prove that when ${\mathscr H}$ is the ideal generated by the collection of all open subspaces of $X$ with pseudocompact closure, then $X$ is connected modulo ${\mathscr H}$ if and only if $\mathrm{cl}_{βX}(βX\setminus\upsilon X)$ is connected, and when $X$ is normal and ${\mathscr H}$ is the ideal generated by the collection of all closed realcompact subspaces of $X$, then $X$ is connected modulo ${\mathscr H}$ if and only if $\mathrm{cl}_{βX}(\upsilon X\setminus X)$ is connected. Here $\upsilon X$ is the Hewitt realcompactification of $X$.

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arXiv2016

Ideals in CB(X) arising from ideals in X

Let $X$ be a completely regular topological space. We assign to each (set theoretic) ideal of $X$ an (algebraic) ideal of $C_B(X)$, the normed algebra of continuous bounded complex valued mappings on $X$ equipped with the supremum norm. We then prove several representation theorems for the assigned ideals of $C_B(X)$. This is done by associating a certain subspace of the Stone--Čech compactification $βX$ of $X$ to each ideal of $X$. This subspace of $βX$ has a simple representation, and in the case when the assigned ideal of $C_B(X)$ is closed, coincides with its spectrum as a $C^*$-subalgebra of $C_B(X)$. This in particular provides information about the spectrum of those closed ideals of $C_B(X)$ which have such representations. This includes the non-vanishing closed ideals of $C_B(X)$ whose spectrums are studied in great detail. Our representation theorems help to understand the structure of certain ideals of $C_B(X)$. This has been illustrated by means of various examples. Our approach throughout will be quite topological and makes use of the theory of the Stone--Čech compactification.

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arXiv2015

Topological extensions with compact remainder

Let $\mathfrak{P}$ be a topological property. We study the relation between the order structure of the set of all $\mathfrak{P}$-extensions of a completely regular space $X$ with compact remainder (partially ordered by the standard partial order $\leq$) and the topology of certain subspaces of the outgrowth $βX\setminus X$. The cases when $\mathfrak{P}$ is either pseudocompactness or realcompactness are studied in more detail.

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arXiv2014

One-point connectifications

A space Y is called an extension of a space X if Y contains X as a dense subspace. An extension Y of X is called a one-point extension if Y-X is a singleton. Compact extensions are called compactifications and connected extensions are called connectifications. It is well known that every locally compact non-compact space has a one-point compactification (known as the Alexandroff compactification) obtained by adding a point at infinity. A locally connected disconnected space, however, may fail to have a one-point connectification. It is indeed a long standing question of Alexandroff to characterize spaces which have a one-point connectification. Here we prove that in the class of completely regular spaces, a locally connected space has a one-point connectification if and only if it contains no compact component.

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arXiv2014

Continuous mappings with null support

Let $X$ be a (topological) space and let ${\mathscr I}$ be an ideal in $X$, that is, a collection of subsets of $X$ which contains all subsets of its elements and is closed under finite unions. The elements of ${\mathscr I}$ are called null. The space $X$ is locally null if each $x$ in $X$ has a null neighborhood in $X$. Let $C_b(X)$ denote the normed algebra of all continuous bounded real-valued mappings on $X$ equipped with the supremum norm, $C_0(X)$ denote the subalgebra of $C_b(X)$ consisting of elements vanishing at infinity and $C_{00}(X)$ the subalgebra of $C_b(X)$ consisting of elements with compact support. We study the normed subalgebra $C^{\mathscr I}_{00}(X)$ of $C_b(X)$ consisting of all $f$ in $C_b(X)$ whose support has a null neighborhood in $X$, and the Banach subalgebra $C^{\mathscr I}_0(X)$ of $C_b(X)$ consisting of all $f$ in $C_b(X)$ such that $|f|^{-1}([1/n,\infty))$ has a null neighborhood in $X$ for all positive integer $n$. We prove that if $X$ is a normal locally null space then $C^{\mathscr I}_{00}(X)$ and $C^{\mathscr I}_0(X)$ are respectively isometrically isomorphic to $C_{00}(Y)$ and $C_0(Y)$ for a unique locally compact Hausdorff space $Y$; furthermore, $C^{\mathscr I}_{00}(X)$ is dense in $C^{\mathscr I}_0(X)$. We construct $Y$ explicitly as a subspace of the Stone--Čech compactification $βX$ of $X$. The space $Y$ is locally compact (and countably compact, in certain cases), contains $X$ densely, and in specific cases turns out to be familiar subspaces of $βX$. The known topological structure of $Y$ enables us to establish several commutative Gelfand--Naimark type theorems and derive results not generally expected to be deducible from the standard Gelfand theory.

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arXiv2013

Representations of certain Banach algebras

For a space $X$ denote by $C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on $X$ and denote by $C_0(X)$ the set of all elements in $C_b(X)$ which vanish at infinity. We prove that certain Banach subalgebras $H$ of $C_b(X)$ are isometrically isomorphic to $C_0(Y)$, for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone--Čech compactification of $X$. The known construction of $Y$ enables us to examine certain properties of either $H$ or $Y$ and derive results not expected to be deducible from the standard Gelfand Theory.

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arXiv2013

The Banach algebra of continuous bounded functions with separable support

We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to $C_0(Y)$, for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$, which we explicitly construct as a subspace of the Stone--Čech compactification of $X$, is countably compact, and if $X$ is non-separable, is moreover non-normal; in addition $C_0(Y)=C_{00}(Y)$. When the underlying field of scalars is the complex numbers, the space $Y$ coincides with the spectrum of the ${C}^*$-algebra $C_s(X)$. Further, we find the dimension of the algebra $C_s(X)$.

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arXiv2013

Representations of certain normed algebras

We show that for a normal locally-${\mathscr P}$ space $X$ (where ${\mathscr P}$ is a topological property subject to some mild requirements) the subset $C_{\mathscr P}(X)$ of $C_b(X)$ consisting of those elements whose support has a neighborhood with ${\mathscr P}$, is a subalgebra of $C_b(X)$ isometrically isomorphic to $C_c(Y)$ for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone--Čech compactification $βX$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, $C_{\mathscr P}(X)$ coincides with the set of those elements of $C_b(X)$ whose support has ${\mathscr P}$, it moreover becomes a Banach algebra, and simultaneously, $Y$ satisfies $C_c(Y)=C_0(Y)$. This includes the cases when ${\mathscr P}$ is the Lindelöf property and $X$ is either a locally compact paracompact space or a locally-${\mathscr P}$ metrizable space. In either of the latter cases, if $X$ is non-${\mathscr P}$, $Y$ is non-normal, and $C_{\mathscr P}(X)$ fits properly between $C_0(X)$ and $C_b(X)$; even more, we can fit a chain of ideals of certain length between $C_0(X)$ and $C_b(X)$. The known construction of $Y$ enables us to derive a few further properties of either $C_{\mathscr P}(X)$ or $Y$. Specifically, when ${\mathscr P}$ is the Lindelöf property and $X$ is a locally-${\mathscr P}$ metrizable space, we show that \[\dim C_{\mathscr P}(X)=\ell(X)^{\aleph_0},\] where $\ell(X)$ is the Lindelöf number of $X$, and when ${\mathscr P}$ is countable compactness and $X$ is a normal space, we show that \[Y=\mathrm{int}_{βX}\upsilon X\] where $\upsilon X$ is the Hewitt realcompactification of $X$.

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arXiv2013

One-point extensions and local topological properties

A space $Y$ is called an extension of a space $X$ if $Y$ contains $X$ as a dense subspace. An extension $Y$ of $X$ is called a one-point extension of $X$ if $Y\backslash X$ is a singleton. P. Alexandroff proved that any locally compact non-compact Hausdorff space $X$ has a one-point compact Hausdorff extension, called the one-point compactification of $X$. Motivated by this, S. Mrówka and J.H. Tsai [On local topological properties. II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 1035-1040] posed the following more general question: For what pairs of topological properties ${\mathscr P}$ and ${\mathscr Q}$ does a locally-${\mathscr P}$ space $X$ having ${\mathscr Q}$ possess a one-point extension having both ${\mathscr P}$ and ${\mathscr Q}$? Here, we provide an answer to this old question.

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arXiv2012

Embedding measure spaces

For a given measure space $(X,{\mathscr B},μ)$ we construct all measure spaces $(Y,{\mathscr C},λ)$ in which $(X,{\mathscr B},μ)$ is embeddable. The construction is modeled on the ultrafilter construction of the Stone--Čech compactification of a completely regular topological space. Under certain conditions the construction simplifies. Examples are given when this simplification occurs.

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arXiv2012

Connectedness modulo a topological property

Let ${\mathscr P}$ be a topological property. We say that a space $X$ is ${\mathscr P}$-connected if there exists no pair $C$ and $D$ of disjoint cozero-sets of $X$ with non-${\mathscr P}$ closure such that the remainder $X\backslash(C\cup D)$ is contained in a cozero-set of $X$ with ${\mathscr P}$ closure. If ${\mathscr P}$ is taken to be "being empty" then ${\mathscr P}$-connectedness coincides with connectedness in its usual sense. We characterize completely regular ${\mathscr P}$-connected spaces, with ${\mathscr P}$ subject to some mild requirements. Then, we study conditions under which unions of ${\mathscr P}$-connected subspaces of a space are ${\mathscr P}$-connected. Also, we study classes of mappings which preserve ${\mathscr P}$-connectedness. We conclude with a detailed study of the special case in which ${\mathscr P}$ is pseudocompactness. In particular, when ${\mathscr P}$ is pseudocompactness, we prove that a completely regular space $X$ is ${\mathscr P}$-connected if and only if $cl_{βX}(βX\backslash\upsilon X)$ is connected, and that ${\mathscr P}$-connectedness is preserved under perfect open continuous surjections. We leave some problems open.

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