Hull-kernel Topology Research Articles

On Prime Ideal Space of a Partially Ordered Ternary Semigroup

In this paper, we introduced the hull-kernel topology $\tau$ on the set $\mathcal P$ of prime ideals in a partially ordered ternary semigroup $T$ and investigated various topological properties of the structure space $(\mathcal P, \tau)$. We also obtained some useful results about compactness and connectedness of the set of all prime full ideals of $T$.

The hull-kernel topology on prime ideals in ordered semigroups

The hull-kernel topology on prime ideals in ordered semigroups

The spectrum of the $C^*$-algebra of singular integral operators with semi-almost periodic coefficients

A $C^*$-algebra generated by one-dimensional singular integral operators with semi-almost periodic coefficients is studied. The primitive spectrum of this algebra is described, that is, all of its primitive ideals are listed and the Jacobson topology is described. Bibliography: 27 titles.

Mp-Residuated Lattices

This paper is devoted to the study of a fascinating class of residuated lattices, the so-called mp-residuated lattice, in which any prime filter contains a unique minimal prime filter. A combination of algebraic and topological methods is applied to obtain new and structural results on mp-residuated lattices. It is demonstrated that mp-residuated lattices are strongly tied up with the dual hull-kernel topology. Especially, it is shown that a residuated lattice is mp if and only if its minimal prime spectrum, equipped with the dual hull-kernel topology, is Hausdorff if and only if its prime spectrum, equipped with the dual hull-kernel topology, is normal. The class of mp-residuated lattices is characterized by means of pure filters. It is shown that a residuated lattice is mp if and only if its pure filters are precisely its minimal prime filters, if and only if its pure spectrum is homeomorphic to its minimal prime spectrum, equipped with the dual hull-kernel topology.

Two topologies on the set of minimal prime elements of a continuous frame

In this paper, we discuss the properties of the hull-kernel topology and the inverse topology on the set [Formula: see text] of minimal prime elements of a continuous frame [Formula: see text]. We prove that the set [Formula: see text] endowed with the hull-kernel topology and the inverse topology, respectively, is a [Formula: see text]-space. Moreover, we obtain that there is a bijective correspondence between the set [Formula: see text] of all maximal Scott open filters of [Formula: see text] and the set [Formula: see text] when [Formula: see text] is a stably continuous frame. By considering the bijective correspondence between the sets [Formula: see text] and [Formula: see text], we propose some sufficient conditions for the topological spaces [Formula: see text] endowed with the hull-kernel topology and the inverse topology, respectively, to be sober, Hausdorff, compact, extremely disconnected and zero-dimensional spaces, respectively, when [Formula: see text] is a stably continuous frame.

The pure spectrum of a residuated lattice

This paper studies a fascinating type of filter in residuated lattices, the so-called pure filters. A combination of algebraic and topological methods on the pure filters of a residuated lattice is applied to obtain some new and structural results. The notion of purely-prime filters of a residuated lattice has been investigated, and a Cohen-type theorem has been obtained. It is shown that the pure spectrum of a residuated lattice is a compact sober space, and a Grothendieck-type theorem has been demonstrated. It is proved that the pure spectrum of a Gelfand residuated lattice is a Hausdorff space, and deduced that the pure spectrum of a Gelfand residuated lattice is homeomorphic to its usual maximal spectrum. Finally, the pure spectrum of an mp-residuated lattice is investigated and verified that a given residuated lattice is mp iff its minimal prime spectrum, equipped with the induced dual hull-kernel topology, and its pure spectrum are homeomorphic.

The structure space of C−(X) via that of Γ-semirings

The structure spaces, endowed with the hull kernel topology, of maximal regular congruences, which are prime on a [Formula: see text]-semiring as well as the structure space of prime congruences on a [Formula: see text]-semiring, have been studied via operator semirings. This study has been used to obtain some important results of the structure space of [Formula: see text] of nonpositive real valued continuous functions over a topological space [Formula: see text]. It has been found that the structure spaces of the semiring [Formula: see text] of nonnegative real valued continuous functions and the [Formula: see text]-semiring [Formula: see text] are homeomorphic. Moreover it has been shown that the structure space of [Formula: see text] is the Stone–Čech compactification of [Formula: see text], where [Formula: see text] is a Tychonoff space. Furthermore, the [Formula: see text]-semiring analogue of the ‘Banach–Stone Theorem’ has been obtained.

The Fuzzy Prime Spectrum of Partially Ordered Sets

We study the space of prime fuzzy ideals (and the space of maximal fuzzy ideals as a subspace) equipped with the hull‐kernel topology in partially ordered sets. Mainly, we investigate the conditions for which the fuzzy prime spectrum of a poset is compact, Hausdorff, and normal, respectively.

On Gelfand residuated lattices

In this paper, a combination of algebraic and topological methods is applied to obtain new and structural results on Gelfand residuated lattices. It is demonstrated that Gelfand’s residuated lattices strongly tied up with the hull–kernel topology. Particularly, it is shown that a residuated lattice is Gelfand if and only if its prime spectrum, equipped with the hull–kernel topology, is normal. The class of soft residuated lattices is introduced, and it is shown that a residuated lattice is soft if and only if it is Gelfand and semisimple. Gelfand residuated lattices are characterized using the pure part of filters. The relation between pure filters and radicals in a Gelfand residuated lattice is described. It is shown that a residuated lattice is Gelfand if and only if its pure spectrum is homeomorphic to its usual maximal spectrum. The pure filters of a Gelfand residuated lattice are characterized. Finally, it is proved that a residuated lattice is Gelfand if and only if the hull–kernel and the \(\mathscr {D}\)-topology coincide on the set of maximal filters.

Mp- and purified residuated lattices

This paper applies a combination of algebraic and topological methods to obtain new and structural results on mp- and purified residuated lattices. It is demonstrated that mp-residuated lattices are strongly tied up with the dual hull-kernel topology. Mainly, it is shown that a residuated lattice is mp if and only if its minimal prime spectrum, equipped with the dual hull-kernel topology, is Hausdorff if and only if its prime spectrum, fitted with the dual hull-kernel topology, is normal. For a residuated lattice \({\mathfrak {A}}\) and a subalgebra \({\mathfrak {S}}\), the notion of disjunctive \({\mathfrak {S}}\)-regular and \({\mathfrak {S}}\)-mp-residuated lattices are introduced and investigated. It is shown that a residuated lattice \({\mathfrak {A}}\) is purified if and only if it is \(\beta ({\mathfrak {A}})\)-mp if and only if it is disjunctive \(\beta ({\mathfrak {A}})\)-regular, where \(\beta ({\mathfrak {A}})\) is the set of complemented elements of \({\mathfrak {A}}\). Some topological characterizations for purified residuated lattices are extracted and proved that a residuated lattice \({\mathfrak {A}}\) is purified if and only if its minimal prime spectrum, equipped with the dual hull-kernel topology, is a profinite space. To support mentioned concepts, we give place to some examples.

Regularity of certain commutative Banach rings

Let (A,‖⋅‖) be a commutative unital Banach ring. Let M(A) be the Berkovich spectrum of A, i.e., the set of all bounded multiplicative non-zero semi-norms defined on A which is endowed with the pointwise convergence topology. As in the case of complex Banach algebras, we introduce the notions of the hull-kernel topology on M(A) and the regularity of a commutative unital Banach ring. We show that the Banach algebra of all bounded continuous k-valued functions defined on a zero-dimensional topological space is regular, where k is a complete valuation field. Hence, we see that the Berkovich spectrum of this algebra is a Hausdorff space in the hull-kernel topology. Next, we use the tool of Gelfand transform to characterize a Banach ring of finite Berkovich spectrum. From this, we see that if A is uniform, i.e., ‖f2‖=‖f‖2 for all f in A, and has finite Berkovich spectrum, then it is regular.

The hull-kernel topology on prime filters in residuated lattices

The hull-kernel topology on prime filters in residuated lattices

The orbit method for locally nilpotent infinite-dimensional Lie algebras

Let n be a locally nilpotent infinite-dimensional Lie algebra over C. Let U(n) and S(n) be its respective universal enveloping algebra and symmetric algebra. Consider the Jacobson topology on the primitive spectrum of U(n), and the Poisson topology on the primitive Poisson spectrum of S(n). We provide a homeomorphism between the corresponding topological spaces (at the level of points, it gives a bijection between the primitive ideals of U(n) and S(n)). We also show that all primitive ideals of S(n) from an open set in a properly chosen topology are generated by their intersections with the Poisson center. Under the assumption that n is a nil-Dynkin Lie algebra, we give two criteria for primitive ideals I(λ)⊂S(n) and J(λ)⊂U(n), λ∈n⁎, to be nonzero. Most of these results generalize known facts about primitive and Poisson spectrum for finite-dimensional nilpotent Lie algebras (but note that for a finite-dimensional nilpotent Lie algebra all primitive ideals I(λ), J(λ) are nonzero).

Alpha Ideals and the Space of Prime Alpha Ideals in Universal Algebras

The purpose of this paper is to study α -ideals in a more general context, in universal algebras having a constant 0 . Several characterizations are obtained for an ideal I of an algebra A to be an α -ideal. It is shown that the class of all α -ideals of an algebra A forms an algebraic lattice. Prime α -ideals and several related properties are investigated. Some properties of the spectral space of prime α -ideals equipped with the hull-kernel topology are derived.

Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras

We describe the Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras under certain conditions.

A Johnson–Kist type representation for truncated vector lattices

We introduce the notion of (maximal) multi-truncations on a vector lattice as a generalization of the notion of truncations, an object of recent origin. We obtain a Johnson–Kist type representation of vector lattices with maximal multi-truncations as vector lattices of almost-finite extended-real continuous functions. The spectrum that allows such a representation is a particular set of prime ideals equipped with the Hull–Kernel topology. Various representations from the existing literature will appear as special cases of our general result.

AF‐embeddability for Lie groups with T1 primitive ideal spaces

We study simply connected Lie groups $G$ for which the hull-kernel topology of the primitive ideal space $\text{Prim}(G)$ of the group $C^*$-algebra $C^*(G)$ is $T_1$, that is, the finite subsets of $\text{Prim}(G)$ are closed. Thus, we prove that $C^*(G)$ is AF-embeddable. To this end, we show that if $G$ is solvable and its action on the centre of $[G, G]$ has at least one imaginary weight, then $\text{Prim}(G)$ has no nonempty quasi-compact open subsets. We prove in addition that connected locally compact groups with $T_1$ ideal spaces are strongly quasi-diagonal.

The ideal structures of self-similar -graph C*-algebras

AbstractLet$(G,\unicode[STIX]{x1D6EC})$be a self-similar$k$-graph with a possibly infinite vertex set$\unicode[STIX]{x1D6EC}^{0}$. We associate a universal C*-algebra${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$to$(G,\unicode[STIX]{x1D6EC})$. The main purpose of this paper is to investigate the ideal structures of${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. We prove that there exists a one-to-one correspondence between the set of all$G$-hereditary and$G$-saturated subsets of$\unicode[STIX]{x1D6EC}^{0}$and the set of all gauge-invariant and diagonal-invariant ideals of${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. Under some conditions, we characterize all primitive ideals of${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar$P$-graph C*-algebras in depth.

Fuzzy prime spectrum of $C$-algebras

In this paper, we define fuzzy prime ideals of $ C- $algebras and investigate some of their properties. Furthermore, we study the topological properties of the space of fuzzy prime ideals of $ C- $algebra equipped with the hull-kernel topology.

On Banach space projective tensor product of $$C^*$$-algebras

For \(C^*\)-algebras A and B, we show that there is a natural homeomorphism from the product of spaces of maximal (resp., maximal modular) ideals of A and B onto the space of maximal (resp., maximal modular) ideals of \(A \otimes ^\gamma B\), with respect to the hull-kernel topology. This is based on and preceded by an analysis of the structure of closed ideals of \(A \otimes ^\gamma B\) in terms of those of A and B. During the process, we also identify the center of \(A \otimes ^\gamma B\) with \({\mathcal {Z}}(A) \otimes ^\gamma {\mathcal {Z}}(B)\).