Extremely Disconnected Research Articles

Fuzzy Nano z-locally Closed Sets, Extremally Disconnected Spaces, Normal Spaces, and Their Application

In this paper, we introduce fuzzy nano (resp. δ, δS, P and Z) locally closed set and fuzzy nano (resp. δ, δS, P and Z) extremally disconnected spaces in fuzzy nano topological spaces. Also, we introduce some new spaces called fuzzy nano (resp. δ, δS, P and Z) normal spaces and strongly fuzzy nano (resp. δ, δS, P and Z) normal spaces with the help of fuzzy nano (resp. δ, δS, P and Z)-open sets in fuzzy nano topological space. Numerical data is used to quantify the provided features. Furthermore, using fuzzy nano topological spaces, an algorithm for multiple attribute decision-making (MADM) with an application in medical diagnosis is devised.

Discrete Subsets in Topological Groups and Countable Extremally Disconnected Groups

It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasov's question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed. It is also proved that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter and, hence, a countable nondiscrete extremally disconnected group cannot be constructed in ZFC.

The Mazur–Ulam property for commutative von Neumann algebras

ABSTRACTLet be a σ-finite measure space. Given a Banach space X, let the symbol stand for the unit sphere of X. We prove that the space of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm satisfies the Mazur–Ulam property, that is, if X is any complex Banach space, every surjective isometry admits an extension to a surjective real linear isometry . This conclusion is derived from a more general statement which assures that every surjective isometry where K is a Stonean space, admits an extension to a surjective real linear isometry from onto X.

Generalised weighted composition operators on Maeda–Ogasawara spaces

Every Archimedean Riesz space can be embedded as an order dense subspace of some C∞(X), the Riesz space of all extended continuous functions on a Stonean space X, called its Maeda–Ogasawara space. Furthermore, it is a fact that every Riesz homomorphism between spaces of ordinary continuous functions on compact Hausdorff spaces is a weighted composition operator. We prove that a generalised statement holds for Maeda–Ogasawara spaces and refine these results in case the homomorphism preserves order limits.

Some geometry of convex bodies in [formula omitted] spaces

We deal with some problems related to vector addition and diametric completion procedures of convex bodies in C(K) spaces. We prove that each of the following properties of convex bodies in C(K) characterizes the underlying compact Hausdorff space K as a Stonean space: (i) C(K) has a generating unit ball; (ii) all Maehara sets in C(K) are complete; (iii) the set Dd of all complete sets of diameter d in C(K) is convex. In contrast to these results, we further show the following for all C(K) spaces: (a) a weaker version of the generating unit ball property is satisfied; (b) there is a Maehara-like completion procedure; (c) Dd is starshaped with respect to any ball of radius d/2. The proofs are based on a systematic investigation of generalized order intervals and intersections of balls, which is carried out in the first part of the paper, and of diametrically complete sets. Furthermore, a new characterization of spaces ℓ∞n among all n-dimensional real normed spaces, in terms of the Minkowski addition of intersections of translates of the unit ball, is provided.

On Kakutani--Krein and Maeda--Ogasawara spaces

Let E be an Archimedean Riesz space. It is shown that the Kakutani--Krein space of the center of the Dedekind completion of E and the Maeda--Ogasawara space of E are homeomorphic. By applying this, we can reprove a Banach Stone type theorem for C\infty(S) spaces, where S is a Stonean space.

On Generalizations of Extremally Disconnected Spaces

On Generalizations of Extremally Disconnected Spaces

INJECTIVE ENVELOPES AND LOCAL MULTIPLIER ALGEBRAS OF SOME SPATIAL CONTINUOUS TRACE C*-ALGEBRAS

A precise description of the injective envelope of a spatial continuous trace C*-algebra A over a Stonean space Delta is given. The description is based on the notion of a weakly continuous Hilbert bundle, which we show to be a Kaplansky--Hilbert module over the abelian AW*-algebra C(Delta). We then use the description of the injective envelope of A to study the first- and second-order local multiplier algebras of A. In particular, we show that the second-order local multiplier algebra of A is precisely the injective envelope of A.

-Fuzzy Extremally Disconnected Ideal Topological Spaces

The notion of ★-Externally disconnected fuzzy ideal topological spaces is introduced and studied. Many characterizations of the space are obtained.

On lattices of continuous functions on a Stonian space

Suppose that Ω is a compact Hausdorff space with a preorder [les ] whose graph is closed, and let Ω∘ be an open subset of Ω. This paper provides conditions sufficient to allow every increasing bounded real continuous function on Ω∘ to be extended to an increasing real continuous function on Ω. These conditions are: (i) that Ω is a Stonian space, and (ii) that the set C↑(Ω, [les ]) of increasing real continuous functions on Ω is a regular Dedekind complete sublattice of C(Ω). Under these conditions it is also shown that C↑(Ω, [les ]) is generated by idempotents, and an extension theorem for idempotents is proved.

Fixed point theorems in hyperconvex metric spaces

The notion of hyperconvex spaces was introduced by Aronszajn and Panitchpakdi [1] in 1956. In 1979, independently Sine [13] and Soardi [16] proved the xed point property for nonexpansive maps on bounded hyperconvex spaces. Since then many interesting works have appeared for hyperconvex spaces. For example, see [2–8,14,15]. It is known that the space C(E) of all continuous real functions on a Stonian space E (extremally disconnected compact Hausdor space) with the usual norm is hyperconvex, and that every hyperconvex real Banach space is a space C(E) for some Stonian space E. Then (Rn; ‖ · ‖∞); l∞ and L∞ are concrete examples of hyperconvex spaces. Until recently, the study of hyperconvex spaces was concentrated to the relationship with nonexpansive maps. However, recently, Khamsi [5] established the Knaster– Kuratowski–Mazurkiewicz theorem (in short, KKM theorem) for hyperconvex spaces and applied it to prove an analogue of Ky Fan’s best approximation theorem extending the Brouwer and the Schauder xed point theorems. This seems to be the rst attempt to prove such results in a hyperconvex space setting. In this paper, we obtain a Ky Fan type matching theorem for open covers, a coincidence theorem, a Fan–Browder type xed point theorem, a Brouwer–Schauder–Rothe type xed point theorem, and other results for hyperconvex spaces. Those are usually

The algebra of unbounded continuous functions on a Stonean space and unbounded operators.

The algebra of unbounded continuous functions on a Stonean space and unbounded operators.

Regular norm balls can be closed in the strong operator topology

The main results obtained are: – A Dedekind complete Banach lattice Y has a Fatou norm if and only if, for any Banach lattice X, the regular-norm unit ball Ur = {T ∈ Lr(X,Y): ||T||r ≤ 1} is closed in the strong operator topology on the space of all regular operators, Lr(X,Y). – A Dedekind complete Banach lattice Y has a norm which is both Fatou and Levi if and only if, for any Banach lattice X, the regular-norm unit ball Ur is closed in the strong operator topology on the space of all bounded operators, L(X,Y). – A Banach lattice Y has a Fatou–Levi norm if and only if for every L-space X the space L(X,Y) is a Banach lattice under the operator norm. – A Banach lattice Y is isometrically order isomorphic to C(S) with the supremum norm, for some Stonean space S, if and only if, for every Banach lattice X, L(X,Y) is a Banach lattice under the operator norm. Several examples demonstrating that the hypotheses may not be removed, as well as some applications of the results obtained to the spaces of operators are also given. For instance: – If X = Lp(μ) and Y = Lq(ν), where 1 < p,q < ∞, then Lr(X,Y) is a first category subset of L(X,Y).

Equivalence Relations on Stonian Spaces

Quotient spaces of locally compact Stonian spaces which generalize in some sense the concept of Stone representation space of a Boolean algebra are investigated emphasizing the measure theoretical point of view, and a representation theorem for finitely additive measures is proved.

Continuous Functions on Extremally Disconnected Spaces

Continuous Functions on Extremally Disconnected Spaces

On Some Homogeneous Extremally Disconnected Spaces

On Some Homogeneous Extremally Disconnected Spaces

Topological function-theoretic proofs in spectral theory

In this paper, the spectral theorem and related characterizations of the spectrum and the spectral projections for bounded self adjoint and normal operators on a Hilbert space, are proved in purely topological —function theoretic terms. The basis for such a development, is the Gelfand—Naimark theorem for commutativeC *-algebras and the fact that the structure space of the (abelian) von Neumann algebra generated by the operator is a Stonean space.

Dense Extremally Disconnected Subspaces

We prove that every compact Basically Disconnected space of $\pi$-weight ${\omega _1}$ has a dense Extremally Disconnected subspace. In Boolean algebraic terms: every $\sigma$-complete Boolean algebra B with density ${\omega _1}$ carries an ultrafilter which generates an ultrafilter in the completion of B. The statement that every compact Basically Disconnected space of weight $\mathfrak {c}$ has a dense Extremally Disconnected subspace is shown to be equivalent to CH.

Measure‐theoretic characterizations of hereditarily‐normal spaces

In this paper we characterize hereditarily‐normal spaces in terms of the measure‐theoretic properties of the lattice of closed sets. We then generalize from that lattice to other lattices. We apply the results to extremally‐disconnected spaces.

Normal Measures on Stonian Spaces with σ‐Compact and Paracompact Support

AbstractIn the first part of the paper the spaces of normal measures with σ‐compact and with paracompact support on some locally compact STONian space are investigated, and their order continuous duals are determined. In the second part abstract measures with corresponding properties are studied.