Similarity invariants of operators on a class of Gowers-Maurey spaces

This paper studies the similarity invariants of operators on a class of Gowers-Maurey spaces, Σ _dc spaces, where an infinite dimensional Banach space X is called a Σ _dc space if for every bounded linear operator on X the spectrum is disconnected unless it is a singleton. It shows that two strongly irreducible operators T ₁ and T ₂ on a Σ _dc space are similar if and only if the K ₀-group of the commutant algebra of the direct sum T ₁⊕T ₂ is isomorphic to the group of integers ?. On a Σ _dc space X, it uses the semigroups of the commutant algebras of operators to give a condition that an operator is similar to some operator in (ΣSI)(X), it further gives a necessary and sufficient condition that two operators in (ΣSI)(X) are similar by using the ordered K ₀-groups. It also proves that every operator in (ΣSI)(X) has a unique (SI) decomposition up to similarity on a Σ _dc space X, where (ΣSI)(X) denotes the class of operators which can be written as a direct sum of finitely many strongly irreducible operators.     

Resolution property in generalized k-spaces

We show that a T₁ space X is resolvable if the set of limit points λ (X) of various simultaneously separated subsets of X is dense in X. Moreover, if λ (X) is open also, then X is ω-resolvable. It follows that a self-dense, Hausdroff space satisfying a generalized k-space (sequential space) condition is resolvable (respectively, ω-resolvable).     

Hyperbolicity in median graphs

If X is a geodesic metric space and x ₁,x ₂,x ₃?∈?X, a geodesic triangle T?=?{x ₁,x ₂,x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e., ${\delta}(X)=\inf\{{\delta}\ge 0: \, X \, \text{is $\delta$-hyperbolic}\}. $ In this paper we study the hyperbolicity of median graphs and we also obtain some results about general hyperbolic graphs. In particular, we prove that a median graph is hyperbolic if and only if its bigons are thin.     

Calibers and point-finite cellularity of the space Cp(X) and some questions of S. Gul'ko and M. Husek

It is well known that ℵ₁ is a precaliber of Cp(X) for every Tychonoff space X. We prove under GCH that a compact space X is metrizable if and only if ℵ₁ and ℵ₂ are calibers of Cp(X). We show also that if X is a compact space then ℵ₁ is a caliber of Cp(X) if and only if its diagonal Δ ⊂ X² is small in the sense of Husek [9]. A similar method is used to establish that if X is an extremally disconnected compact space then Cp(X) admits a continuous injection into Σ₁ (τ) (for some τ) if and only if the space X is separable.     

Injective Banach spaces of typeC(T)

A Banach spaceX is aP _λ-space if wheneverX is isometrically embedded in another Banach spaceY there is a projection ofY ontoX with norm at most λ.C(T) denotes the Banach space of continuous real-valued functions on the compact Hausdorff spaceT. T satisfies the countable chain condition (CCC) if every family of disjoint non-empty open sets inT is countable.T is extremally disconnected if the closure of every open set inT is open. The main result is that ifT satisfies the CCC andC(T) is aP _λ-space, thenT is the union of an open dense extremally disconnected subset and a complementary closed setT _Asuch thatC(T_A) is aP _λ?1-space.     

Descriptions of equivalence of O-connectivity

We first study some properties of the subspace, and investigate into the relationship of separation between a fuzzy topological space (fts) and its subspace. Then we obtain the equivalence conditions for O-connectivity. The results on O-connectivity and separation are very similar to those in general topology. Finally we discuss the relationship of connectivity between an O-connected set A in the fts (X, ω ($\text{T}$)) induced by the crisp topological space (X, $\text{T}$) and the crisp set A₀ (=supp A) in (X, $\text{T}$).     

Unification Approach to the Separation Axioms Between T 0 and Completely Hausdorff

The aim of this paper is to introduce a new weak separation axiom that generalizes the separation properties between T ₁ and completely Hausdorff. We call a topological space (X, τ) a T _κ,ξ-space if every compact subset of X with cardinality ≦ κ is ξ-closed, where ξ is a general closure operator. We concentrate our attention mostly on two new concepts: kd-spaces and T _1/3-spaces.     

The spaces Cp(X): decomposition into a countable union of bounded subspaces and completeness properties

A Tychonoff space X has to be finite if C_p(X) is σ-countably compact [23]. However, this is not true if only σ-pseudocompactness of C_p(X) is assumed. It is proved that C_p(X) is σ-pseudocompact iff X is pseudocompact and b-discrete. The technique developed yields an example showing that the theorem of Grothendieck [7] cannot be extended over the class of pseudocompact spaces. Some generalizations of the results of Lutzer and McCoy [9] are obtained. We establish also that ∏{C_p(X_t):tϵT} is a Baire space in case C_p(X_t) is Baire for each t ∈ T.     

Fuzzy measures and discreteness

The main result is to show that the space of nonmonotonic fuzzy measures on a measurable space (X,X) with total variation norm is separable if and only if the σ-algebra X is a finite set. Our result is related to fuzzy analysis, functional spaces and discrete mathematics.     

On the direct product of fuzzy topological groups

The connections between some countability properties of a topological space (X, $\text{T}$) and its generated fuzzy topological space (X, ω($\text{T}$)) are investigated.     

Random products of contractions in metric and Banach Spaces

Suppose (X, d) is a metric space and {T₀,…, T_N} is a family of quasinonexpansive self-mappings on X. We give conditions sufficient to guarantee that every possible iteration of mappings drawn from {T₀,…, T_N} converges. As a consequence, if C₀,…, C_N are closed convex subsets of a Hilbert space with nonempty intersection, one of which is compact, and the proximity mappings are iterated in any order (provided only that each is used infinitely often), then the resulting sequence converges strongly to a point of the common intersection.     

On similarity between topologies

Let T ₁ and T ₂ be topologies defined on the same set X and let us say that (X, T ₁) and (X, T ₂) are similar if the families of sets which have nonempty interior with respect to T ₁ and T ₂ coincide. The aim of the paper is to study how similar topologies are related with each other.     

Collectionwise normality and selections into Hilbert spaces

A T₁-space X is countably paracompact and collectionwise normal if and only if every l.s.c. mapping from X into a Hilbert space with closed and convex point-images has a continuous selection. This settles a conjecture posed by M. Choban, V. Gutev and S. Nedev [M. Choban, S. Nedev, Continuous selections for mappings with generalized ordered domain, Math. Balkanica (N.S.) 11 (1-2) (1997) 87-95].     

Covering compacta by discrete subspaces

For any space X, denote by dis(X) the smallest (infinite) cardinal κ such that κ many discrete subspaces are needed to cover X. It is easy to see that if X is any crowded (i.e. dense-in-itself) compactum then dis(X)?m, where m denotes the additivity of the meager ideal on the reals. It is a natural, and apparently quite difficult, question whether in this inequality m could be replaced by c. Here we show that this can be done if X is also hereditarily normal.Moreover, we prove the following mapping theorem that involves the cardinal function dis(X). If is a continuous surjection of a countably compact T₂ space X onto a perfect T₃ space Y then .     

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

The pseudoweights of a space

Let X be a T_i-space, i ⩽ 2. We define the T_i-pseudoweight of X, ψ _i(X), to be the least weightof a coarser T_i topology on X. Reed and Zenor have shown that if X is a Moore space, and |X| ⩽ 2^ω, then ψ₁(X) = ω, but there is a Moore space, X, such that ψ₂(X) = w(X) = |X| = ω₁.Theorem 1: If X is metric, ψ₀(X) = log w(X), where log κ = min{λ:2^λ ⩾ κ}. Theorem 2: If X is compact and T₂, then ψ₁(X) = ψ₂(X) = w(X) (but it is possible to have ψ₀(X) = log w (X)< w(X)). Theorem 3: If X is a GO-space, then ψ₁(X) = ψ₂(X) (but it is possible to have ψ₀(X) =log ψ₁(X) < ψ₁(X) < w(X) even if X is a LOTS). Finally, Hart has shown that if X is an infinite LOTS, then w(X) = c (X) · ψ₁(X). Theorem 4: If X is an infinite LOTS, then w(X) =c(X) · ψ₀ (X).     

Resolvability of spaces having small spread or extent

In a recent paper O. Pavlov proved the following two interesting resolvability results:

(1)

If a T₁-space X satisfies Δ(X)>ps(X) then X is maximally resolvable.

(2)

If a T₃-space X satisfies Δ(X)>pe(X) then X is ω-resolvable.

Here ps(X) (pe(X)) denotes the smallest successor cardinal such that X has no discrete (closed discrete) subset of that size and Δ(X) is the smallest cardinality of a non-empty open set in X.In this note we improve (1) by showing that Δ(X)>ps(X) can be relaxed to Δ(X)?ps(X), actually for an arbitrary topological space X. In particular, if X is any space of countable spread with Δ(X)>ω then X is maximally resolvable.The question if an analogous improvement of (2) is valid remains open, but we present a proof of (2) that is simpler than Pavlov's.     

Additions to the Periodic Decomposition Theorem

A pair of linear bounded commuting operators T₁, T₂ in a Banach space is said to possess a decomposition property (DePr) if Ker (I-T₁)(I-T₂) = Ker (I-T₁) + Ker (I-T₂). A Banach space X is said to possess a 2-decomposition property (2-DePr) if every pair of linear power bounded commuting operators in X possesses the DePr. It is known from papers of M. Laczkovich and Sz. Révész that every reflexive Banach space X has the 2-DePr. In this paper we prove that every quasi-reflexive Banach space of order 1 has the 2-DePr but not all quasi-reflexive spaces of order 2. We prove that a Banach space has no 2-DePr if it contains a direct sum of two non-reflexive Banach spaces. Also we prove that if a bounded pointwise norm continuous operator group acts on X then every pair of operators belonging to it has a DePr. A list of open problems is also included. This revised version was published online in August 2006 with corrections to the Cover Date.     

The optional stopping theorem for quantum martingales

In classical probability theory, a random time T is a stopping time in a filtration (Ft)_t?0 if and only if the optional sampling holds at T for all bounded martingales. Furthermore, if a process (Xt)_t?0 is progressively measurable with respect to (Ft)_t?0, then XT is FT-measurable. Unfortunately, this is not the case in noncommutative probability with the definition of stopped process used until now. It is shown in this article that we can define the stopping of noncommutative processes in Fock space in such a way that all the bounded martingales can be stopped at any stopping time T, are adapted to the filtration of the past before T and satisfy the optional stopping theorem.     

Cosmic spaces which are not μ-spaces among function spaces with the topology of pointwise convergence

A space is called a μ-space if it can be embedded in a countable product of paracompact F_σ-metrizable spaces. The following are shown:(1) For a Tychonoff space X, if C_p(X,R) is a μ-space, then X is a countable union of compact metrizable subspaces.(2) For a zero-dimensional space X, C_p(X,2) is a μ-space if and only if X is a countable union of compact metrizable subspaces.In particular, let P be the space of irrational numbers. Then C_p(P,2) is a cosmic space (i.e., a space with a countable network) which is not a μ-space.