A note on realizing homomorphisms of category algebras

The complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a complete matrix space X to the category algebra $\text{C}$(Y) of a Baire topological space Y are characterized as those σ-homomorphisms which are induced by continuous maps from dense G₈-subsets of Y into X. This result is used to deduce a series of related results in topology and measure theory (some of which are well-known). Finally a similar result for the complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a compact Hausdorff space X tothe category algebra $\text{C}$(Y) of a Baire topological space Y is proved.     

A Nagata-like theorem for certain function spaces

Let X be a space, and let A be a zero-dimensional topological ring. In this paper we will consider a few natural questions that arise when studying the space C _p (X, A), the ring of continuous functions from X to A, endowed with the topology of pointwise convergence. It will be shown that the zero-dimensionality of the codomain plays a vital role in this study. An upper and lower bound will be determined for the density of C _p (X, A) using the density of A and the weight of X. The character of C _p (X, A) will be computed, thus characterizing when C _p (X, A) is metrizable. Lastly, we will consider the topological dual space of C _p (X, A) and use it to prove a Nagata-like theorem.     

On countable bounded tightness for spaces Cp(X)

It is well known that the space C_p([0,1]) has countable tightness but it is not Fréchet-Urysohn. Let X be a Cech-complete topological space. We prove that the space C_p(X) of continuous real-valued functions on X endowed with the pointwise topology is Fréchet-Urysohn if and only if C_p(X) has countable bounded tightness, i.e., for every subset A of C_p(X) and every x in the closure of A in C_p(X) there exists a countable and bounding subset of A whose closure contains x. We study also the problem when the weak topology of a locally convex space has countable bounded tightness. Additional results in this direction are provided.     

On boundaries on the predual of the Lorentz sequence space

Let X be the canonical predual of the Lorentz sequence space and let Au(BX) be the Banach algebra of all complex valued functions defined on the closed unit ball BX of X which are uniformly continuous on BX and holomorphic on the interior of BX, endowed with the sup norm. A characterization of the boundaries for Au(BX) is given in terms of the distance to the strong peak sets of this algebra.     

Vector measures and strict topologies

Let X be a completely regular Hausdorff space and Cb(X) be the space of all real-valued bounded continuous functions on X, endowed with the strict topology βσ. We study topological properties of continuous and weakly compact operators from Cb(X) to a locally convex Hausdorff space in terms of their representing vector measures. In particular, Alexandrov representation type theorems are derived. Moreover, a Yosida-Hewitt type decomposition for weakly compact operators on Cb(X) is given.     

Fragmentability of groups and metric-valued function spaces

Let (X,τ) be a topological space and let ρ be a metric defined on X. We shall say that (X,τ) is fragmented by ρ if whenever ε>0 and A is a nonempty subset of X there is a τ-open set U such that U∩A≠∅ and ρ−diam(U∩A)<ε. In this paper we consider the notion of fragmentability, and its generalisation σ-fragmentability, in the setting of topological groups and metric-valued function spaces. We show that in the presence of Baireness fragmentability of a topological group is very close to metrizability of that group. We also show that for a compact Hausdorff space X, σ-fragmentability of (C(X),‖⋅_‖∞) implies that the space Cp(X;M) of all continuous functions from X into a metric space M, endowed with the topology of pointwise convergence on X, is fragmented by a metric whose topology is at least as strong as the uniform topology on C(X;M). The primary tool used is that of topological games.     

Closed ideals in topological algebras: A characterization of the topological Φ-algebra C k (x)

Let A be a uniformly closed and locally m-convex Φ-algebra. We obtain internal conditions on A stated in terms of its closed ideals for A to be isomorphic and homeomorphic to C _k (X), the Φ-algebra of all the real continuous functions on a normal topological space X endowed with the compact convergence topology.     

There do not exist minimal algebras between C*(X) and C(X) with prescribed real maximal ideal spaces

Let C(X) be the algebra of all real-valued continuous functions on a completely regular Hausdorff space X, and C*(X) the subalgebra of bounded functions. We prove that for any intermediate algebra A between C*(X) and C(X), other than C*(X), there exists a smaller intermediate algebra with the same real maximal ideals as in A. The space X is called A-compact if any real maximal ideal in A corresponds to a point in X. It follows that, for a noncompact space X, there does not exist any minimal intermediate algebra A for which A is A-compact. This completes the answer to a question raised by Redlin and Watson in 1987.     

Topological-algebraic properties of function spaces with set-open topologies

For a Tychonoff space X, we denote by Cλ(X) the space of all real-valued continuous functions on X with set-open topology. In this paper, we study the topological-algebraic properties of Cλ(X). Our main results state that (1) Cλ(X) is a topological vector space (a topological group) iff λ is a family of C-compact sets and Cλ(X)=Cλ^′(X), where λ^′ consists of all C-compact subsets of every set of λ. In particular, if Cλ(X) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family λ; (2) a topological group Cλ(X) is ω-narrow iff λ is a family of metrizable compact subsets of X.     

Description of Closed Maximal Ideals in Topological Algebras of Continuous Vector-Valued Functions

Let X be a completely regular Hausdorff space, A be a unital locally convex algebra with jointly continuous multiplication and C(X,A) be the algebra of all continuous A-valued functions on X equipped with the topology of \({\mathcal{K}(X)}\) -convergence. Moreover, let \({\mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(A)}\) denote the set of all closed maximal left and two-sided ideals in A, respectively. In this note, we describe all closed maximal left and two-sided ideals in C(X,A) and show that there exist bijections from \({\mathfrak{M}_{\ell}(C(X, A))}\) onto \({X \times \mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(C(X, A))}\) onto \({X \times \mathfrak{M}(A)}\) . We also present new characterizations of closed maximal ideals in C(X, A) when A is a unital commutative locally convex Gelfand–Mazur algebra with jointly continuous multiplication.     

Spaces of maps into topological group with the Whitney topology

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps endowed with the Whitney (graph) topology and by Cc(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X,G) is an l₂-manifold. In this article we show that if X is non-compact and not end-discrete then Cc(X,G) is an (R^∞×l₂)-manifold, and moreover the pair (C(X,G),Cc(X,G)) is locally homeomorphic to the pair of the box and the small box powers of l₂.     

When is the Isbell topology a group topology?

Conditions on a topological space X under which the space C(X,R) of continuous real-valued maps with the Isbell topology κ is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in Cκ(X,R) if and only if X is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in Cκ(X,R) if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if X is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces.     

Commutative Abelian Galois extensions of a Banach algebra

Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C(X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X.     

Automatic continuity and representation of group homomorphisms defined between groups of continuous functions

Let C(X,G) be the group of continuous functions from a topological space X into a topological group G with pointwise multiplication as the composition law, endowed with the uniform convergence topology. To what extent does the group structure of C(X,G) determine the topology of X? More generally, when does the existence of a group homomorphism H between the groups C(X,G) and C(Y,G) implies that there is a continuous map h of Y into X such that H is canonically represented by h? We prove that, for any topological group G and compact spaces X and Y, every non-vanishing C-isomorphism (defined below) H of C(X,G) into C(Y,G) is automatically continuous and can be canonically represented by a continuous map h of Y into X. Some applications to specific groups and examples are given in the paper.     

On derivable mappings

A linear mapping δ from an algebra A into an A-bimodule M is called derivable at c∈A if δ(a)b+aδ(b)=δ(c) for all a,b∈A with ab=c. For a norm-closed unital subalgebra A of operators on a Banach space X, we show that if C∈A has a right inverse in B(X) and the linear span of the range of rank-one operators in A is dense in X then the only derivable mappings at C from A into B(X) are derivations; in particular the result holds for all completely distributive subspace lattice algebras, J-subspace lattice algebras, and norm-closed unital standard algebras of B(X). As an application, every Jordan derivation from such an algebra into B(X) is a derivation. For a large class of reflexive algebras A on a Banach space X, we show that inner derivations from A into B(X) can be characterized by boundedness and derivability at any fixed C∈A, provided C has a right inverse in B(X). We also show that if A is a canonical subalgebra of an AF C^∗-algebra B and M is a unital Banach A-bimodule, then every bounded local derivation from A into M is a derivation; moreover, every bounded linear mapping from A into B that is derivable at the unit I is a derivation.     

On the supercyclicity and hypercyclicity of the operator algebra

Let B（X） be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or ＊-strong topology. We give sufficient conditions for a continuous linear mapping L ： B（X） →B（X） to be supercyclic or ,-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied＇ by Chan and here we obtain an analogous result in the case of ＊-strong operator topology.     

Proper actions, fixed-point algebras and naturality in nonabelian duality

Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on C₀(X). We consider a category in which the objects are C^∗-dynamical systems (A,G,α) for which there is an equivariant homomorphism of (C₀(X),γ) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra Aα which is Morita equivalent to A_×α,rG. We show that the assignment (A,α)?Aα is functorial, and that Rieffel's Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural.     

Group topologies coarser than the Isbell topology

The Isbell, compact-open and point-open topologies on the set C(X,R) of continuous real-valued maps can be represented as the dual topologies with respect to some collections α(X) of compact families of open subsets of a topological space X. Those α(X) for which addition is jointly continuous at the zero function in Cα(X,R) are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections α(X) for which Cα(X,R) is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if X is infraconsonant. Examples based on measure theoretic methods, that Cα(X,R) can be strictly finer than the compact-open topology, are given. To our knowledge, this is the first example of a splitting group topology strictly finer than the compact-open topology.