Diagonalizing matrices over C(X)

A complete characterization of those compact Hausdorff spaces is given such that for every n, each normal element in the algebra C(X)?M_n of continuous functions from X to $\text{M}$_n can be continuously diagonalized. The conditions are that X be a sub-Stonean space with dim X ? 2 and carries no nontrivial G-bundles over any closed subset, for G a symmetric group or the circle group. In particular, diagonalization is assured on every totally disconnected sub-Stonean space, but also on connected spaces of the form β(Y)/Y, where Y is a simply-connected (noncompact) graph.     

The central configurations of four masses x, −x, y, −y

The configuration of a homothetic motion in the N-body problem is called a central configuration. In this paper, we prove that there are exactly three planar non-collinear central configurations for masses x, −x, y, −y with x≠y (a parallelogram and two trapezoids) and two planar non-collinear central configurations for masses x, −x, x, −x (two diamonds). Except the case studied here, the only known case where the four-body central configurations with non-vanishing masses can be listed is the case with equal masses (A. Albouy, 1995-1996), which requires the use of a symbolic computation program. Thanks to a lemma used in the proof of our result, we also show that a co-circular four-body central configuration has non-vanishing total mass or vanishing multiplier.     

Property C. Wallman basis and S-metrizability

In this present paper we prove that every Lindelof space which has a perfect locally connected Hausdorff compactification, has property C. (This latter concept was introduced by R.F. Dickman Jr). We make clear that this class of Lindelöf spaces properly contains the class of paracompact, connected, locally compact and locally connected spaces, as well as the class of those spaces whose topology can be induced by a metric with property S (or S-metrizable spaces). In this fashion, we simultaneously generalize two previous results of Dickman on spaces with property C. The use of Wallman basis with certain connectedness properties turns out to be a very convenient tool in the construction of locally connected compactifications as well as in characterizing S-metrizable spaces.     

A second-dual method for C(X) isomorphisms

Let X and Y denote compact Hausdorff spaces, C(X) and C(Y) the sup-norm Banach spaces of continuous real-valued functions on X and Y respectively, and φ: C(X) → C(Y) an onto isomorphism whose bound is less than three. With the aid of second duals, we construct a natural function from X into the set of finite subsets of Y. This provides a new approach to results of Amir and Cambern. We then obtain necessary conditions on X and Y for the existence of a bound-two isomorphism; for instance, the existence of open sets U and V of X and Y respectively such that U and V are homeomorphic and XßU and YßV are homeomorphic.     

Weak barrelledness for C(X) spaces

Buchwalter and Schmets reconciled C_c(X) and C_p(X) spaces with most of the weak barrelledness conditions of 1973, but could not determine if -barrelled ⇔ ?^∞-barrelled for C_c(X). The areas grew apart. Full reconciliation with the fourteen conditions adopted by Saxon and Sánchez Ruiz needs their 1997 characterization of Ruess' property (L), which allows us to reduce the C_c(X) problem to its 1973 status and solve it by carefully translating the topology of Kunen (1980) and van Mill (1982) to find the example that eluded Buchwalter and Schmets. The more tractable C_p(X) readily partitions the conditions into just two equivalence classes, the same as for metrizable locally convex spaces, instead of the five required for C_c(X) spaces. Our paper elicits others, soon to appear, that analytically characterize when the Tychonov space X is pseudocompact, or Warner bounded, or when C_c(X) is a df-space (Jarchow's 1981 question).     

Cyclic vectors and cellular indecomposable operators on Qp spaces

We identify the functions whose polynomial multiples are weak* dense in Q p spaces and prove that if | f (z) | ≥ | g(z) | and g is cyclic in Q p , then f is cyclic in Q p . We also show that the multiplication operator M z on Q p spaces is cellular indecomposable.     

Explicit characterizations of orthogonality in lS(C)

In a previous paper, the author used a notion of orthogonality introduced in another article to establish characterizations for orthogonality in the spaces l_S^p(C), 1?p<∞, thus obtaining generalizations of the usual characterization of orthogonality in the Hilbert spaces l_S²(C), via inner products. In this paper we make explicit these characterizations for some of the spaces l_S^p(C). We finish by presenting some remarks and open problems.     

Constructing aD, non-D-spaces

We introduce a general method to construct 0-dimensional, scattered T₂ spaces which are not linearly D. The construction is used to show that there are aD, non-D-spaces, answering a question of Arhangel?skii. The latter example is achieved using Shelah?s club guessing principles.     

Characterizations of spaces by embeddings in βX

In this paper we obtain characterizations of metrizable spaces, paracompact M-spaces, Moore spaces and semimetrizable spaces in terms of the way those spaces are embedded in their Stone-?ech compactification. In addition, we give an internal characterization of paracompact M-spaces which we use in the proof of the embedding characterization.     

Generalized G-KKM Theorems in Generalized Convex Spaces and Their Applications

Some new generalized G-KKM and generalized S-KKM theorems are proved under the noncompact setting of generalized convex spaces. As applications, some new minimax inequalities, saddle point theorems, a coincidence theorem, and a fixed point theorem are given in generalized convex spaces. These theorems improve and generalize many important known results in recent literature.     

Configuration spaces are not homotopy invariant

We present a counterexample to the conjecture on the homotopy invariance of configuration spaces. More precisely, we consider the lens spaces L_7,1 and L_7,2, and prove that their configuration spaces are not homotopy equivalent by showing that their universal coverings have different Massey products.     

An operator space approach to condition C

We show that there exists a natural embedding from the tensor product V∗∗⊗W∗∗ of the biduals of operator spaces V and W into the bidual of the injective tensor product of V and W, which is separately weak^∗ continuous. From this, we define condition C for operator spaces.     

Vanishing exponential integrability for functions whose gradients belong to L(log(e+L))

If the gradient of u(x) is nth power locally integrable on Euclidean n-space, then the integral average over a ball B of the exponential of a constant multiple of |u(x)−u_B|^n/(n−1), u_B=average of u over B, tends to 1 as the radius of B shrinks to zero—for quasi almost all center points. This refines a result of N. Trudinger (1967). We prove here a similar result for the class of gradients in Lⁿ(log(e+L))^α, 0?α?n−1. The results depend on a capacitary strong-type inequality for these spaces.     

Smoothness and duality in Lp(E, μ)

Let (T, Σ, μ) be a measure space, E a Banach space, and L_p(E, μ) the Lebesque-Bochner function spaces for 1 < p < ∞. It is shown that L_p(E, μ) is smooth if and only if E is smooth. From this result a Radon-Nikodym theorem for conjugates of smooth Banach spaces is established, and thus a general geometric condition on E sufficient to ensure that $\text{L}{}_{\mathrm{p}}\text{(E, μ)}{}^1\text{? L}{}_{\mathrm{q}}\text{(E}{}^1\text{, μ)}$ for all p, 1 < p < ∞. Alternate proofs of certain known results concerning the duals of L_p(E, μ) spaces are provided.     

Translation-invariant linear forms on C0(G), C(G), Lp(G) for noncompact groups

Suppose G is a locally compact noncompact group. For abelian such G's, it is shown in this paper that L¹(G), C(G), and L^∞(G) always have discontinuous translation-invariant linear forms(TILF's) while C₀(G) and L^p(G) for 1 < p < ∞ have such forms if and only if $\text{G}\text{H}$ is a torsion group for some open σ-compact subgroup H of $\text{G}$. For σ-compact amenable G's, all the above spaces have discontinuous left TILF's.     

On functions in p-adic BMO and the distribution of prime integers

We study the spaces BMOp of functions of bounded mean oscillation modeled on a p-adic martingale, and determine their relationship with the ordinary, continuous space BMO of functions of bounded mean oscillation. Somewhat surprisingly, these results are related to information about the distribution of primes.     

Minimisation of functions of a positive semidefinite matrix A subject to AX = 0

A common problem in multivariate analysis is that of minimising or maximising a function f of a positive semidefinite matrix A subject possibly to AX = 0. Typically A is a variance-covariance matrix. Using the theory of nearest point projections in Hilbert spaces, it is shown that the solution satisfies the equation f′(A) + M ? A = 0, where A = P₀(M) and P₀ is a certain projection operator.     

On Banach-Stone type theorems in uniform multiplicative subgroups of C(K)-spaces

In this article, we study the preservation properties of(ilov) boundary of multiplicative subgroups in C(X) spaces for non-surjective norm-preserving multiplicative maps.We also show a sufficient condition for surjective maps between groups of positive continuous functions to be a composition operator.