Let E be a Banach sequence space with the property that if (αi) ∈ E and |βi|≦|αi| for all i then (βi) ∈ E and ‖(βi)‖E≦‖(αi)‖E. For example E could be co, lp or some Orlicz sequence space. If (Xn) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum of the Xn's and symbolize by ⊕EXn. Specifically, ⊕EXn = {(xn)|(xn)∈Xn and (‖xn‖)∈E}. The E sum is a Banach space with norm defined by: ‖(xn)‖ = ‖(‖xn‖)‖E. This type of space has long been the source of examples and counter-examples in the geometric theory of Banach spaces. For instance, Day [7] used E=lp and Xk=lqk, with appropriate choice of qk, to give an example of a reflexive Banach space not isomorphic to any uniformly conves Banach space. Recently VanDulst and Devalk [33] have considered Orlicz sums of Banach spaces in their studies of Kadec-Klee property.

Throughout this paper all spaces are T1 and N will denote the set of all positive integer numbers.

A quasi-metric on a set X is a non-negative real-valued function d on X × X such that, for all x, y, z ∈ X, (i) d(x, y) = 0 if, and only if, x = ysemicolon (ii) d(x, y)≦d(x, z) + d(z, y).

Let T2 = {(eix1, eix2):0 ≦ xj<2π, j=1,2} be a two dimensional torus and r, s, t and k be positive integers with k>r+s+t–2. Our main object is to study the approximation and interpolation properties of a class of smooth functions whose restrictions to each triangle of a three direction mesh lie in the linear span of or 0≦μ≦r–1, r+s–l≦μ+ν≦r+s+t–2, or 0≦ν≦s–1, r+s–1≦μ+ν≦r+s+t–2} Where (z1, z2) ∈ T2.

Throughout the paper, the symbols G1 and G2 will denote two locally compact abelian groups with character groups X1 and X2, respectively. Haar measures on Gj are denoted by μj; the ones on Xj are denoted by θj (j=1,2). The measures μj and θj are normalized so that the Plancherel Theorem holds (see [7, p. 226, Theorem 31.18]).

If G is an additive (but not necessarily abelian) group and S is a semigroup of endomorphisms of G, the endomorphism near-ring R of G generated by S consists of all the expressions of the form ɛ1s1+…+ɛnsnwhere ɛi=±1 and si∈S for each i. When functions are written on the right, R forms a distributively generated left near-ring under pointwise addition and composition of functions. A basic reference on near-rings which has a substantial treatment of endomorphism near-rings is [6].

Zemanian [17] obtained abelian theorems for the Hankel and K-transforms of functions and then extended his results to the corresponding transforms of distributions in the sense of Schwartz [11]. Jones [6] has discussed at length asymptotic behaviours of transforms generalized in his sense. Following the technique of Zemanian many authors have obtained abelian theorems for more general transforms of functions and distributions in the sense of Schwartz. Mention may be made of the works of Joshi and Saxena [7], Lavoine and Misra [8] and Pathak [10]. However, these authors were confined to the transforms of real variables only.

In [7, Section 5], Glimm showed that if φ and ψ are inequivalent pure states of a liminal C*-algebra A such that the Gelfand-Naimark-Segal (GNS) representations πφ and πψ cannot be separated by disjoint open subsets of the spectrum  then ½ (φ+ψ) is a weak*-limit of pure states. We extend this to arbitrary C*-algebras (and more general convex combinations) by means of what we hope will be regarded as a transparent proof based on the notion of transition probabilities. As an application, we show that if J is a proper primal ideal of a separable C*-algebra A then there exists a state φ in (the pure state space) such that J=ker πφ (Theorem 3). The significance of this is discussed below after the introduction of further notation and terminology.

Let X be a complex Banach space, and let and denote respectively the algebras of bounded and compact operators on X. The quotient algebra is called the Calkin algebra associated with X. It is known that both and are complex Banach algebras with unit e. For such unital Banach algebras B, set

and define the numerical range of x ∈ B as

x is said to be hermitian if W(x)⊆R. It is known that

Fact 1. ([4 vol. I, p. 46]) x is hermitian if and only if ‖eiαx‖ = (or ≦)1 for all α ∈ R, where ex is defined by

Multipoint Padé fractions were introduced in [2]. They are continued fractions defined in the following way:

Let {a1,a2,…,ap} be given fixed points in the complex plane. For each n ≧1 let an = am where 1≦m≦p and n≡m (mod p).

We are concerned with the following classical version of the Borsuk–Ulam theorem: Let f:Sn→Rk be a map and let Af = {x∈Sn|fx= f(−x)}. Then, if k≦n, Af≠φ. In fact, theorems due to Yang [17] give an estimation of the size of Af in terms of the cohomology index. This classical theorem concerns the antipodal action of the group G=ℤ2 on Sn. It has been generalized and extended in many ways (see a comprehensive expository article by Steinlein [16]). This author ([9, 10)] and Nakaoka [14] proved “continuous” or “parameterized” versions of the theorem. Analogous theorems for actions of the groups G=S1 or S3 have been proved in [11], and [12]; compare also [4, 5, 6].

By the work of Taskinen (see [4, 5]), we know that there is a Fréchet space E such that Lb(E, l2) is not a (DF)-space. Moreover there is a Fréchet–Montel space F such that is not (DF). In this second example, the duality theorem of Buchwalter (cf. [2, §45.3]) can be applied to obtain that and hence is a (gDF)-space (cf. [1, Ch. 12 or 3, Ch. 8]). The (gDF)-spaces were introduced by several authors to extend the (DF)-spaces of Grothendieck and to provide an adequate frame to consider strict topologies.

In 1982, the first exotic ℝ4 was discovered—a smooth manifold homeomorphic to ℝ4, but not diffeomorphic to it. The object shocked topologists by its open defiance of the rules of high-dimensional smoothing theory. The exotic ℝ4 was constructed by connecting the two powerful machines of Freedman [4] and Donaldson [2] to earlier work of Casson [1].

When the function f(u) is of “bistable type’, i.e. has two zeros h̲ and h+ at which f' is negative and (for simplicity) has only one other zero between them, then the constant functions u = h± are L∞-stable solutions of the nonlinear diffusion equation

In addition, there are travelling wave solutions u+(x, t) and u̲(x, t) which, if

connect h+ to h̲ in the sense that

the convergence being uniform on bounded x-intervals. These solutions are of the form

where U(z) is a monotone function (the wave's profile), U(±∞) = h±, and the velocity c is a specific positive number depending on the function f.

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaces

where H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).