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.

The definitions of the various proper homotopy groups correspond to three main geometrical ideas: sequences of spheres converging to a Freudenthal end (Brown groups); infinite cylinders giving the mobility of spheres towards a proper end (Čerin-Steenrod groups); sequences of spheres, each one movable to the next one following a proper end (Čech groups). The Brown and Čech groups have a rather complex structure and the calculations of these groups are very difficult (see [4]). The Čerin-Steenrod groups have a much simpler structure and this fact eases the computations.

Consider the initial boundary value problem

In the context of the heat conduction problem, this models the case where the heat flux across the ends at the rod is a function of the temperature. If the heat exchange between the rod and its surroundings is purely by convection, then one commonly assumes that f is a linear function of the difference in temperatures between the ends of the rod and that of the surroundings, (Newton's law of cooling). For the case of purely radiative transfer of energy a fourth power law for the function f is usual, (Stefan's law).

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

In [5], Ky Fan proved the following remarkable amenability “invariant subspace” theorem:

Let G be an amenable group of continuous, invertible linear operators acting on a locally convex space E. Let H be a closed subspace of finite codimension n in E and X⊂E be such that:

(i) H and X are G-invariant;

(ii) (e + H) ∩X is compact convex for all e ∈ E;

(iii) X contains an n-dimensional subspace V of E. Then there exists an n-dimensional subspace of E contained in X and invariant under G.

As usual in the theory of polynomial near-rings, we deal with right near-rings. If N = (N, +,·) is a near-ring, the set of distributive elements of N will be denoted by Nd;

It is easy to check that, if N is an abelian near-ring (i.e., r + s = s + r, for all r, s∈N), then Nd is a subring of N.

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).

In “Viability Theory”, we select trajectories which are viable in the sense that they always satisfy a given constraint. Since the fundamental work of Nagumo [26], we know that in order to guarantee existence of viable trajectories, we need to satisfy certain tangential conditions. In the case of differential inclusions and using the modern terminology and notation of tangent cones, this condition takes the form F(t, x) ∩ TK#φ, where F(.,.) is the orientor field involved in the differential inclusion, K is the viability (constraint) set and TK(x) is the tangent cone to K at x. Results on the existence of viable solutions for differential inclusions can be found in Aubin–Cellina [2] and Papageorgiou [30,32].

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].

For z in D and ζ in ∂D, we denote by pz(ζ) the Poisson kernel (1 − │z│2)│1 − ζ−2 for the open unit disc D. We ask for what countable sets {an:n∈ℕ} of points of D there exist complex numbers λn with

by which we mean that the series converges to zero in the norm of L1(∂D).

Let V and W be finite dimensional real vector spaces, k≧0 an integer. We write L(V, W) for the space of all linear maps V→W and Lk(V, W) for the subspace of maps with kernel of dimension k; in particular, L0(V, W) is the open subspace of injective linear maps. Thus Lk(ℝn, ℝn) is the space of n × n-matrices of rank n – k in the title. We also need the notation Gk(V) for the Grassmann manifold of K-dimensional subspaces of V.

This paper contains results related to Titchmarsh's convolution theorem and valid for , the additive group of Rn with the discrete topology. The method of proof consists in transferring the problem to Rn with the usual topology by a procedure which has been used earlier, for instance in Helson [3].

In Section 1, the classical support theorems are generalized to . In [1], Titchmarsh's convolution theorem [6] on R was generalized to convolutions of functions belonging to certain weighted Lp-spaces on R. Section 2 contains a corresponding generalization to weighted l2(Rd).

It should be observed that convolutions of elements f and g in l1() can be interpreted as convolutions of bounded discrete measures on Rn. Hence, in that case the support theorem (Theorem 4.33 of Hörmander [5]) is directly applicable to give the results of our Theorems 1 and 3. So the novelty in our theorems lies in the fact that they apply for instance to the case when it is only assumed f, g ∈l2(), together with support conditions. It is not known whether it suffices to assume f∈l1(), g∈lp(), when p > 2.

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].

Throughout this paper, we work in the category of (p-localized) spaces having the homotopy type of connected CW-complexes of finite type with base point. We consider a principal bundle

where Gn = SU(n), U(n) or Sp(n) and d = 1, 1 or 2 respectively. In this case, the bundle is obtained as an induced bundle by a mapping f of base space S2dn−1 from the classical group extension as follows:

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinates

where C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equation

in D can be approximated in the space by an entire Herglotz wave function

with kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

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.

The matricial Nevanlinna–Pick interpolation criterion determines when there is an analytic matrix contraction valued function on the complex unit disc which assumes preassigned n × n matrix values w1,…,wm at preassigned interpolation points z1,…,zm. Taking ∥wi∥ < 1, for i = 1,…,m, the necessary and sufficient condition is the positivity of the nm × nm matricial Pick matrix,

1. The subject-matter of this paper is in some sense known; but we will try to organise, explain and reprove it, and to give examples.

A collection = {Gα|α∈A} of proper subgroups Gα of a group G is a fibration of G if

It is of geometric interest to associate two semigroups to a group G with fibration :