We extend the famous Kodaira-Spencer's completeness theorem for a family of deformations of complex structures (see [12]). As an application, we show that the canonical family constructed in [9] is versai.

This paper defines three simplicial approximation properties for maps of 2-cells and 2-spheres into spaces, each providing homotopical tameness conditions on the approximating images. These are the general position properties used in the two main results. The first shows that a resolvable generalized 3-manifold is a genuine 3- manifold if and only if it has the weakest of these approximation properties as well as a mild 3-dimensional disjoint disks condition known as the Light Map Separation Property. The second shows a resolvable generalized 3-manifold to be a 3-manifold if and only if it satisfies the strongest of these approximation properties.

This paper examines the simultaneous doubling of multiple intervals of a lattice in great detail. In the case of a finite set of W-failure intervals, it is shown that there in a unique smallest lattice mapping homomorphically onto the original lattice, in which the set of W-failures is removed. A nice description of this new lattice is given. This technique is used to show that every lattice that is a bounded homomorphic image of a free lattice has a projective cover. It is also used to give a sufficient condition for a fintely presented lattice to be weakly atomic and shows that the problem of which finitely presented lattices are finite is closely related to the problem of characterizing those finite lattices with a finite W-cover.

Let ƒ = (ƒ1,… ,ƒn): (ℝn, 0) → (ℝn, 0) be a real analytic mapping and 0 is isolated in ƒ-1 (0). The aim of this paper is to describe the degree deg0ƒ in terms of parametrizations of irreducible components of the real analytic curve given by the equations ƒ1(x) = ̇̇̇= ƒn-1 (x) = 0 near 0 ∈ ℝn.

C* -convex sets in matrix algebras are convex sets of matrices in which matrix-valued convex coefficients are admitted along with the usual scalar-valued convex coefficients. A Carathéodory-type theorem is developed for C*-convex hulls of compact sets of matrices, and applications of this theorem are given to the theory of matricial ranges. If T is an element in a unital C*-algebra , then for every n ∈ N, the n x n matricial range Wn(T) of T is a compact C* -convex set of n x n matrices. The basic relation W1(T) = conv σ-(T) is well known to hold if T exhibits the normal-like quality of having the spectral radius of β T + μ 1 coincide with the norm ||β T + μ 1|| for every pair of complex numbers β and μ. An extension of this relation to the matrix spaces is given by Theorem 2.6: Wn (T) is the C*-convex hull of the n x n matricial spectrum σn(T) of T if, for every B,M ∈ ℳn, the norm of T ⊗ B + 1 ⊗ M in ⊗ ℳn is the maximum value in {||∧⊗B + 1 ⊗M|| : Λ ∈ σn (T)}. The spatial matricial range of a Hilbert space operator is the analogue of the classical numerical range, although it can fail to be convex if n > 1. It is shown in § 3 that if T has a normal dilation N with σ (N) ⊂ σ (T), then the closure of the spatial matricial range of T is convex if and only if it is C*-convex.

The sequence of spaces of Maddox, c0(p), c(p)and l∞(p), are investigated. Here, p — (pk) is a bounded sequence of strictly positive numbers. It is observed that C0(P) is an echelon space of order 0 and that l∞(p) is a co-echelon space of order ∞, while clearly c(p) = c0(p) ⊗ 〈 (1,1,1,…) 〉. This sheds a new light on the topological and sequence space structure of these spaces: Based on the highly developed theory of (co-) echelon spaces all known and various new structural properties are derived.

We give a characterization of univalent positively oriented harmonic mappings ƒ defined on an exterior neighbourhood of the closed unit disk { z: | z| ≤1} such that .

We study skeleton C*-subalgebras of a given C*-algebra. We show that if A is a unital (non-unital but σ-unital) simple C*-algebra, ℳ is any unital (nonunital) matroid C* -algebra, then A contains a skeleton C*-subalgebra B with a quotient which is isomorphic to ℳ. Other results for skeleton C*-subalgebras are also obtained. Applications of these results to the structure of quasi-multipliers and perturbations of C*-algebras are given.

We are concerned here with the development of a more general real case of the classical theorem of Gelfand ([5], 3.1.20), which represents a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space.

In § 1 we point out that when looking at real algebras there is not always a one-to-one correspondence between the maximal ideals of the algebra B, denoted ℳ, and the set of unital (real) algebra homomorphisms from B into C, denoted by ΦB. This simple point and subsequent observations lead to a theory of representations of real commutative unital Banach algebras where elements are represented as sections of a bundle of real fields associated with the algebra (Theorem 3.5). After establishing this representation theorem, we look into the question of when a real commutative Banach algebra is already complex. There is a natural topological obstruction which we delineate. Theorem 4.8 gives equivalent conditions which determine whether such an algebra is already complex.

Finally, in § 5 we abstractly characterize those section algebras which appear as the target algebras for our Gelfand transform. We dub these algebras “almost complex C*- algebras” and provide a natural classification scheme.

We shall show that if a Riemann surface is continuable, then it admits one of three types of continuations. Using this classification of continuations, we construct two nontrivial examples of two-sheeted unlimited covering Riemann surfaces of the unit disk one of which is continuable and the other is not.

In this note we give a symplectic approach to Yang Mills theory for non commutative n-tori, inspired by the classical theory of Atiyah and Bott.

Let G be a profinite group. We define an extension (E, J) of G by a group A to consist of an exact sequence of groups together with a section j : G → E of K satisfying: for some open normal subgroup Sof G, and the map is continuous (A being discrete).

In [Con2] Connes introduced cyclic cohomology HC*(A) for an associative algebra A. When A is a complex algebra he constructed a Chern character for p-summable Fredholm modules over A taking values in HC*(A). As a very special case, when X is a closed C∞-manifold and A = C∞ (X), this construction recovers the usual Chern character, which is a rational isomorphism from the K-homology K0(X) to , the even dimensional deRham homology of X.

In this paper the question of weak and strong local optimality of a Lipschitz (as opposed to C1 ) extremal is addressed. We show that the classical Jacobi sufficient conditions can be extended to the case of Lipschitz candidates. The key idea for this achievement lies in proving that the “generalized” strengthened Weierstrass condition is equivalent to the existence of a “feedback control” function at which the maximum in the “true” Hamiltonian is attained. Then the Hamilton-Jacobi approach is pursued in order to conclude the result.