Some Algebraic Features of Z2-graded KK-Theory

We study the relation of Z₂equivariant and Z₂graded KK-theory. The former is the universal stable, split exact and homotopy invariant theory on the category of Z₂graded C^*algebras and graded homomorphisms (Theorem 1). We obtain an abstract characterization for the product of the graded KK-functor (Theorem 2). We give generalizations to Z₂graded C^*algebras of the Universal Coefficent Theorem, Künneth Theorem and Künneth Theorem for tensor products. We prove some results about graded crossed products of Thom isomorphism and Pimsner-Voiculescu type (Theorem 3 and Corollary 2) and compute an example. We obtain a split surjective map KK(A,B) KK(A⁰,B⁰) commuting with products, where A⁰is a canonically defined trivially graded algebra for any Z₂-graded A.     

Zur Idealtheorie von Segal-Algebren

It is known that for every Segal algebra S¹(G) in L¹(G) with right approximate units there is a bijective correspondence between the closed right ideals of S¹(G) and those of L¹(G) ([3], §9, Theorem 1). For abelian groups H. Reiter showed that under this correspondence also the existence of approximate units is preserved ([3], §16, Theorem 1). Here among similar results a very simple proof of this fact is given for right approximate units in two-sided ideals which works without the assumption that G be abelian. In fact, the result can be established for abstract Segal algebras, in the sense of J. Burnham [1].     

On Infinite Camina Groups

A group G is called a Camina group if G′ ≠ G and each element x ∈ G?G′ satisfies the equation x ^G  = xG′, where x ^G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).     

Embeddings in Lattice-O *-Groups

In the paper, the following results are obtained: the existence of simple divisible lattice-O ^*-groups is established (Theorem 2.1) and it is proved that any countable lattice-orderable or right-orderable group can be isomorphically embedded in a simple divisible lattice-O ^*-group (Theorem 2.2 and Corollary 2.3).     

Unique continuation for solutions to pde's; between hörmander's theorem and holmgren' theorem

We prove a new unique continuation result for solutions to partial differential equations, “interpolating” between Holmgren's Theorem and Hörmander's Theorem. More precisely, under some partial analyticity assumptions on the coefficients we obtain an intermediate unique continuation result which is in general weaker than Holmgren's Theorem (which applies to problems with analytic coefficients) but stronger than Hörmander's Theorem (which applies to problems with C¹ coefficients). Some applications to the wave and the Schroedinger equation are considered next. In particular we obtain a result conjectured by Hörmander, namely that for the wave equation with C¹ but time independent coefficients one has unique continutaion across any noncharacteristic surface.     

Über den Typenkonvergenzsatz auf zusammenhängenden Lie Gruppen

The Convergence of Types Theorem on ^d is wellknown as an important tool for investigations on the limit behaviour of normalized sums or r.v. It is natural to look for a generalization for group-valued r.v. While for simply connected nilpotent Lie groups the Theorem is valid in general the existence of non-trivial compact subgroups causes problems. For compact extensions of nilpotent groups we prove restricted versions of the Convergence of Types Theorem.

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Herrn Prof. Dr. L. Schmetterer zum 70. Geburtstag gewidmet     

Restricted versions of the Tukey-Teichmüller theorem that are equivalent to the Boolean prime ideal theorem

We formulate a restricted version of the Tukey-Teichmüller Theorem that we denote by (rTT). We then prove that (rTT) and (BPI) are equivalent in ZF and that (rTT) applies rather naturally to several equivalent forms of (BPI): Alexander Subbase Theorem, Stone Representation Theorem, Model Existence and Compactness Theorems for propositional and first-order logic. We also give two variations of (rTT) that we denote by (rTT)⁺ and (rTT)⁺⁺; each is equivalent to (rTT) in ZF. The variation (rTT)⁺⁺ applies rather naturally to various Selection Lemmas due to Cowen, Engeler, and Rado.Dedicated to W.W. Comfort on the occasion of his seventieth birthday.     

A Banach-lattice version of the Josefson-Nissenzweig theorem

Let E, F be two Banach lattices with E order continuous. If F can be mapped positively onto E then the dual F^* contains a weak^* -null sequence of positive and norm-one elements (Theorem 1). This is a Banach-lattice version of the classical Josefson-Nissenzweig theorem. It is an immediate consequence of the dual characterization of order continuity: E is order continuous iff E is Dedekind complete and every norm-one and pairwise disjoint sequence in E^* is weak^*-null (Theorem 2).     

Combinatorics of open covers (XI): Menger- and Rothberger-bounded groups

We examine Menger-bounded (=o-bounded) and Rothberger-bounded groups. We give internal characterizations of groups having these properties in all finite powers (Theorems 6 and 7, and Theorem 15). In the metrizable case we also give characterizations in terms of measure-theoretic properties relative to left-invariant metrics (Theorems 12 and 19). Among metrizable σ-totally bounded groups we characterize the Rothberger-bounded groups by the corresponding game (Theorem 22).     

On delaunay oriented matroids for convex distance functions

For any finite point setS inE ^d, an oriented matroid DOM (S) can be defined in terms of howS is partitioned by Euclidean hyperspheres. This oriented matroid is related to the Delaunay triangulation ofS and is realizable, because of thelifting property of Delaunay triangulations. We prove that the same construction of aDelaunay oriented matroid can be performed with respect to any smooth, strictly convex distance function in the planeE ² (Theorem 3.5). For these distances, the existence of a Delaunay oriented matroid cannot follow from a lifting property, because Delaunay triangulations might be nonregular (Theorem 4.2(i). This is related to the fact that the Delaunay oriented matroid can be nonrealizable (Theorem 4.2(ii). This research was partially supported by the Spanish Grant DGICyT PB 92/0498-C02 and the David and Lucile Packard Foundation.     

Non-Liouville numbers and a Theorem of Hörmander

It is shown in this paper that Theorem 1 of [G. H. Meisters, “Translation-invariant linear forms and a formula for the Dirac measure,” J. Functional Analysis 8 (1971), 173–188] can be deduced from a very general result of Lars Hörmander, namely, Theorem 1 of “Generators for some rings of analytic functions” [Bull. Amer. Math. Soc.73 (1967), 943–949]. However, Hörmander's theorem is evidently not applicable in several other cases where Meisters'-type results have been obtained (e.g., Theorem 1 of G.H. Meisters and Wolfgang M. Schmidt, “Translation-invariant linear forms on L²(G) for compact abelian groups G,” J. Functional Analysis11 (1972), 407–424).     

On a generalized Minkowski inequality and its relation to dominates for t-norms

Leth:?⁺ → ?⁺ be a continuous strictly increasing function withh(0) = 0. Such functionsh give rise to a generalization of the Minkowski inequality; namely, (1) $$h^{ - 1} (h(a + b) + h(c + d)) \leqq h^{ - 1} (h(a + c) + h(b + d))$$ wherea, b, c, andd are arbitrary non-negative real numbers. Theorem 1 shows that, ifh and logh′ (e ^x ) are both convex functions, thenh satisfies (1). Theorem 2, the major result, demonstrates that, if bothh ₁ andh ₂ satisfy the hypotheses of Theorem 1, then the composition ofh ₁ withh ₂ also satisfies the hypotheses of Theorem 1 and hence the inequality (1). The remainder of the paper shows how (1) and Theorems 1 and 2 impinge on the dominates relation for strict t-norms. In particular, Theorem 3 shows how (1) can be interpreted as equivalent to the dominates relation for two strict t-norms. Theorem 4 shows how to use Theorems 1 and 3 to construct a strict t-norm which dominates a given strict t-norm. And, Theorem 5 shows how Theorem 2 can be used to give a qualified answer of yes to the open question of whether or not the dominates relation is a transitive relation.     

On Existence Theorems for Solutions of Non‐Linear Systems

An equivalent formulation of a recent result in [1] states that if the conditions of the Theorem of Newton‐Kantorovich are satisfied in a slightly modified (but equivalent) form in the maximum norm for a function g : G → ℝⁿ, G ⊆ ℝⁿ, guaranteeing the existence of a zero in D, then the conditions of Miranda's Theorem are automatically satisfied. We prove that this result holds for arbitrary norms if the conditions of the Theorem of Newton‐Kantorovich are suitable strengthened and Miranda's Theorem is suitably generalized.     

On the span of some three valued martingale difference sequences inL p (1<p<∞) andH 1

LetX ^p denote the span of a three valued martingale difference sequence with nested supports inL ^p. The spacesL ^{p, l p , l 2 , l p} ⊗l ², (Σ ⊗l ²) _p are the only isomorphic types which can be obtained in this way (Theorem 2). This result is obtained by interpolation from the correspondingH ¹ result (Theorem 1).     

Structure of finite nondispersible groups each nonmetacyclic subgroup of which is normal

We determine the structure of finite minimal nondispersible groups each nonmetacyclic subgroup of which is normal (Theorem 2) and describe all finite nondispersible groups each nonmetacyclic subgroup of which is normal (Theorem 3).     

Non-CLT groups of order pq 3

In this note we give a characterization of finite groups of order pq ³ (p, q primes) that fail to satisfy the Converse of Lagrange’s Theorem.     

The generalized analytic signal

This paper is an explication of the analytic signal in the generalized case, i.e., the analytic signal of a generalized function and of a generalized stochastic process. The contributions of the author are: (1) an L²-theory of distributions which, in the study of the analytic signal, has an advantage over the usual Schwartz-Itô-Gel'fand theory because the Cauchy representation is defined; (2) a proof (Theorem 2.5) that the Schwartz distributions δ, δ⁺, δ^? and ? may be extended to the L₂-case, expressions (Theorems 2.6 and 2.7) for their Hilbert and Fourier transforms in the L₂-case, and expressions (Section 2.1) for their analytic signals; (3) a proof (Theorem 3.3) that an orthogonal L₂-process, and therefore the Fourier transform of a second-order stationary stochastic process (Theorem 3.4), is strictly generalized; (4) a representation theorem (Theorem 3.5) which extends the Itô spectral representation theorem for stationary random distributions to the nonspectral, nonstationary, L₂-case; (5) expressions for the Cauchy representation (Theorem 3.6) and the analytic signal (Theorem 3.7) of an L₂-process; (6) an expression for and the covariance kernel of the analytic signal of white noise (Section 3.4). The word application in the text refers to the application of previously developed concepts.     

Automorphism groups of compact projective planes

Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hähl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).