I2-lacunary strongly summability for multidimensional measurable functions

Let I2 ? P(N ? N) be a nontrivial ideal. We provide a new approach to the concept of I2-double lacunary statistical convergence and I2-lacunary strongly double summable by taking f(?,?), which is a multidimensional measurable real valued function on (1,?) ? (1,?). Additionally, we examine the relation between these two new methods.

An Application of Collapsing Levels to the Representation Theory of Affine Vertex Algebras

We discover a large class of simple affine vertex algebras $V_{k} ({\mathfrak{g}})$, associated to basic Lie superalgebras ${\mathfrak{g}}$ at non-admissible collapsing levels $k$, having exactly one irreducible ${\mathfrak{g}}$-locally finite module in the category ${\mathcal O}$. In the case when ${\mathfrak{g}}$ is a Lie algebra, we prove a complete reducibility result for $V_k({\mathfrak{g}})$-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra $V^k ({\mathfrak{g}})$ at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras $V_{-1/2} (C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one nontrivial ideal.

Index maps in the K-theory of graph algebras

Let C*(E) be the graph C*-algebra associated to a graph E and let J be a gauge-invariant ideal in C*(E). We compute the cyclic six-term exact sequence in K-theory associated to the extensionin terms of the adjacency matrix associated to E. The ordered six-term exact sequence is a complete stable isomorphism invariant for several classes of graph C*-algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, finite collections of such sequences constitute complete invariants.Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of E.

NONSTABILITY OF THE IDEALS OF THE CORONA ALGEBRA OF A SEPARABLE STABLE PRIME C*-ALGEBRA

Let [Formula: see text] be a separable stable and prime C*-algebra. We show that every proper nontrivial ideal in [Formula: see text] (and hence every proper ideal in [Formula: see text] which properly contains [Formula: see text]) is not stable. Some consequences are counterexamples naturally occuring, in the non-σ-unital case, to Zhang's Dichotomy as well as the Hjelmborg–Rørdam theory of stability.

Centraliser near-rings determined by fixed point free automorphism groups

SynopsisLet G be a group and let A be a fixed point free group of automorphisms of G. It is shown that the centraliser near-ring MA(G) has at most one nontrivial ideal. Conditions on the pair (A, G) are given which force MA(G) to be simple. It is shown that if a nonsimple near-ring MA(G) exists, then A and G have unusual properties.

Chang's conjecture and powers of singular cardinals

In [2] Galvin and Hajnal showed, as a corollary to a more general result, that if , is a strong limit cardinal, then . They established similar bounds for powers of singular cardinals of cofinality greater than ω. Jech and Prikry in [3] showed that the Galvin-Hajnal bound can be improved if we assume that ω1 carries an ω2 saturated ω1 complete, nontrivial ideal. (See [7] for definitions), namely: under the given assumption provided is a strong limit cardinal.In this paper we show that the same conclusion can be derived from Chang's Conjecture (see below) which is, at least consistencywise, a weaker assumption than the existence of an ω2 saturated ideal on ω1. We do not know if assumptions like these are necessary for obtaining the result.Our notations and terminology should be understood by any reader acquainted with set theory. Chang's Conjecture is the following model theoretic assumption introduced by C. C. Chang:which is deciphered as follows: Every structure 〈A, R,…〉 in a countable type where ∣A∣ = ω2, R ⊆ A, ∣R∣ = ω1 has an elementary substructure: 〈A′,R′,…〉 where ∣A′∣ = ω1 and ∣R′∣ = ω0. The consistency of Chang's Conjecture modulo the existence of Ramsey cardinals is claimed in [5].