Parallel to the first two authors’ earlier classification of separable unital oneparameter continuous fields of Kirchberg algebras with torsion free K-groups supported in one dimension, one-parameter separable unital continuous fields of AF-algebras are classified by their ordered K0-sheaves. We prove Effros-Handelman-Shen type theorems for separable unital one-parameter continuous fields of AF...

We define natural A ∞ -transformations and construct A ∞ -category of A ∞ -functors. The notion of non-strict units in an A ∞ -category is introduced. The 2-category of (unital) A ∞ -categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta6...

The article presents some results concerning H-bases and theirs applications in multivariate interpolation. We derived the space of reduced polynomials with respect to a particular inner product. We made some connections with least interpolation and presented two application of the connection between spaces of reduced polynomials modulo a H-basis and spaces of ideal interpolation. 2000 Mathemat...

A unital l-group (G,u) is an abelian group G equipped with a translation-invariant lattice-order and a distinguished element u, called orderunit, whose positive integer multiples eventually dominate each element of G. We classify finitely generated unital l-groups by sequences W = (W0,W1, . . .) of weighted abstract simplicial complexes, where Wt+1 is obtained from Wt either by the classical Al...

We show that morphisms from n homotopy unital A∞-algebras to a single one are maps over an operad module with n + 1 commuting actions of the operad A ∞, whose algebras are homotopy unital A∞-algebras. The operad A∞ and modules over it have two useful gradings related by isomorphisms which change the degree. The composition of A ∞-morphisms with several entries is presented as a convolution of a...

All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R(M), the idealization of M . Homogenous ideals of R(M) have the form I (+)N , where I is an ideal of R and N a submodule of M such that IM ⊆ N . A ring R (M) is called a homogeneous ring if every ideal of R (M) is homogeneous. In this paper we continue our recent work on the idealization of m...

theory of schemes led to progress on classical and concrete problems about curves and surfaces. 3. A theorem of Farah The following theorem appears in a paper by by Ilijas Farah (“All automorphisms of the Calkin algebra are inner”, Annals of Mathematics 173 (2011) 619–661) If Todorcevic’s axiom (TA) holds then all automorphisms of the Calkin algebra are inner. Since the focus of this lecture se...

For a quartic primitive CM field K, we say that a rational prime p is evil if at least one of the abelian varieties with CM by K reduces modulo a prime ideal p|p to a product of supersingular elliptic curves with the product polarization. We call such primes evil primes for K. In [GL], we showed that for fixed K, such primes are bounded by a quantity related to the discriminant of the field K. ...

The existence of exact upper bounds for increasing sequences of ordinal functions modulo an ideal is discussed. The main theorem (Theorem 18 below) gives a necessary and suucient condition for the existence of an exact upper bound f for a < I-increasing sequence f = hf : < i On A where > jAj + is regular: an eub f with lim inf I cff(a) = exists if and only if for every regular 2 (jAj;) the set ...

We prove that under a mild hypothesis, the infinite tensor power of a unital separable C∗-algebra absorbs the Jiang–Su algebra Z tensorially. Combining this result with a recent theorem of Winter, we complete Elliott’s classification program for strongly self-absorbing ASH algebras. We also give a succinct universal property for Z in an ambient category so large that there are no unital separab...