Topic
Direct limit
About: Direct limit is a research topic. Over the lifetime, 551 publications have been published within this topic receiving 7319 citations. The topic is also known as: inductive limit.
Papers published on a yearly basis
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Papers
TL;DR: In this article, it was shown that for any connected CWcomplex X and for 5*(X) denoting any of the four functors Y., Y3 if, *, and *, which map from the category of topological spaces X to the category diff of Q-spectra, it is possible to verify the conjecture.
Abstract: duff In this paper we are concerned with the four functors Y., Y3 if, *, and *, which map from the category of topological spaces X to the category diff of Q-spectra. The functor 3*( ) (or ?if ( )) maps the space X to the Qspectrum of stable topological (or smooth) pseudoisotopies of X. (The spaces diff dif 3i(X), id9 (X), i > 0, are Hatcher's deloopings of 970(X), % Yd(X) (cf. [29])). The functor J*( ) maps a path-connected space X to the algebraic Ktheoretic S2-spectrum for the integral group ring Z7r1X. (The spaces X(X), i > 0, are the Gersten-Wagoner deloopings of Z%O(X); cf. [27, 52].) 5*(X) is also defined on a nonpath-connected space X to be () (X1), where {Xi: i e I} are the path components of X and iI (Xi) indicates the direct limit of all finite products of the {I (Xi): i E I}. If X is path connected then the functor *-' ( ) maps X to the L -surgery classifying spaces for oriented surgery problems with fundamental group r I1X identified by their fourfold periodicity k 90(X) = k+4j (X) (cf. [35, 39, 44, 47]). In addition, if X is not path connected then we set Y77(X) = tiEI5*fJ'(Xi), where {Xi: i E i} are the path components of X. Results obtained by the authors over the past five years (cf. [15-25] and, in particular, [19; 16, A.1, A.18]), together with many earlier results and conjectures which are reviewed in 1.6.1-1.6.6, suggest that, for any connected CWcomplex X and for 5* () denoting any of the above functors, the Q-spectrum *(X) should be computable in a simple way from the values taken by 9'*() on covering spaces of X, which have very simple fundamental groups. In more detail, we let 9'(X) denote the collection of all subgroups H c ir IX, which are either finite or virtually infinite cyclic. A group H is virtually infinite cyclic if there is a short exact sequence of groups 1 -+ Z -+ H -+ F -+ 1 with F equal to a finite group and Z equal to the infinite cyclic group. For each such group we let XH -+ X denote the covering space projection corresponding to H. Then the Q-spectrum 5? (X) should be computable in a simple way from the Q-spectra {5 (XH): H E 9 (X)}. The purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any X
365 citations
TL;DR: In this paper, Anderson-Fuller et al. extended the theory of Morita equivalence to rings without identity, and showed that a set of commuting idempotents is equivalent to a ring with unitary left modules.
Abstract: In the paper [1] Abrams made a first step in extending the theory of Morita equivalence to rings without identity. He considered rings in which a set of commuting idempotents is given such that every element of the ring admits one of these idempotents as a two-sided unit, and the categories of all left modules over these rings which are unitary in a natural sense. He proved that two such module categories over the rings R and S, say, are equivalent if and only if there exists a unitary left i?-module P which is a generator, the direct limit of a given kind of system of finitelygenerated projective modules, and such that S is isomorphic to the ring of certain endomorphisms of P. The aim of the present paper is to extend this theory in two ways: to cover a wider range of rings, and to transfer more of the classical Morita theory. Firstly, one can weaken the condition of commutativity of the idempotents in question: it sufficesto require that any two of them have a common upper bound under the natural partial order (i.e., any two elements of the ring admit a common two-sided identity), a condition which is fulfilledby all regular rings (regular in the sense of Neumann). Whenever one has such a system of idempotents, then any larger system, in particular, the set of all idempotents, is also such, which is not the case for the systems of Abrams. Secondly, by a suitable modification of some homological lemmas we obtain also the two-sided characterizations of Morita equivalence, arriving thus at a complete analogy to the classicalcase of rings with identity. Our presentation is a combination of those in Anderson-Fuller [2], §§21-22,and Bass [5] (see also [6], Chapter II). This machinery allows us to avoid the elaborate construction of Abrams. As examples we describe, among others, those rings with local units which are Morita equivalent to division rings and primary rings, respectively. The Rees matrix rings studied in [4] turn out to have a natural place in this theory. The theory we present here is a counterpart of the theory of Morita duality developed by Yamagata [10]. On the one hand, we shall use the same modified Hom-functors but for projective and not injective modules, and on the
137 citations
Journal Article•
TL;DR: In this article, it was shown that A can be written as an inductive limit of direct sums of matrix algebras over certain special 3-dimensional spaces, namely the tracial state space and the ordered K-group.
Abstract: Suppose that formula math. is a simple C*-algebra, where X n,i are compact metrizable spaces of uniformly bounded dimensions (this restriction can be relaxed to a condition of very slow dimension growth). It is proved in this article that A can be written as an inductive limit of direct sums of matrix algebras over certain special 3-dimensional spaces. As a consequence it is shown that this class of inductive limit C*-algebras is classified by the Elliott invariant - consisting of the ordered K-group and the tracial state space - in a subsequent paper joint with G. Elliott and L. Li (Part II of this series). (Note that the C*-algebras in this class do not enjoy the real rank zero property.).
130 citations
TL;DR: In this paper, it was shown that if A is the direct limit of finite-dimensional CSL algebras via *-extendable embeddings (e.g., a triangular AF algebra), then a local derivation on A must be a derivation.
117 citations
TL;DR: In this article, the authors expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two opera- tions: (1) extension and (2) direct limit.
Abstract: The technical lemma underlying the 5-dimensional topological s-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating ~-null immersions of disks. These conjectures are theorems precisely for those fundamental groups ("good groups") where the ~l-null disk lemma (NDL) holds. We expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two opera- tions: (1) extension and (2) direct limit. The finitely generated groups in this class are amenable and no amenable group is known to lie outside this class.
115 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 5 |
| 2024 | 6 |
| 2023 | 7 |
| 2022 | 27 |
| 2021 | 28 |
| 2020 | 22 |