scholarly journals Rank zero and rank one sets of harmonic maps

2000 ◽  
Vol 7 (2) ◽  
pp. 417-442 ◽  
Author(s):  
Qing Han ◽  
Fanghua Lin
Keyword(s):  
Harmonic Maps ◽  
Rank Zero ◽  
Rank One

scholarly journals Harmonic maps into singular spaces andp-adic superrigidity for lattices in groups of rank one

1992 ◽  
Vol 76 (1) ◽  
pp. 165-246 ◽  
Author(s):  
Mikhail Gromov ◽  
Richard Schoen
Keyword(s):  
Harmonic Maps ◽  
Singular Spaces ◽  
Rank One

scholarly journals Inductive limit of direct sums of simple TAI algebras

2019 ◽  
pp. 1-26
Author(s):  
Bo Cui ◽  
Chunlan Jiang ◽  
Liangqing Li
Keyword(s):  
Inductive Limit ◽  
Direct Sums ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Rank One ◽  
Tracial Rank ◽  
Ideal Property ◽  
Funct Anal

An ATAI (or ATAF, respectively) algebra, introduced in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (or in [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], respectively) is an inductive limit [Formula: see text], where each [Formula: see text] is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404], the second author classified all ATAI algebras by an invariant consisting orderd total [Formula: see text]-theory and tracial state spaces of cut down algebras under an extra restriction that all element in [Formula: see text] are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic [Formula: see text]-group as an addition to the invariant used in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404]. The theorem is proved by reducing the class to the classification theorem of [Formula: see text] algebras with ideal property which is done in [G. Gong, C. Jiang and L. Li, A classification of inductive limit C*-algebras with ideal property, preprint (2016), arXiv:1607.07681]. Our theorem generalizes the main theorem of [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (see Corollary 4.3).

On Legendre transformations and umbilic catastrophes

1978 ◽  
Vol 83 (2) ◽  
pp. 273-288 ◽  
Author(s):  
M. J. Sewell
Keyword(s):  
Elementary Mathematics ◽  
Hessian Matrix ◽  
Full Rank ◽  
Rank Zero ◽  
Rank One ◽  
Dimension One

In a previous paper this author used elementary mathematics to describe some simple connexions between multi-valued Legendre transformations and those elementary catastrophes having one ‘behaviour’ variable, which belong to a family of polynomials called cuspoids (see Sewell (13)). The Legendre singularities permitted there are isolated and of dimension one, i.e. they exist where a Hessian matrix exceptionally has co-rank one (not more) and elsewhere has full rank (co-rank zero).

Ideal Structure of Multiplier Algebras of Simple C*-algebras With Real Rank Zero

2001 ◽  
Vol 53 (3) ◽  
pp. 592-630 ◽  
Author(s):  
Francesc Perera
Keyword(s):  
Stable Rank ◽  
Real Rank ◽  
Ideal Structure ◽  
Real Rank Zero ◽  
Von Neumann ◽  
Rank Zero ◽  
Rank One ◽  
A Chain ◽  
Multiplier Algebras ◽  
The Ideal

AbstractWe give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of σ-unital simple C*-algebras A with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of themultiplier algebra , is therefore analyzed. In important cases it is shown that, if A has finite scale then the quotient of modulo any closed ideal I that properly contains A has stable rank one. The intricacy of the ideal structure of is reflected in the fact that can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.

Homomorphisms From C(X) Into C*-Algebras

1997 ◽  
Vol 49 (5) ◽  
pp. 963-1009 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Countable Rank ◽  
Rank One ◽  
Dimensional Range

AbstractLet A be a simple C*-algebra with real rank zero, stable rank one and weakly unperforated K0(A) of countable rank. We show that a monomorphism Φ: C(S2) → A can be approximated pointwise by homomorphisms from C(S2) into A with finite dimensional range if and only if certain index vanishes. In particular,we show that every homomorphism ϕ from C(S2) into a UHF-algebra can be approximated pointwise by homomorphisms from C(S2) into the UHF-algebra with finite dimensional range.As an application, we show that if A is a simple C*-algebra of real rank zero and is an inductive limit of matrices over C(S2) then A is an AF-algebra. Similar results for tori are also obtained. Classification of Hom (C(X), A) for lower dimensional spaces is also studied.

scholarly journals On the Weak Order of Coxeter Groups

2019 ◽  
Vol 71 (2) ◽  
pp. 299-336 ◽  
Author(s):  
Matthew Dyer
Keyword(s):  
Complete Lattice ◽  
Coxeter Groups ◽  
Closure Operator ◽  
Root Systems ◽  
Bruhat Order ◽  
Weak Order ◽  
Maximal Subgroups ◽  
Rank Zero ◽  
Power Set ◽  
Rank One

AbstractThis paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

scholarly journals Strict Comparison of Positive Elements in Multiplier Algebras

2017 ◽  
Vol 69 (02) ◽  
pp. 373-407 ◽  
Author(s):  
Victor Kaftal ◽  
Ping Wong Ng ◽  
Shuang Zhang
Keyword(s):  
Positive Element ◽  
Real Rank ◽  
Real Rank Zero ◽  
Multiplier Algebra ◽  
Rank Zero ◽  
C Algebra ◽  
Positive Elements ◽  
Rank One ◽  
Multiplier Algebras ◽  
Positive Linear Combination

AbstractMain result: If a C*-algebrais simple,σ-unital, hasfinitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebraalso has strict comparison of positive elements by traces. The same results holds if finitely many extremal traces is replaced byquasicontinuous scale. A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary σ-unital C* -algebra can be approximated by a bi-diagonal series. As an application of strict comparison, ifis a simple separable stable C* -algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

scholarly journals Rank Growth of Elliptic Curves in Non-Abelian Extensions

2019 ◽  
Author(s):  
Robert J Lemke Oliver ◽  
Frank Thorne
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Positive Constant ◽  
Rank Zero ◽  
Abelian Extensions ◽  
Quadratic Twists ◽  
Rank One ◽  
Higher Rank ◽  
Parity Conjecture

Abstract Given an elliptic curve $E/\mathbb{Q}$, it is a conjecture of Goldfeld that asymptotically half of its quadratic twists will have rank zero and half will have rank one. Nevertheless, higher rank twists do occur: subject to the parity conjecture, Gouvêa and Mazur constructed $X^{1/2-\epsilon }$ twists by discriminants up to $X$ with rank at least two. For any $d\geq 3$, we build on their work to consider the rank growth of $E$ in degree $d$$S_d$-extensions of $\mathbb{Q}$ with discriminant up to $X$. We prove that there are at least $X^{c_d-\epsilon }$ such fields where the rank grows, where $c_d$ is a positive constant that tends to $1/4$ as $d\to \infty $. Moreover, subject to a suitable parity conjecture, we obtain the same result for fields for which the rank grows by at least two.

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