scholarly journals Noncommutative Lattices and the Algebras of Their Continuous Functions

1998 ◽  
Vol 10 (04) ◽  
pp. 439-466 ◽  
Author(s):  
Elisa Ercolessi ◽  
Giovanni Landi ◽  
Paulo Teotonio-Sobrinho
Keyword(s):  
Quantum Physics ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Hausdorff Topology ◽  
Topological Information ◽  
C Algebras ◽  
Finite Dimensional ◽  
Noncommutative Algebras ◽  
Af Algebras ◽  
The Continuum

Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces of noncommutative C*-algebras. These noncommutative algebras play the same rôle as the algebra of continuous functions [Formula: see text] on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C*-algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.

scholarly journals MORPHISMS OF SIMPLE TRACIALLY AF ALGEBRAS

2004 ◽  
Vol 15 (09) ◽  
pp. 919-957 ◽  
Author(s):  
MARIUS DADARLAT
Keyword(s):  
Existence And Uniqueness ◽  
Residually Finite ◽  
C Algebras ◽  
Finite Dimensional ◽  
K Theory ◽  
Inner Automorphisms ◽  
Af Algebras ◽  
Universal Coefficient Theorem

Let A, B be separable simple unital tracially AF C*-algebras. Assuming that A is exact and satisfies the Universal Coefficient Theorem (UCT) in KK-theory, we prove the existence, and uniqueness modulo approximately inner automorphisms, of nuclear *-homomorphisms from A to B with prescribed K-theory data. This implies the AF-embeddability of separable exact residually finite-dimensional C*-algebras satisfying the UCT and reproves Huaxin Lin's theorem on the classification of nuclear tracially AF C*-algebras.

scholarly journals A Continuous Field of Projectionless C*-Algebras

2001 ◽  
Vol 53 (1) ◽  
pp. 51-72 ◽  
Author(s):  
Andrew Dean
Keyword(s):  
Continuous Functions ◽  
Unit Interval ◽  
Type I ◽  
Crossed Products ◽  
Inductive Limits ◽  
Direct Sums ◽  
Matrix Algebras ◽  
C Algebras ◽  
Finite Dimensional ◽  
Continuous Field

AbstractWe use some results about stable relations to show that some of the simple, stable, projectionless crossed products of O2 by considered by Kishimoto and Kumjian are inductive limits of type C*-algebras. The type I C*-algebras that arise are pullbacks of finite direct sums of matrix algebras over the continuous functions on the unit interval by finite dimensional C*-algebras.

scholarly journals σ-Continuous Functions and Related Cardinal Characteristics of the Continuum

2020 ◽  
Vol 76 (1) ◽  
pp. 1-10
Author(s):  
Taras Banakh
Keyword(s):  
Continuous Functions ◽  
Topological Spaces ◽  
Cardinal Characteristics ◽  
The Continuum

AbstractA function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous) if there exists a (closed) cover {Xn}n∈ω of X such that for every n ∈ ω the restriction f ↾ Xn is continuous. By 𝔠 σ (resp. 𝔠¯σ)we denote the largest cardinal κ ≤ 𝔠 such that every function f : X → ℝ defined on a subset X ⊂ ℝ of cardinality |X| <κ is σ-continuous (resp. ¯σ-continuous). It is clear that ω1 ≤ 𝔠¯σ ≤ 𝔠 σ ≤ 𝔠.We prove that 𝔭 ≤ 𝔮0 = 𝔠¯σ =min{𝔠 σ, 𝔟, 𝔮 }≤ 𝔠 σ ≤ min{non(ℳ), non(𝒩)}.

C*-ALGEBRAIC METHODS IN STATISTICAL MECHANICS AND FIELD THEORY

1990 ◽  
Vol 04 (05) ◽  
pp. 1069-1118 ◽  
Author(s):  
David E. EVANS
Keyword(s):  
Field Theory ◽  
Statistical Mechanics ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Inductive Limits ◽  
Von Neumann ◽  
C Algebras ◽  
Conformal Field ◽  
Finite Dimensional ◽  
Af Algebras

We survey the recent work in non-commutative operator algebras (especially AF-algebras, those which are inductive limits of finite dimensional C*-algebras) and which arise in studying critical phenomena in classical statistical mechanics and conformal field theory, from a C*- or topological viewpoint, rather than a von Neumann algebra/measure theoretic one.

scholarly journals Classification of Integrodifferential C-Algebras*

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1900
Author(s):  
Anton A. Kutsenko
Keyword(s):  
Infinite Product ◽  
Complete Characterization ◽  
Difference Operators ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Differential Algebras ◽  
Af Algebras ◽  
Product Of Matrices

The infinite product of matrices with integer entries, known as a modified Glimm–Bratteli symbol n, is a new, sufficiently simple, and very powerful tool for the characterization of approximately finite-dimensional (AF) algebras. This symbol provides a convenient algebraic representation of the Bratteli diagram for AF algebras in the same way as was previously performed by J. Glimm for more simple uniformly hyperfinite (UHF) algebras. We apply this symbol to characterize integrodifferential algebras. The integrodifferential algebra FN,M is the C*-algebra generated by the following operators acting on L2([0,1)N→CM): (1) operators of multiplication by bounded matrix-valued functions, (2) finite-difference operators, and (3) integral operators. Most of the operators and their approximations studying in physics belong to these algebras. We give a complete characterization of FN,M. In particular, we show that FN,M does not depend on M, but depends on N. At the same time, it is known that differential algebras HN,M, generated by the operators (1) and (2) only, do not depend on both dimensions N and M; they are all *-isomorphic to the universal UHF algebra. We explicitly compute the Glimm–Bratteli symbols (for HN,M, it was already computed earlier) which completely characterize the corresponding AF algebras. This symbol n is an infinite product of matrices with nonnegative integer entries. Roughly speaking, all the symmetries appearing in the approximation of complex infinite-dimensional integrodifferential and differential algebras by finite-dimensional ones are coded by a product of integer matrices.

INDUCTIVE LIMITS OF C*-ALGEBRAS AND COMPACT QUANTUM METRIC SPACES

2020 ◽  
pp. 1-24
Author(s):  
KONRAD AGUILAR
Keyword(s):  
Inductive Limit ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Ideal Space ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Af Algebras ◽  
The Ideal

Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

Crossed products of C∗-algebras C(X,A) and their applications

2016 ◽  
Vol 27 (03) ◽  
pp. 1650029
Author(s):  
Jiajie Hua
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Crossed Products ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
C Algebras ◽  
Finite Dimensional ◽  
Tracial Rank ◽  
Stably Finite ◽  
Unital Simple

Let [Formula: see text] be an infinite compact metric space with finite covering dimension, let [Formula: see text] be a unital separable simple AH-algebra with no dimension growth, and denote by [Formula: see text] the [Formula: see text]-algebra of all continuous functions from [Formula: see text] to [Formula: see text] Suppose that [Formula: see text] is a minimal group action and the induced [Formula: see text]-action on [Formula: see text] is free. Under certain conditions, we show the crossed product [Formula: see text]-algebra [Formula: see text] has rational tracial rank zero and hence is classified by its Elliott invariant. Next, we show the following: Let [Formula: see text] be a Cantor set, let [Formula: see text] be a stably finite unital separable simple [Formula: see text]-algebra which is rationally TA[Formula: see text] where [Formula: see text] is a class of separable unital [Formula: see text]-algebras which is closed under tensoring with finite dimensional [Formula: see text]-algebras and closed under taking unital hereditary sub-[Formula: see text]-algebras, and let [Formula: see text]. Under certain conditions, we conclude that [Formula: see text] is rationally TA[Formula: see text] Finally, we classify the crossed products of certain unital simple [Formula: see text]-algebras by using the crossed products of [Formula: see text].

scholarly journals A Characterization of the discontinuity point set of strongly separately continuous functions on products

2016 ◽  
Vol 66 (6) ◽  
Author(s):  
Olena Karlova ◽  
Volodymyr Mykhaylyuk
Keyword(s):  
Continuous Mapping ◽  
Discontinuity Point ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Finite Dimensional ◽  
Continuous Mappings ◽  
Countable Product ◽  
Topology Of Pointwise Convergence ◽  
Point Set ◽  
Necessary And Sufficient

AbstractWe study properties of strongly separately continuous mappings defined on subsets of products of topological spaces equipped with the topology of pointwise convergence. In particular, we give a necessary and sufficient condition for a strongly separately continuous mapping to be continuous on a product of an arbitrary family of topological spaces. Moreover, we characterize the discontinuity point set of strongly separately continuous function defined on a subset of countable product of finite-dimensional normed spaces.

scholarly journals R-continuous functions

1992 ◽  
Vol 15 (1) ◽  
pp. 57-64 ◽  
Author(s):  
Ch. Konstadilaki-Savvapoulou ◽  
D. Janković
Keyword(s):  
Continuous Functions ◽  
Strong Form ◽  
Topological Spaces ◽  
Strong Continuity ◽  
Closed Graph ◽  
Continuity Properties

A strong form of continuity of functions between topological spaces is introduced and studied. It is shown that in many known results, especially closed graph theorems, functions under consideration areR-continuous. Several results in the literature concerning strong continuity properties are generalized and/or improved.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

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