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.

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].

Some C*-Algebras with Outer Derivations, II

1974 ◽  
Vol 26 (1) ◽  
pp. 185-189 ◽  
Author(s):  
George A. Elliott
Keyword(s):  
Tensor Product ◽  
Finite Number ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Direct Sum ◽  
Finite Dimensional

In this paper we shall consider the class of C*-algebras which are inductive limits of sequences of finite-dimensional C*-algebras. We shall give a complete description of those C*-algebras in this class every derivation of which is inner.Theorem. Let A be a C*-algebra. Suppose that A is the inductive limit of a sequence of finite-dimensional C*-algebras. Then the following statements are equivalent:(i) every derivation of A is inner;(ii) A is the direct sum of a finite number of algebras each of which is either commutative, the tensor product of a finite-dimensional and a commutative with unit, or simple with unit.

scholarly journals CERTAIN APERIODIC AUTOMORPHISMS OF UNITAL SIMPLE PROJECTIONLESS C*-ALGEBRAS

2009 ◽  
Vol 20 (10) ◽  
pp. 1233-1261 ◽  
Author(s):  
YASUHIKO SATO
Keyword(s):  
Inductive Limit ◽  
Crossed Products ◽  
Inductive Limits ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
Tracial State ◽  
C Algebras ◽  
Cyclic Groups ◽  
Unital Simple

Let G be an inductive limit of finite cyclic groups, and A be a unital simple projectionless C*-algebra with K1(A) ≅ G and a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut (A)/ WInn (A) are conjugate, where WInn (A) means the subgroup of Aut (A) consisting of automorphisms which are inner in the tracial representation.In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple A𝕋-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and crossed products by actions of ℤ with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism ᾶ of A with the Rohlin property such that ᾶ and α are asymptotically unitarily equivalent. For the proof we use an aperiodic automorphism of the Jiang-Su algebra.

scholarly journals $K$-Continuity Is Equivalent To $K$-Exactness

2016 ◽  
Vol 118 (1) ◽  
pp. 95
Author(s):  
Otgonbayar Uuye
Keyword(s):  
Tensor Product ◽  
Approximation Theory ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation

Let $A$ be a $C^{*}$-algebra. It is well known that the functor $B \mapsto A \otimes B$ of taking the minimal tensor product with $A$ preserves inductive limits if and only if it is exact. $C^{*}$-algebras with this property play an important role in the structure and finite-dimensional approximation theory of $C^{*}$-algebras. We consider a $K$-theoretic analogue of this result and show that the functor $B \mapsto K_{0}(A \otimes B)$ preserves inductive limits if and only if it is half-exact.

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.

scholarly journals Classification of certain inductive limit actions of compact groups on af algebras

2020 ◽  
pp. 2050128
Author(s):  
Qingyun Wang
Keyword(s):  
Compact Group ◽  
Inductive Limit ◽  
Complete Classification ◽  
Compact Groups ◽  
C Algebra ◽  
Finite Dimensional ◽  
K Theory ◽  
Af Algebras

Let [Formula: see text] be an AF algebra, [Formula: see text] be a compact group. We consider inductive limit actions of the form [Formula: see text], where [Formula: see text] is an action on the finite-dimensional C*-algebra [Formula: see text] which fixes each matrix summand. We give a complete classification up to conjugacy of such actions using twisted equivariant K-theory.

scholarly journals Nontrivially Noetherian $C^*$-algebras

2012 ◽  
Vol 111 (1) ◽  
pp. 135 ◽  
Author(s):  
Taylor Hines ◽  
Erik Walsberg
Keyword(s):  
Chain Condition ◽  
Ideal Structure ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Primitive Ideals ◽  
Descending Chain Condition ◽  
Ascending Chain Condition ◽  
The Ideal

We say that a $C^*$-algebra is Noetherian if it satisfies the ascending chain condition for two-sided closed ideals. A nontrivially Noetherian $C^*$-algebra is one with infinitely many ideals. Here, we show that nontrivially Noetherian $C^*$-algebras exist, and that a separable $C^*$-algebra is Noetherian if and only if it contains countably many ideals and has no infinite strictly ascending chain of primitive ideals. Furthermore, we prove that every Noetherian $C^*$-algebra has a finite-dimensional center. Where possible, we extend results about the ideal structure of $C^*$-algebras to Artinian $C^*$-algebras (those satisfying the descending chain condition for closed ideals).

scholarly journals Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

2017 ◽  
Vol 69 (3) ◽  
pp. 548-578 ◽  
Author(s):  
Michael Hartglass
Keyword(s):  
Differential Calculus ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Weighted Graph ◽  
Positive Cone ◽  
Von Neumann ◽  
Planar Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

scholarly journals Nilpotent Group C*-algebras as Compact Quantum Metric Spaces

2017 ◽  
Vol 60 (1) ◽  
pp. 77-94 ◽  
Author(s):  
Michael Christ ◽  
Marc A. Rieòel
Keyword(s):  
Finite Groups ◽  
Word Length ◽  
Polynomial Growth ◽  
Metric Spaces ◽  
Length Function ◽  
C Algebra ◽  
Pointwise Multiplication ◽  
C Algebras ◽  
Definition Of ◽  
Compact Quantum Metric Space

AbstractLet be a length function on a group G, and let M denote the operator of pointwise multiplication by on l2(G). Following Connes, M𝕃 can be used as a “Dirac” operator for the reduced group C*-algebra (G). It deûnes a Lipschitz seminorm on (G), which defines a metric on the state space of (G). We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-* topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on ûnitely generated nilpotent-by-finite groups.

Sign in / Sign up

Export Citation Format

Share Document