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.

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 HILBERT C*-BIMODULES OF FINITE INDEX AND APPROXIMATION PROPERTIES OF C*-ALGEBRAS

2017 ◽  
Vol 60 (2) ◽  
pp. 321-331
Author(s):  
MARZIEH FOROUGH ◽  
MASSOUD AMINI
Keyword(s):  
Finite Index ◽  
Morita Equivalence ◽  
Discrete Groups ◽  
Crossed Products ◽  
Approximation Properties ◽  
C Algebras ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
The Stability

AbstractLet A and B be arbitrary C*-algebras, we prove that the existence of a Hilbert A–B-bimodule of finite index ensures that the WEP, QWEP, and LLP along with other finite-dimensional approximation properties such as CBAP and (S)OAP are shared by A and B. For this, we first study the stability of the WEP, QWEP, and LLP under Morita equivalence of C*-algebras. We present examples of Hilbert A–B-bimodules, which are not of finite index, while such properties are shared between A and B. To this end, we study twisted crossed products by amenable discrete groups.

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.

Finite-dimensional approximation of the scattering matrix

1986 ◽  
Vol 64 (5) ◽  
pp. 611-616 ◽  
Author(s):  
Helmut Kröger ◽  
Anais Smailagic ◽  
Ralph Girard
Keyword(s):  
Weak Coupling ◽  
Weak Coupling Regime ◽  
S Wave ◽  
Proton Proton ◽  
Time Evolution Operator ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
S Matrix ◽  
Computer Calculations

A finite-dimensional nonperturbative approximation scheme of the time-evolution operator and the S matrix for relativistic field theories is discussed. It is amenable to computer calculations. Parallels with lattice-field theory are drawn. The method is outlined for the ϕ4 theory. Equivalence to standard perturbation theory in the weak-coupling regime is obtained in the limit of the approximation parameters. The method is tested numerically for nonrelativistic proton–proton s-wave scattering and the the ϕ4 model in the weak-coupling regime in 1 + 1 dimensions. In both examples, convergence to the reference solution is found.

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