The range of the invariant for ring and C*-algebra direct limits of finite-dimensional semisimple real algebras

2013 ◽  
Vol 69 (2) ◽  
pp. 327-338
Author(s):  
P.J. Stacey
Keyword(s):  
C Algebra ◽  
Finite Dimensional ◽  
Direct Limits ◽  
Real Algebras

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].

scholarly journals K-loop Structures Raised by the Direct Limits of Pseudo Unitary $U(p, a_{n})$ and Pseudo Orthogonal $O(p, a_{n})$ Groups

2017 ◽  
Vol 9 (5) ◽  
pp. 37
Author(s):  
ALPER BULUT
Keyword(s):  
Relativistic Theory ◽  
Direct Limit ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Inverse Property ◽  
Finite Dimensional ◽  
Direct Limits ◽  
Bol Loop

A left Bol loop satisfying the automorphic inverse property is called a K-loop or a gyrocommutative gyrogroup. K-loops have been in the centre of attraction since its first discovery by A.A. Ungar in the context of Einstein's 1905 relativistic theory. In this paper some of the infinite dimensional K-loops are built from the direct limit of finite dimensional group transversals.

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 Conditions for primitivity of unital amalgamated full free products of finite-dimensional C*-algebras

2014 ◽  
Vol 25 (09) ◽  
pp. 1450086
Author(s):  
Francisco Torres-Ayala
Keyword(s):  
Free Products ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Trivial Center

We consider amalgamated unital full free products of the form A1*D A2, where A1, A2 and D are finite-dimensional C*-algebras and there are faithful traces on A1 and A2 whose restrictions to D agree. We provide several conditions on the matrices of partial multiplicities of the inclusions D ↪ A1 and D ↪ A2 that guarantee that the C*-algebra A1*D A2 is primitive. If the ranks of the matrices of partial multiplicities are one or all entries are 0 or ≥ 2, we prove that the algebra A1*D A2 is primitive if and only if it has a trivial center.

scholarly journals REALIZATION OF MINIMAL C*-DYNAMICAL SYSTEMS IN TERMS OF CUNTZ–PIMSNER ALGEBRAS

2009 ◽  
Vol 20 (06) ◽  
pp. 751-790 ◽  
Author(s):  
FERNANDO LLEDÓ ◽  
EZIO VASSELLI
Keyword(s):  
Dynamical Systems ◽  
Group Action ◽  
Unitary Irreducible Representation ◽  
Full Spectrum ◽  
C Algebra ◽  
Finite Dimensional ◽  
Relative Commutant ◽  
Chain Group ◽  
Fixed Point Algebra ◽  
Discrete Abelian Group

In the present article, we provide several constructions of C*-dynamical systems [Formula: see text] with a compact group [Formula: see text] in terms of Cuntz–Pimsner algebras. These systems have a minimal relative commutant of the fixed-point algebra [Formula: see text] in [Formula: see text], i.e. [Formula: see text], where [Formula: see text] is the center of [Formula: see text], which is assumed to be non-trivial. In addition, we show in our models that the group action [Formula: see text] has full spectrum, i.e. any unitary irreducible representation of [Formula: see text] is carried by a [Formula: see text]-invariant Hilbert space within [Formula: see text]. First, we give several constructions of minimal C*-dynamical systems in terms of a single Cuntz–Pimsner algebra [Formula: see text] associated to a suitable [Formula: see text]-bimodule ℌ. These examples are labelled by the action of a discrete Abelian group ℭ (which we call the chain group) on [Formula: see text] and by the choice of a suitable class of finite dimensional representations of [Formula: see text]. Second, we present a more elaborate contruction, where now the C*-algebra [Formula: see text] is generated by a family of Cuntz–Pimsner algebras. Here, the product of the elements in different algebras is twisted by the chain group action. We specify the various constructions of C*-dynamical systems for the group [Formula: see text], N ≥ 2.

LIMITS OF HOMOMORPHISMS WITH FINITE-DIMENSIONAL RANGE

2005 ◽  
Vol 16 (07) ◽  
pp. 807-821 ◽  
Author(s):  
SHANWEN HU ◽  
HUAXIN LIN ◽  
YIFENG XUE
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
C Algebra ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Unital Simple ◽  
Dimensional Range

Let X be a compact metric space and A be a unital simple C*-algebra with TR (A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.

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 On Darboux Theorem for symplectic forms on direct limits of symplectic Banach manifolds

2018 ◽  
Vol 15 (12) ◽  
pp. 1850206
Author(s):  
Fernand Pelletier
Keyword(s):  
Direct Limit ◽  
Symplectic Forms ◽  
Banach Manifolds ◽  
Finite Dimensional ◽  
Direct Limits ◽  
Darboux Theorem

Given an ascending sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true, we can ask about conditions under which the Darboux Theorem is also true on the direct limit. We will show that, in general, without very strong conditions, the answer is negative. In particular, we give an example of an ascending symplectic Banach manifolds on which the Darboux Theorem is true but not on the direct limit. In the second part, we illustrate this discussion in the context of an ascending sequence of Sobolev manifolds of loops in symplectic finite-dimensional manifolds. This context gives rise to an example of direct limit of weak symplectic Banach manifolds on which the Darboux Theorem is true around any point.

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