Real Af C*-Algebras With K0 of Small Rank

1989 ◽  
Vol 41 (5) ◽  
pp. 786-807
Author(s):  
T. Giordano ◽  
D. E. Handelman
Keyword(s):  
Algebraic Number ◽  
Direct Limit ◽  
Algebraic Number Field ◽  
Morita Equivalence ◽  
Equivalence Classes ◽  
C Algebras ◽  
Finite Dimensional ◽  
Real Algebra ◽  
Rank 2 ◽  
Prime 2

A real AF C*-algebra is the norm closure of a direct limit of finite dimensional real C*-algebras (with real *-algebra maps). When we use the unadorned “AF C*-algebra”, we mean the usual complex version.Let R be a simple AF C*-algebra such that K0(R) is free of rank 2 or 3. The problem is to find (up to Morita equivalence) all real AF C*-algebras A such that AꕕC≅R. This is closely related to the problem of finding all involutions on R [3], [10].For example, when the rank is 2, generically there are 8 such classes. The exceptional cases arise when the ratio of the two generators in K0(R) is a quadratic (algebraic) number, and here there are 4, 5, or 8 Morita equivalence classes, the number depending largely on the behaviour of the prime 2 in the relevant algebraic number field.

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.

scholarly journals MORITA EQUIVALENCE FOR C*-ALGEBRAS WITH THE WEAK BANACH–SAKS PROPERTY. II

2007 ◽  
Vol 50 (1) ◽  
pp. 185-195
Author(s):  
Masaharu Kusuda
Keyword(s):  
Morita Equivalence ◽  
C Algebras ◽  
Morita Equivalent ◽  
Finite Dimensional

AbstractLet $C^*$-algebras $A$ and $B$ be Morita equivalent and let $X$ be an $A$–$B$-imprimitivity bimodule. Suppose that $A$ or $B$ is unital. It is shown that $X$ has the weak Banach–Saks property if and only if it has the uniform weak Banach–Saks property. Thus, we conclude that $A$ or $B$ has the weak Banach–Saks property if and only if $X$ does so. Furthermore, when $C^*$-algebras $A$ and $B$ are unital, it is shown that $X$ has the Banach–Saks property if and only if it is finite dimensional.

Strong Morita Equivalence for Heisenberg C*-Algebras and the Positive Cones of Their K 0-Groups

1988 ◽  
Vol 40 (04) ◽  
pp. 833-864 ◽  
Author(s):  
Judith A. Packer
Keyword(s):  
Heisenberg Group ◽  
Morita Equivalence ◽  
Equivalence Classes ◽  
Crossed Products ◽  
C Algebras ◽  
Projective Representations ◽  
Discrete Heisenberg Group ◽  
Strong Morita Equivalence

In [14] we began a study of C*-algebras corresponding to projective representations of the discrete Heisenberg group, and classified these C*-algebras up to *-isomorphism. In this sequel to [14] we continue the study of these so-called Heisenberg C*-algebras, first concentrating our study on the strong Morita equivalence classes of these C*-algebras. We recall from [14] that a Heisenberg C*-algebra is said to be of class i, i ∊ {1, 2, 3}, if the range of any normalized trace on its K 0 group has rank i as a subgroup of R; results of Curto, Muhly, and Williams [7] on strong Morita equivalence for crossed products along with the methods of [21] and [14] enable us to construct certain strong Morita equivalence bimodules for Heisenberg C*-algebras.

A Remark on Separable Orders

1969 ◽  
Vol 12 (4) ◽  
pp. 453-455 ◽  
Author(s):  
Klaus W. Roggenkamp
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Finitely Generated ◽  
Algebraic Integers ◽  
Finite Dimensional ◽  
Torsion Free

K = algebraic number field,R = algebraic integers in K,A = finite dimensional semi-simple K-algebra, A. = simple K-algebra,i = 1,…, n,Ki = center of Ai, = 1,…, n,G = R-order in A,Ri = G ∩ ki.All modules under consideration are finitely generated left modules. A G-lattice is a G-module which is R-torsion-free.

Dispersive and explosive mappings

1975 ◽  
Vol 20 (1) ◽  
pp. 33-37
Author(s):  
T. K. Sheng
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Rational Numbers ◽  
Inner Product ◽  
Product Spaces ◽  
Real Numbers ◽  
Inner Product Spaces ◽  
Finite Dimensional ◽  
Image Set

Let Q, R be rational numbers and real numbers respectively. We use V(F) and W(F) to denote finite dimensional inner product spaces over F. Given V(Q), we use V(R) for the smallest inner space over R containing V(Q). It is known that an R-homomorphism of V(R) to W(R) is continous. We prove that if a Q-homomorphism f: V(R) → W(R), then f is dispersive, i.e., given any v0 ∈ V(Q) and ε > 0, the image set f[D(v0, ε)], where D(v0, ε) = [v: v ∈ V(Q), ¦v – v0¦ < ε], is not bounded. It is also shown that some Q-homomorphism f: V(Q) → W(Q) can be explosive in the sense that for any v0 ∈ V(Q) and ε > 0, the set f[D[v0, ε)] is dense in W(Q). As a particular case of dispersive and explosive Q-homomorphisms, we show that the algebraic number field isomorphism f: Q(a) → Q(β), where f(a) = β and α ≠ β or βmacr; (βmacr; being complex conjugates of β) is explosive.

Some Results on the Schur Index of a Representation of a Finite Group

1970 ◽  
Vol 22 (3) ◽  
pp. 626-640 ◽  
Author(s):  
Charles Ford
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Complex Field ◽  
Complex Vector Space ◽  
Linear Transformations ◽  
Complex Vector ◽  
Finite Dimensional ◽  
A Finite Group

Let ℭ be a finite group with a representation as an irreducible group of linear transformations on a finite-dimensional complex vector space. Every choice of a basis for the space gives the representing transformations the form of a particular group of matrices. If for some choice of a basis the resulting group of matrices has entries which all lie in a subfield K of the complex field, we say that the representation can be realized in K. It is well known that every representation of ℭ can be realized in some algebraic number field, a finitedimensional extension of the rational field Q.

C*-ALGEBRAS OF TRIVIAL K-THEORY AND SEMILATTICES

1999 ◽  
Vol 10 (01) ◽  
pp. 93-128 ◽  
Author(s):  
HUAXIN LIN
Keyword(s):  
Direct Limit ◽  
Tensor Products ◽  
Equivalence Classes ◽  
Direct Sums ◽  
C Algebras ◽  
Direct Limits ◽  
One To One ◽  
K Theory ◽  
Essential Extensions ◽  
Distributive Semilattices

We give a class of nuclear C*-algebras which contains [Formula: see text] and is closed under stable isomorphism, ideals, quotients, hereditary subalgebras, tensor products, direct sums, direct limits as well as extensions. We show that this class of C*-algebras is classified by their equivalence classes of projections and there is a one to one correspondence between (unital) C*-algebras in the class and countable distributive semilattices (with largest elements). One of the main results is that essential extensions of a C*-algebras which is a direct limit of finite direct sums of corners of [Formula: see text] by the same type of C*-algebras are still direct limits of finite direct sums of corners of [Formula: see text].

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