Direct limit decomposition for $\mathrm{C}^*$-algebras of minimal diffeomorphisms

AbstractWe prove the following result. Let A be a direct limit of direct sums of C*-algebras of the form C(X) ⊗ Mn, with the spaces X being compact metric. Suppose that there is a finite upper bound on the dimensions of the spaces involved, and suppose that A is simple. Then the C* exponential rank of A is at most 1 + ε, that is, every element of the identity component of the unitary group of A is a limit of exponentials. This is true regardless of whether the real rank of A is 0 or 1.

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Real Af C*-Algebras With K0 of Small Rank

Canadian Journal of Mathematics ◽

10.4153/cjm-1989-036-6 ◽

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.

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C*-ALGEBRAS OF TRIVIAL K-THEORY AND SEMILATTICES

International Journal of Mathematics ◽

10.1142/s0129167x99000057 ◽

1999 ◽

Vol 10 (01) ◽

pp. 93-128 ◽

Cited By ~ 1

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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Limit $C^*$-algebras associated with an automorphism

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-14451 ◽

2004 ◽

Vol 95 (1) ◽

pp. 101 ◽

Cited By ~ 1

Author(s):

Baruch Solel

Keyword(s):

Fock Space ◽

Direct Limit ◽

Complete Classification ◽

C Algebra ◽

C Algebras ◽

Product Presentation ◽

Necessary And Sufficient ◽

Deddens Algebras ◽

Creation Operators

We present and study $C^*$-algebras generated by "periodic weighted creation operators" on the Fock space associated with an automorphism $\alpha$ on a $C^*$-algebra $A$. These algebras can be viewed as generalized Bunce-Deddens algebras associated with the automorphism and can be written as a certain direct limit. We prove a crossed product presentation for such an algebra and find a necessary and sufficient condition for it to be simple. In the case where the automorphism is induced by an irrational rotation (on C(T)) we compute the K-theory groups and obtain a complete classification of these algebras.

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