Dimension Groups and Their Affine Representations

Effros, Edward G.; Handelman, David; Shen, Chao-Liang

doi:10.2307/2374244

Cited by 205 publications

(189 citation statements)

References 6 publications

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“…It is easy to see that all these properties pass to inductive limits, whence they are satisfied by all dimension monoids. The converse is known as the Effros-Handelman-Shen theorem, [EHS80], which is formulated for partially ordered groups. The version given here for a positively ordered monoid M follows by passing to the Grothendieck completion G, from which M can be recovered as M = G + .…”

Section: Proof Let Us Show (1) By Theorem 44(1) Cu(m ) Satisfies mentioning

confidence: 99%

Tensor products and regularity properties of Cuntz semigroups

2018

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Abstract. The Cuntz semigroup of a C * -algebra is an important invariant in the structure and classification theory of C * -algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups.Given a C * -algebra A, its (concrete) Cuntz semigroup Cu(A) is an object in the category Cu of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu in [CEI08]. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter Cu-semigroups.We establish the existence of tensor products in the category Cu and study the basic properties of this construction. We show that Cu is a symmetric, monoidal category and relate Cu(A ⊗ B) with Cu(A) ⊗ Cu Cu(B) for certain classes of C * -algebras.As a main tool for our approach we introduce the category W of precompleted Cuntz semigroups. We show that Cu is a full, reflective subcategory of W. One can then easily deduce properties of Cu from respective properties of W, for example the existence of tensor products and inductive limits. The advantage is that constructions in W are much easier since the objects are purely algebraic.For every (local) C * -algebra A, the classical Cuntz semigroup W (A) together with a natural auxiliary relation is an object of W. This defines a functor from C * -algebras to W which preserves inductive limits. We deduce that the assignment A → Cu(A) defines a functor from C * -algebras to Cu which preserves inductive limits. This generalizes a result from [CEI08].We also develop a theory of Cu-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C * -algebra has a natural product giving it the structure of a Cu-semiring. For C * -algebras, it is an important regularity property to tensorially absorb a strongly self-absorbing C * -algebra. Accordingly, it is of particular interest to analyse the tensor products of Cu-semigroups with the Cu-semiring of a strongly self-absorbing C * -algebra. This leads us to define 'solid' Cu-semirings (adopting the terminology from solid rings), as those Cu-semirings S for which the product induces an isomorphism between S ⊗ Cu S and S. This can be considered as an analog of being strongly self-absorbing for Cu-semirings. As it turns out, if a strongly self-absorbing C * -algebra satisfies the UCT, then its Cu-semiring is solid. We prove a classification theorem for solid Cu-semirings. This raises the question of whether the Cuntz semiring of every strongly self-absorbing C * -algebra is solid.If R is a solid Cu-semiring, then a Cu-semigroup S is a semimodule over R if and only if R ⊗ Cu S is isomorphic to S. Thus, analogous to the case for C * -algebras, we can think of semimodules over R as Cu-semigroups that tensorially absorb R. We give explicit characterizations when a Cu-semigroup is such a semimodule for the cases that R is the Cu-semiring of a strongly self-absorbing C * -algebra satisfying the UCT. For instance, we show that ...

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Section: Proof Let Us Show (1) By Theorem 44(1) Cu(m ) Satisfies mentioning

confidence: 99%

Tensor products and regularity properties of Cuntz semigroups

2018

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“…Namely, the framework of pre-ordered groups we develop here gives an affirmative answer to Open Problems 1, 7 and 8 of [17]. Moreover, in [30], we obtain, as an application of the main result of this paper, a version of the theorem of representation of dimension groups ( [15]) for simple Riesz groups, as well as an affirmative answer to Open Problem 2 of [17] for the case of simple groups.…”

Section: E Pardomentioning

confidence: 55%

Metric completions of ordered groups and 𝐾₀ of exchange rings

Pardo

1998

Trans. Amer. Math. Soc.

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Abstract. We give a description of the closure of the natural affine continuous function representation of K 0 (R) for any exchange ring R. This goal is achieved by extending the results of Goodearl and Handelman, about metric completions of dimension groups, to a more general class of pre-ordered groups, which includes K 0 of exchange rings. As a consequence, the results about K + 0 of regular rings, which the author gave in an earlier paper, can be extended to a wider class of rings, which includes C * -algebras of real rank zero, among others. Also, the framework of pre-ordered groups developed here allows other potential applications.

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“…It was soon recognized [19] that this new invariant is the K 0 functor from ring theory. A very striking converse to Elliott's theorem was found by Effros, Handelmann and Shen [17] which characterized those groups which arise as the K 0 group of an AF algebra.…”

Section: Introductionmentioning

confidence: 84%

Essentially Normal Operators

Davidson

2010

A Glimpse at Hilbert Space Operators

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Abstract. This is a survey of essentially normal operators and related developments. There is an overview of Weyl-von Neumann theorems about expressing normal operators as diagonal plus compact operators. Then we consider the Brown-Douglas-Fillmore theorem classifying essentially normal operators. Finally we discuss almost commuting matrices, and how they were used to obtain two other proofs of the BDF theorem. Mathematics Subject Classification (2000). 47-02,47B15,46L80.

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Dimension Groups and Their Affine Representations

Cited by 205 publications

References 6 publications

Tensor products and regularity properties of Cuntz semigroups

Tensor products and regularity properties of Cuntz semigroups

Metric completions of ordered groups and 𝐾₀ of exchange rings

Essentially Normal Operators

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