A Survey on Classification of C∗-Algebras with the Ideal Property

Gong, Guihua; Jiang, Chunlan; Wang, Kun

doi:10.1007/978-3-030-43380-2_12

Cited by 4 publications

(9 citation statements)

References 41 publications

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“…• The invariant Inv introduced by Gong, Jiang and Li in [15] and [16]. This invariant based on the original Elliott invariant together with total K-Theory, affine functions from the tracial space of corner algebras and their Hausdorffified K 1 -groups, is a complete invariant for AH-algebras with the ideal property and no dimension growth.…”

Section: Further Remarksmentioning

confidence: 99%

“…(See e.g. [18], [16].) Among these new components, we may find the affine maps from the tracial space of corner algebras (with compatibility axioms) and we observe that the total K-Theory comes to replace the original K * -group.…”

Section: Introductionmentioning

confidence: 96%

“…A first version obtained, denoted by Inv 0 , partially classifies AH-algebras with ideal property and no dimension growth, a class that contains both simple and real rank zero AH-algebras. Later, in [15] and [16], they round off the classification of all AHalgebras with ideal property and no dimension growth by further refining of their invariant, now termed Inv, by adding Hausdorffified K 1 -groups of corner algebras (with some more compatibility axioms).…”

Section: Introductionmentioning

confidence: 99%

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The unitary Cuntz semigroup on the classification of non-simple C*-algebras

Cantier¹

2021

Preprint

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This paper argues that the unitary Cuntz semigroup, introduced in [9] and termed Cu 1 , contains crucial information regarding the classification of non-simple C * -algebras. We exhibit two (non-simple) C * -algebras that agree on their Cuntz semigroups, termed Cu, and their K 1 -groups and yet disagree at level of their unitary Cuntz semigroups. In the process, we establish that the unitary Cuntz semigroup contains rigorously more information about non-simple C * -algebras than Cu and K 1 alone.

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Section: Further Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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The unitary Cuntz semigroup on the classification of non-simple C*-algebras

Cantier¹

2021

Preprint

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“…And we will compare the decompositions of two different homomorphisms in the last part of chapter 4. Such decomposition and comparison results will be used in the proof of the uniqueness theorem for AH algebras with the ideal property in [19] by Gong, Jiang and Li.…”

Section: Introductionmentioning

confidence: 99%

On the Decomposition Theorems for C*-algebras

Jiang

Wang

2019

Preprint

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Elliott dimension drop interval algebra is an important class among all C * -algebras in the classification theory. Especially, they are building stones of AHD algebra and the latter contains all AH algebras with the ideal property of no dimension growth. In this paper, we will show two decomposition theorems related to the Elliott dimension drop interval algebra. Our results are key steps in classifying all AH algebras with the ideal property of no dimension growth.

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“…We call it Stevens invariant. Stevens invariant of a C * -algebra A consists of the K 0 -group of A, the K 1 -group of A and the tracial state spaces of all hereditary C * -subalgebras of the form eAe with certain compatibility conditions, where e is any projection in A. Stevens invariant is also used to classify AH-algebras with the ideal property (see [19]).…”

Section: Introductionmentioning

confidence: 99%

On invariants of C*-algebras with the ideal property

Wang

2018

J. Noncommut. Geom.

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In this paper, we study the relation between the extended Elliott invariant and the Stevens invariant of C * -algebras. We show that in general the Stevens invariant can be derived from the extended Elliott invariant in a functorial manner. We also show that these two invariants are isomorphic for C * -algebras satisfying the ideal property. A C * -algebra is said to have the ideal property if each of its closed two-sided ideals is generated by projections inside the ideal. Both simple, unital C * -algebras and real rank zero C * -algebras have the ideal property. As a consequence, many classes of non-simple C * -algebras can be classified by their extended Elliott invariants, which is a generalization of Elliott's conjecture.

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A Survey on Classification of C∗-Algebras with the Ideal Property

Cited by 4 publications

References 41 publications

The unitary Cuntz semigroup on the classification of non-simple C*-algebras

The unitary Cuntz semigroup on the classification of non-simple C*-algebras

On the Decomposition Theorems for C*-algebras

On invariants of C*-algebras with the ideal property

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