Non-simple purely infinite C *-algebras

Kirchberg, Eberhard; Rørdam, Mikael

doi:10.1353/ajm.2000.0021

Cited by 186 publications

(337 citation statements)

References 22 publications

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“…Moreover, in the stably finite case, equivalence of projections in the Cuntz semigroup is the same as ordinary equivalence of projections, in matrix algebras over A (see, for example, [45] (p. 641)). Let us now make the slightly stronger assumption of stable rank 1.…”

Section: Isomorphism Results and Remarks On K-theorymentioning

confidence: 99%

Cuntz Semigroups of Compact-Type Hopf C*-Algebras

Kucerovsky

2017

Axioms

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Abstract:The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups. We show that in many cases, isomorphisms of Cuntz semigroups that respect this additional structure can be lifted to Hopf algebra (bi)isomorphisms, up to a possible flip of the co-product. This shows that the Cuntz semigroup provides an interesting invariant of C*-algebraic quantum groups.

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Section: Isomorphism Results and Remarks On K-theorymentioning

confidence: 99%

Cuntz Semigroups of Compact-Type Hopf C*-Algebras

Kucerovsky

2017

Axioms

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“…Following [18], we denote the set of positive elements in a C * -algebra A by A + . The ideal in A generated by an element b is denoted AbA.…”

Section: Purely Infinite K-graph C * -Algebrasmentioning

confidence: 99%

“…Recall that for positive elements a ∈ M n (A) and b ∈ M m (A), we say that a is Cuntz below b, denoted a b, if there exists a sequence of elements x k in M m,n (A) such that x * k bx k → a in norm. We say A is purely infinite if there are no characters on A and for all a, b ∈ A + , we have a b if and only if a ∈ AbA (see [18,Definition 4.1]). …”

Section: Purely Infinite K-graph C * -Algebrasmentioning

confidence: 99%

Real rank and topological dimension of higher rank graph algebras

Pask¹,

Sierakowski²,

Sims³

2017

Indiana Univ. Math. J.

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Abstract. We study dimension theory for the C * -algebras of row-finite k-graphs with no sources. We establish that strong aperiodicity-the higher-rank analogue of condition (K)-for a k-graph is necessary and sufficient for the associated C * -algebra to have topological dimension zero. We prove that a purely infinite 2-graph algebra has real-rank zero if and only if it has topological dimension zero and satisfies a homological condition that can be characterised in terms of the adjacency matrices of the 2-graph. We also show that a k-graph C * -algebra with topological dimension zero is purely infinite if and only if all the vertex projections are properly infinite. We show by example that there are strongly purely infinite 2-graphs algebras, both with and without topological dimension zero, that fail to have real-rank zero.

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“…Proof. Supposing that ii) holds, we thus have a sequence of copies of O 2 , with generators a n and b n such that a n and b n asymptotically commute with the given element c. Recall that Kirchberg and Rørdam defined [13] a positive element c in the corona Q to be properly infinite if there exists a sequence of elements R n in M 2 (Q) such that R n ( c 0 0 0 ) R * n goes in norm to ( c 0 0 c ) . It is straightforward to check that R n := a n 0 b n 0 will do the job.…”

Section: Remarks On Full Extensions (The Corona Factorization Property)mentioning

confidence: 99%

When are Fredholm triples operator homotopic?

Kucerovsky

2006

Proc. Amer. Math. Soc.

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Abstract. Fredholm triples are used in the study of Kasparov's KK-groups, and in Connes's noncommutative geometry. We define an absorption property for Fredholm triples, and give an if and only if condition for a Fredholm triple to be absorbing. We study the interaction of the absorption property with several of the more common equivalence relations for Fredholm triples. In general these relations are coarser than homotopy in the norm topology. We give simple conditions for an equivalence of triples to be implemented by an operator homotopy (i.e. a homotopy with respect to the norm topology). This can be expected to have applications in index theory, as we illustrate by proving two theorems of Pimsner-Popa-Voiculescu type. We show that there is some relationship with the interesting Toms-Winter characterization of D-absorbing algebras, recently obtained as part of Elliott's classification program.

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Non-simple purely infinite C *-algebras

Cited by 186 publications

References 22 publications

Cuntz Semigroups of Compact-Type Hopf C*-Algebras

Cuntz Semigroups of Compact-Type Hopf C*-Algebras

Real rank and topological dimension of higher rank graph algebras

When are Fredholm triples operator homotopic?

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