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On certain Cuntz-Pimsner algebras

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Alex Kumjian

Operator Algebras · Submitted 29 Aug 2001

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that $\Oc_E$, the Cuntz-Pimsner algebra associated to the Hilbert $A$-bimodule $E = \Hf \ot_{\C} A$, is simple and purely infinite.

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arXiv IDmath/0108194 (version 1)
Pith IDZTTYN2LA, Pith's permanent record of this paper and its review
CitationAlex Kumjian. "On certain Cuntz-Pimsner algebras." arXiv:math/0108194 (2001). https://pith.science/paper/ZTTYN2LA
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arXiv listing abstract v1. Results and the main claim are underlined; .

Let $A$ be a separable unital C*-algebra and let $\pi : A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a separable Hilbert space $\Hf$ such that $\pi(A) \cap \Kc(\Hf) = \{0 \}$. We show that $\Oc_E$, the Cuntz-Pimsner algebra associated to the Hilbert $A$-bimodule $E = \Hf \ot_{\C} A$, is simple and purely infinite. If $A$ is nuclear and belongs to the bootstrap class to which the UCT applies, then the same applies to $\Oc_E$. Hence by the Kirchberg-Phillips Theorem the isomorphism class of $\Oc_E$ only depends on the $K$-theory of $A$ and the class of the unit.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $A$ be a separable unital C*-algebra and let $\pi : A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a separable Hilbert space $\Hf$ such that $\pi(A) \cap \Kc(\Hf) = \{0 \}$. background 0.95 0.09 0.02 asserted 0.72
s1 We show that $\Oc_E$, the Cuntz-Pimsner algebra associated to the Hilbert $A$-bimodule $E = \Hf \ot_{\C} A$, is simple and purely infinite. result 0.99 0.95 0.86 asserted 1.00
s2 If $A$ is nuclear and belongs to the bootstrap class to which the UCT applies, then the same applies to $\Oc_E$. result 0.92 0.73 0.08 asserted 0.97
s3 Hence by the Kirchberg-Phillips Theorem the isomorphism class of $\Oc_E$ only depends on the $K$-theory of $A$ and the class of the unit. interpretation 0.82 0.38 0.32 asserted 1.00

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

Let $A$ be a separable unital C*-algebra and let $\pi : A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a separable Hilbert space $\Hf$ such that $\pi(A) \cap \Kc(\Hf) = \{0 \}$. We show that $\Oc_E$, the Cuntz-Pimsner algebra associated to the Hilbert $A$-bimodule $E = \Hf \ot_{\C} A$, is simple and purely infinite. If $A$ is nuclear and belongs to the bootstrap class to which the UCT applies, then the same applies to $\Oc_E$. Hence by the Kirchberg-Phillips Theorem the isomorphism class of $\Oc_E$ only depends on the $K$-theory of $A$ and the class of the unit.

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