## Just-infinite C^{∗}-algebras and Their Invariants

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**Just-infinite C ^{∗}-algebras and Their Invariants.** / Rørdam, Mikael.

Research output: Contribution to journal › Journal article › Research › peer-review

#### Harvard

^{∗}-algebras and Their Invariants',

*International Mathematics Research Notices*, vol. 2019, no. 12, pp. 3621-3645. https://doi.org/10.1093/imrn/rnx227

#### APA

^{∗}-algebras and Their Invariants.

*International Mathematics Research Notices*,

*2019*(12), 3621-3645. https://doi.org/10.1093/imrn/rnx227

#### Vancouver

^{∗}-algebras and Their Invariants. International Mathematics Research Notices. 2019;2019(12):3621-3645. https://doi.org/10.1093/imrn/rnx227

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#### Bibtex

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#### RIS

TY - JOUR

T1 - Just-infinite C∗-algebras and Their Invariants

AU - Rørdam, Mikael

PY - 2019

Y1 - 2019

N2 - Just-infinite C∗-algebras, that is, infinite dimensional C∗-algebras, whose proper quotients are finite dimensional, were investigated in [3]. One particular example of a just-infinite residually finite dimensional AF-algebras was constructed in [3]. In this article, we extend that construction by showing that each infinite dimensional metrizable Choquet simplex is affinely homeomorphic to the trace simplex of a just-infinite residually finite dimensional C∗-algebra. The trace simplex of any unital residually finite dimensional C∗-algebra is hence realized by a just-infinite one. We determine the trace simplex of the particular residually finite dimensional AF-algebras constructed in [3], and we show that it has precisely one extremal trace of type II1. We give a complete description of the Bratteli diagrams corresponding to residually finite dimensional AF-algebras. We show that a modification of any such Bratteli diagram, similar to the modification that makes an arbitrary Bratteli diagram simple, will yield a just-infinite residually finite dimensional AF-algebra.

AB - Just-infinite C∗-algebras, that is, infinite dimensional C∗-algebras, whose proper quotients are finite dimensional, were investigated in [3]. One particular example of a just-infinite residually finite dimensional AF-algebras was constructed in [3]. In this article, we extend that construction by showing that each infinite dimensional metrizable Choquet simplex is affinely homeomorphic to the trace simplex of a just-infinite residually finite dimensional C∗-algebra. The trace simplex of any unital residually finite dimensional C∗-algebra is hence realized by a just-infinite one. We determine the trace simplex of the particular residually finite dimensional AF-algebras constructed in [3], and we show that it has precisely one extremal trace of type II1. We give a complete description of the Bratteli diagrams corresponding to residually finite dimensional AF-algebras. We show that a modification of any such Bratteli diagram, similar to the modification that makes an arbitrary Bratteli diagram simple, will yield a just-infinite residually finite dimensional AF-algebra.

UR - http://www.scopus.com/inward/record.url?scp=85072080615&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnx227

DO - 10.1093/imrn/rnx227

M3 - Journal article

AN - SCOPUS:85072080615

VL - 2019

SP - 3621

EP - 3645

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 12

ER -

ID: 230391787