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A simple separable exact C*-algebra not anti-isomorphic to itself

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Abstract

We give an example of an exact, stably finite, simple, separable C*-algebra D which is not isomorphic to its opposite algebra. Moreover, D has the following additional properties. It is stably finite, approximately divisible, has real rank zero and stable rank one, has a unique tracial state, and the order on projections over D is determined by traces. It also absorbs the Jiang-Su algebra Z, and in fact absorbs the 3∞ UHF algebra. We can also explicitly compute the K-theory of D, namely \({K_0 (D) \cong {\mathbb{Z}} [ \tfrac{1}{3}]}\) with the standard order, and K 1 (D) =  0, as well as the Cuntz semigroup of D, namely \({W (D) \cong {\mathbb{Z}} [ \tfrac{1}{3} ]_{+} \sqcup (0, \infty).}\)

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    N. Christopher Phillips

  2. Department of Mathematics and Interdisciplinary Studies, Lakehead University-Orillia, 500 University Ave, Orillia, ON, L3V09B, Canada

    Maria Grazia Viola

Authors
  1. N. Christopher Phillips
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  2. Maria Grazia Viola
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Correspondence to Maria Grazia Viola.

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N. C. Phillips was partially supported by NSF Grant DMS-0701076, by the Fields Institute for Research in Mathematical Sciences, Toronto, Canada, and by an Elliott Distinguished Visitorship at the Fields Institute. M. G. Viola was partly supported by the Fields Institute for Research in Mathematical Sciences, and by Queen’s University, Kingston, Canada.

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Phillips, N.C., Viola, M.G. A simple separable exact C*-algebra not anti-isomorphic to itself. Math. Ann. 355, 783–799 (2013). https://doi.org/10.1007/s00208-011-0755-z

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