A simple separable C*-algebra not isomorphic to its opposite algebra

Abstract

We give an example of a simple separable C*-algebra that is not isomorphic to its opposite algebra. Our example is nonnuclear and stably finite, has real rank zero and stable rank one, and has a unique tracial state. It has trivial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 1"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">K_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 0"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">K_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group is order isomorphic to a countable subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.