TY - JOUR
T1 - Central sequence C∗-algebras and tensorial absorption of the Jiang-Su algebra
AU - Kirchberg, Eberhard
AU - Rørdam, Mikael
PY - 2014/10/1
Y1 - 2014/10/1
N2 - We study properties of the central sequence algebra Aω A″ of a C∗-algebra A, and we present an alternative approach to a recent result of Matui and Sato. They prove that every unital separable simple nuclear C∗-algebra, whose trace simplex is finitedimensional, tensorially absorbs the Jiang-Su algebra if and only if it has the strict comparison property. We extend their result to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension. We are also able to relax the assumption on the C∗-algebra of having the strict comparison property to a weaker property that we call local weak comparison. Namely, we prove that a unital separable simple nuclear C∗-algebra, whose trace simplex has finite-dimensional closed extreme boundary, tensorially absorbs the Jiang-Su algebra if and only if it has the local weak comparison property. We can also eliminate the nuclearity assumption if instead we assume the (SI) property of Matui and Sato, and, moreover, that each II1-factor representation of the C∗-algebra is a McDuff factor.
AB - We study properties of the central sequence algebra Aω A″ of a C∗-algebra A, and we present an alternative approach to a recent result of Matui and Sato. They prove that every unital separable simple nuclear C∗-algebra, whose trace simplex is finitedimensional, tensorially absorbs the Jiang-Su algebra if and only if it has the strict comparison property. We extend their result to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension. We are also able to relax the assumption on the C∗-algebra of having the strict comparison property to a weaker property that we call local weak comparison. Namely, we prove that a unital separable simple nuclear C∗-algebra, whose trace simplex has finite-dimensional closed extreme boundary, tensorially absorbs the Jiang-Su algebra if and only if it has the local weak comparison property. We can also eliminate the nuclearity assumption if instead we assume the (SI) property of Matui and Sato, and, moreover, that each II1-factor representation of the C∗-algebra is a McDuff factor.
U2 - 10.1515/crelle-2012-0118
DO - 10.1515/crelle-2012-0118
M3 - Journal article
SN - 0075-4102
SP - 175
EP - 214
JO - Journal fuer die Reine und Angewandte Mathematik
JF - Journal fuer die Reine und Angewandte Mathematik
IS - 695
ER -