Some Results on Inner Quasidiagonal C *-algebras
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Some Results on Inner Quasidiagonal C*-algebras Qi Hui LI1) School of Science, East China University of Science and Technology, Shanghai 200237, P. R. China E-mail : qihui [email protected]
Rui WANG Shanghai Aerospace Control Technology Institute, Shanghai 200211, P. R. China E-mail : [email protected] Abstract In the current article, we prove the crossed product C*-algebra by a Rokhlin action of finite group on a strongly quasidiagonal C*-algebra is strongly quasidiagonal again. We also show that a just-infinite C*-algebra is quasidiagonal if and only if it is inner quasidiagonal. Finally, we compute the topological free entropy dimension in just-infinite C*-algebras. Keywords Inner quasidiagonal C*-algebras, crossed product C*-algebras, strongly quasidiagonal C*-algebras, just-infinite C*-algebras, topological free entropy dimension MR(2010) Subject Classification
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46L05, 46L35
Introduction
To distinguish the class of NF algebras and the class of strong NF algebras, Blackadar and Kirchberg introduced the concept of inner quasidiagonal C*-algebras in [3]. From its definition, it is apparent that the class of inner quasidiagonal C*-algebras is a subclass of quasidiagonal C*-algebras. Many basic properties of inner quasidiagonal C*-algebras have been discussed in [3] and [4]. It was also shown that a separable C*-algebra is a strong NF algebra if and only if it is nuclear and inner quasidiagonal. Therefore the class of all strong NF algebras is strictly contained in the class of nuclear and quasidiagonal C*-algebras (i.e., NF algebras). Examples of separable nuclear C*-algebras which are quasidiagonal but not inner quasidiagonal were given in the same article. And we also know that all separable simple quasidiagonal C*-algebras are inner quasidiagonal, all strongly quasidiagonal C*-algebras are inner quasidiagonal. Recall that a C*-algebra is called strongly quasidiagonal if it is separable and all its representations are quasidiagonal. Since not all C*-subalgebras of inner quasidiagonal C*-algebras are inner, the crossed product C*-algebra by an action of a finite group on an inner quasidiagonal C*-algebra may not be inner quasidiagonal again. In [22], it was shown that the crossed product C*-algebra by a Rokhlin action of a finite group on a unital inner quasidiagonal C*-algebra is inner again. So Received January 28, 2020, revised May 17, 2020, accepted June 5, 2020 The first author was partially supported by NSFC (Grant No. 11671133) 1) Corresponding author
Some Results on Inner Quasidiagonal C*-algebras
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it is natural to ask whether the same conclusion still holds for non-unital inner quasidiagonal C*-algebras. In the current paper, we will prove that the crossed product C*-algebra by a Rokhlin action of finite group on a strongly quasidiagonal C*-algebra (may not be unital) is strongly quasidiagonal again. Just-infinite C*-algebras were first introduced by Grigorchuk, Musat and Rørdam, in [9] as an analogo
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