Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles

作者：Kwasniewski Bartosz Kosma^*; Meyer Ralf

来源：Studia Mathematica, 2018, 241(3): 257-303.

DOI：10.4064/sm8762-5-2017

• group action
• Fell bundle
• aperiodic
• topologically free
• purely outer
• Connes spectrum
• purely infinite
• strongly purely infinite
• filling family

We generalise various non-triviality conditions for group actions to Fell bundles over discrete groups and prove several implications between them. We also study sufficient criteria for the reduced section C* -algebra C-r* (B) of a Fell bundle B - (B-g)(g is an element of G) to be strongly purely infinite. If the unit fibre A : = B-e contains an essential ideal that is separable or of Type I, then B is aperiodic if and only if B is topologically free. If, in addition, G = Z or G = Z/p for a square-free number p, then these equivalent conditions are satisfied if and only if A detects ideals in C-r* (B), if and only if A(+){0} supports C-r* (B)(+) \ {0} in the Cuntz sense. For G as above and for arbitrary A, C-r* (B) is simple if and only if B is minimal and pointwise outer. In general, B is aperiodic if and only if each of its non- trivial fibres has a non-trivial Connes spectrum. If G is finite or if A contains an essential ideal that is of Type I or simple, then aperiodicity is equivalent to pointwise pure outerness.

• 出版日期2018