Sylvester rank functions for amenable normal extensions

We introduce a notion of amenable normal extension S of a unital ring R with a finite approximation system F, encompassing the amenable algebras over a field of Gromov and Elek, the twisted crossed product by an amenable group, and the tensor product with a field extension. It is shown that every Sy...

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Bibliographic Details
Published in:Journal of functional analysis 2021-03, Vol.280 (6), p.108913, Article 108913
Main Authors: Jiang, Baojie, Li, Hanfeng
Format: Article
Language:English
Subjects:
Amenability
Crossed product
Infimum rule
Sylvester rank function
Tracial state
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Summary:We introduce a notion of amenable normal extension S of a unital ring R with a finite approximation system F, encompassing the amenable algebras over a field of Gromov and Elek, the twisted crossed product by an amenable group, and the tensor product with a field extension. It is shown that every Sylvester matrix rank function rk of R preserved by S has a canonical extension to a Sylvester matrix rank function rkF for S. In the case of twisted crossed product by an amenable group, and the tensor product with a field extension, it is also shown that rkF depends on rk continuously. When an amenable group has a twisted action on a unital C⁎-algebra preserving a tracial state, we also show that two natural Sylvester matrix rank functions on the algebraic twisted crossed product constructed out of the tracial state coincide.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2020.108913