Symmetries of simple A T A\mathbb{T} A T -algebras

[摘要] Let AAA be a unital simple ATA\mathbb{T}AT-algebra of real rank zero. Given an order two automorphism h:K1(A)→K1(A)h: K_1(A)\to K_1(A)h:K1​(A)→K1​(A), we show that there is an order two automorphism α\alphaα: A→AA\to AA→A such that α∗0=id\alpha_{*0}=\mathrm {id}α∗0​=id, α∗1=h\alpha_{*1}=hα∗1​=h and the action of Z2\mathbb{Z}_2Z2​ generated by α\alphaα has the tracial Rokhlin property. Consequently, C∗(A,Z2,α)C^*(A,\mathbb{Z}_2,\alpha)C∗(A,Z2​,α) is a simple unital AH-algebra with no dimension growth, and with tracial rank zero. Thus our main result can be considered as the Z2\mathbb{Z}_2Z2​-action analogue of the Lin-Osaka theorem. As a consequence, a positive answer to a lifting problem of Blackadar is also given for certain split case.

[学科分类] 神经科学

[关键词] AT algebra;symmetries;real rank zero;tracial Rokhlin property