For a dense $G_delta$ set of real parameters $ heta$ in $[0,1]$(containing the rationals) it is shown that the group $K_0 (A_ heta times_sigma mathbb{Z}_4)$ is isomorphic to $mathbb{Z}^9$, where$A_ heta$ is the rotation C*-algebra generated by unitaries $U$, $V$satisfying $VU = e^{2pi i heta} UV$ and $sigma$ is the Fourierautomorphism of $A_ heta$ defined by $sigma(U) = V$, $sigma(V) =U^{-1}$. More precisely, an explicit basis for $K_0$ consisting ofnine canonical modules is given. (A slight generalization of thisresult is also obtained for certain separable continuous fields ofunital C*-algebras over $[0,1]$.) The Connes Chern character $chcolon K_0 (A_ heta times_sigma mathbb{Z}_4) o H^{ev} (A_ heta times_sigma mathbb{Z}_4)^*$ is shown to be injective for a dense$G_delta$ set of parameters $ heta$. The main computational tool inthis paper is a group homomorphism $vtr colon K_0 (A_ heta times_sigma mathbb{Z}_4) o mathbb{R}^8 imes mathbb{Z}$obtained from the Connes Chern character by restricting thefunctionals in its codomain to a certain nine-dimensional subspace of$H^{ev} (A_ heta times_sigma mathbb{Z}_4)$. The range of $vtr$is fully determined for each $ heta$. (We conjecture that thissubspace is all of $H^{ev}$.)
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机译：对于$ [0,1] $（包含有理）中的密集实参$ G_delta $集$ heta $，表明组$ K_0（A_ heta times_sigma mathbb {Z} _4）$与$ mathbb是同构的{Z} ^ 9 $，其中$ A_ heta $是由aries $ U $生成的旋转C *-代数，$ V $满足$ VU = e ^ {2pi i heta} UV $和$ sigma $是的傅里叶自同构由$ sigma（U）= V $，$ sigma（V）= U ^ {-1} $定义的$ A_ heta $。更精确地，给出了由九个规范模块组成的$ K_0 $的显式基础。 （对于$ [0,1] $以上的单位C *-代数的某些可分离连续字段，也对此结果进行了概括。）Connes Chern字符$ chcolon K_0（A_ heta times_sigma mathbb {Z} _4）o H ^ { ev}（A_ heta times_sigma mathbb {Z} _4）^ * $对于密集的$ G_delta $参数集$ heta $显示为内射。本文的主要计算工具是群同态$ vtr冒号K_0（A_ heta times_sigma mathbb {Z} _4）o mathbb {R} ^ 8 imes mathbb {Z} $是从Connes Chern角色获得的，方法是将其共域的功能限制为$ H ^ {ev}（A_ heta times_sigma mathbb {Z} _4）$的某个九维子空间。 $ vtr $的范围已完全确定为每个$ heta $。 （我们推测此子空间是$ H ^ {ev} $的全部。）
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