论文标题
在AF-Elgebras上的游戏
Games on AF-algebras
论文作者
论文摘要
我们分析$ \ mathrm {c}^\ ast $ -Algebras，尤其是AF-Algebras及其$ K_0 $ -Groups，在无限逻辑$ \ Mathcal {l} _ {ω_1ω} $的上下文中。给定了两个可分离的Unital Af-egras $ a $和$ b $，并考虑其$ k_0 $ groups作为订购的订单群组，我们证明$ k_0（a）\ equiv_ {ω\cdotα} k_0（b）$ ymif quant $ quant $ equiv_emiif in $ equif n $ equiif in $ mif n $ equif n in $ equif n $ n $ n $ mif n $ mif n $ mif n $ mif n $ mif n $ n $ n最多$β$。通过埃利奥特（Elliott）对可分离的AF-Elgebras的分类，以及将Ehrenfeucht-Fraïssé游戏改编为度量设置的技术证明了这一含义。此外，我们还使用此结果来构建一个家庭$ \ {a_α\} _ {α<ω_1} $的成对非同态可分离的简单的unital af-algebras，该$a_α\equiv_αa_β$满足每个$α<β$。特别是，我们获得了一组任意高的Scott等级的可分离的简单的Unital AF-Elgebras。接下来，我们对上述含义进行部分交谈，这表明$ a \ otimes \ Mathcal {k} \ equiv_ {ω+ 2 \ cdotα+ 2} b \ otimes \ otimies \ nathcal {k} $ Ingrime $ \ mathrm {c}^\ ast $ -algebras $ a $和$ b $。
We analyze $\mathrm{C}^\ast$-algebras, particularly AF-algebras, and their $K_0$-groups in the context of the infinitary logic $\mathcal{L}_{ω_1 ω}$. Given two separable unital AF-algebras $A$ and $B$, and considering their $K_0$-groups as ordered unital groups, we prove that $K_0(A) \equiv_{ω\cdot α} K_0(B)$ implies $A \equiv_αB$, where $M \equiv_βN$ means that $M$ and $N$ agree on all sentences of quantifier rank at most $β$. This implication is proved using techniques from Elliott's classification of separable AF-algebras, together with an adaptation of the Ehrenfeucht-Fraïssé game to the metric setting. We use moreover this result to build a family $\{ A_α\}_{α< ω_1}$ of pairwise non-isomorphic separable simple unital AF-algebras which satisfy $A_α\equiv_αA_β$ for every $α< β$. In particular, we obtain a set of separable simple unital AF-algebras of arbitrarily high Scott rank. Next, we give a partial converse to the aforementioned implication, showing that $A \otimes \mathcal{K} \equiv_{ω+ 2 \cdot α+2} B \otimes \mathcal{K}$ implies $K_0(A) \equiv_αK_0(B)$, for every unital $\mathrm{C}^\ast$-algebras $A$ and $B$.
