We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C\(^*\)-algebras of countable groups with (relative) property (T). We derive that the full C\(^*\)-algebras of the groups \(\mathbb {Z}^2\times \text {SL}_2({\mathbb {Z}})\) and \(\text {SL}_n({\mathbb {Z}})\), for \(n\ge 3\), do not have the local lifting property (LLP). We also prove that the full C\(^*\)-algebras of a large class of groups \(\Gamma \) with property (T), including those such that \(\text {H}^2(\Gamma ,{\mathbb {R}})\not =0\) or \(\text {H}^2(\Gamma ,\mathbb {Z}\Gamma )\not =0\), do not have the lifting property (LP). More generally, we show that the same holds if \(\Gamma \) admits a probability measure preserving action with non-vanishing second \({\mathbb {R}}\)-valued cohomology. Finally, we prove that the full C\(^*\)-algebra of any non-finitely presented property (T) group fails the LP.

我们开发了一种基于第二次同调子组不消失的新方法，用于证明具有（相对）性质（T）的可数组的完全C \（^ * \）-代数的提升性质失败。我们得出\\\\ mathbb {Z} ^ 2 \ times \ text {SL} _2（{\ mathbb {Z}}）\\和\（\ text的全C \（^ * \）-代数{SL} _n（{\ mathbb {Z}}）\）对于\（n \ ge 3 \）而言，不具有局部提升属性（LLP）。我们还证明了具有属性（T）的大型组\（\ Gamma \）的完整C \（^ * \）-代数，包括\（\ text {H} ^ 2（\ Gamma， {\ mathbb {R}}）\ not = 0 \）或\（\ text {H} ^ 2（\ Gamma，\ mathbb {Z} \ Gamma）\ not = 0 \），不具有提升属性（LP）。更普遍地说，我们证明如果\（\ Gamma \）承认概率度量保持动作且第二个\（{\ mathbb {R}} \）的同调性不变，则成立。最后，我们证明了任何非限定表示的属性（T）组的完整C \（^ * \）-代数不通过LP。