We introduce a new iterative amalgamated free product construction of II\(_1\) factors, and use it to construct a separable II\(_1\) factor which does not have property Gamma and is not elementarily equivalent to the free group factor \(\text {L}(\mathbb F_n)\), for any \(2\le n\le \infty \). This provides the first explicit example of two non-elementarily equivalent II\(_1\) factors without property Gamma. Moreover, our construction also provides the first explicit example of a II\(_1\) factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu’s free entropy theory and Popa’s deformation/rigidity theory.

我们引入了一种新的 II \(_1\)因子的迭代合并自由乘积构造，并用它构造了一个可分离的 II \(_1\)因子，该因子不具有属性 Gamma 并且基本上不等价于自由群因子\( \text {L}(\mathbb F_n)\)，对于任何\(2\le n\le \infty \)。这提供了两个非基本等价的没有属性 Gamma的 II \(_1\)因子的第一个明确示例。此外，我们的构造还提供了第一个没有属性 Gamma 的 II \(_1\)因子的明确示例，它本质上也不等于矩阵代数的任何超乘积。我们的证明结合了 Voiculescu 的自由熵理论和 Popa 的变形/刚性理论的技术。