In this paper we study modular extendability and equimodularity of endomorphisms and E$_0$-semigroups on factors with respect to f.n.s. weights. We show that modular extendability is a property that does not depend on the choice of weights, it is a cocycle conjugacy invariant and it is preserved under tensoring. We say that a modularly extendable E$_0$-semigroup is of type EI, EII or EIII if its modular extension is of type I, II or III, respectively. We prove that all types exist on properly infinite factors. We also compute the coupling index and the relative commutant index for the CAR flows and $q$-CCR flows. As an application, by considering repeated tensors of the CAR flows we show that there are infinitely many non cocycle conjugate non-extendable $E_0$-semigroups on the hyperfinite factors of types II$_1$, II$_{\infty}$ and III$_\lambda$, for $\lambda \in (0,1)$.

中文翻译：

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关于因子E 0 -半群的分类和模可扩展性

在本文中，我们研究了关于 fns 权重的因子的自同态和 E$_0$-半群的模扩展性和等模性。我们表明模块化可扩展性是一种不依赖于权重选择的属性，它是一个共循环共轭不变量，并且在张量下保持不变。我们说一个可模扩展的 E$_0$-半群是 EI、EII 或 EIII 类型，如果它的模扩展分别是类型 I、II 或 III。我们证明所有类型都存在于适当的无限因子上。我们还计算了 CAR 流和 $q$-CCR 流的耦合指数和相对交换指数。作为一个应用，通过考虑 CAR 流的重复张量，我们表明在类型为 II$_1$、II$_{\infty}$ 和III$_\lambda$,