Analogous to subfactor theory, employing Watatani's notions of index and $C^{\ast }$-basic construction of certain inclusions of $C^{\ast }$-algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital $C^{\ast }$-algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate $C^{\ast }$-subalgebras of any inclusion of unital $C^{\ast }$-algebras admitting a finite index conditional expectation. Then, on the lines of Bakshi et al. (Trans. Amer. Math. Soc. 371 (2019) 5973–5991), we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate $C^{\ast }$-subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an irreducible inclusion of type $III$ factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite.

中文翻译：

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中间子代数的格

类似于子因子理论，采用 Watatani 的指数概念和$C^{*}$- 某些夹杂物的基本构造$C^{*}$-代数，（a）我们开发了一个傅里叶理论（由傅里叶变换，旋转映射和移位算子组成）关于任何包含简单单位的相对交换$C^{*}$-具有有限 Watatani 指数的代数，以及 (b) 我们引入了中间角之间的内角和外角的概念$C^{*}$-任何包含单位的子代数$C^{*}$-承认有限指数条件期望的代数。然后，根据 Bakshi等人的说法。（Trans. Amer. Math. Soc . 371 (2019) 5973–5991），我们应用这些概念来获得中间格的基数的界限$C^{*}$-任何不可约包含的子代数，如 (a) 中的，并改进 Longo 对不可约包含类型的中间子因子的基数的界$\mathrm{一世}\mathrm{一世}\mathrm{一世}$具有有限指数的因子。此外，我们还证明了对于相当大的有限冯诺依曼代数包含类，中间冯诺依曼子代数的格总是有限的。