中文摘要：

定义并研究了冯-代数．条件期望基于冯-代数的描述。即作为初等算子的良好性质，会较之一般代数简洁许多．这是因为冯-代数包含一些特殊的子代数．给出了此类代数上置信的初等条件期望的描述及其最小存在的充要条件．并且定义了指标冯一有限条件期望．作为以上结果的推论，得出了条件期望指标有限的充分必要条件和一个重要不等式．

英文摘要：

In this work, we define FW-algebras, which contain some special elements, and prove that if A is a FW-algebra, then we can give necessary and sufficient conditions for the minimal conditional expectation to be a faithful elementary conditional expectation of A. We give a description of all faithful elementary conditional expectations on FW-algebras. We then use these results to give necessary and sufficient conditions for certain conditional expectations to be index-finite and we derive an inequality for the index.