Summary: For a conditional expectation \(E\) on a (unital) \(C^*\)-algebra \(A\) there exists a real number \(K \geq 1\) such that the mapping \(K \cdot E -\text{id}_A\) is positive if and only if there exists a real number \(L \geq 1\) such that the mapping \(L \cdot E - \text{id}_A\) is completely positive, among other equivalent conditions. The estimate \((\min K) \leq (\min L) \leq (\min K) [\min K]\) is valid, where \([\cdot]\) denotes the entire part of a real number. As a consequence the notion of a “conditional expectation of finite index” is identified with that class of conditional expectations, which extends and completes results of M. Pimsner and S. Popa [Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 57-106 (1986; Zbl 0646.46057)], S. Popa [Regional Conference Series in Mathematics. 86. Providence, RI: x, 110 p. (1995; Zbl 0865.46044)], M. Baillet, Y. Denizeau and J.-F. Havet [Compos. Math. 66, No. 2, 199-236 (1988; Zbl 0657.46041)] and Y. Watatani [Mem. Am. Math. Soc. 424, 117 p. (1990; Zbl 0697.46024)] and others.