Ideal $F$-norms on $C^*$-algebras

We show that every noncompactness measure on a $W^*$-algebra is an ideal $F$-pseudonorm. We establish the criterion of right Fredholm property of an element with respect to $W^*$-algebra. We prove that the element $-I$ realizes maximum distance from the positive element to the subset of all isometries of unital $C^*$-algebra, here $I$ is the unit of $C^*$-algebra. We also consider differences of two finite products of elements from the unit ball of $C^*$-algebra and obtain an estimate of their ideal $F$-pseudonorms. The paper is concluded with the convergence criterion in complete ideal $F$-norm for two series of elements from $W^*$-algebra.