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

We study ideal $F$-norms $\|\cdot\|_p$, $0 < p <+\infty$ associated with a trace $\varphi$ on a $C^*$-algebra $\mathcal{A}$. If $A, B$ of $\mathcal{A}$ are such that $|A|\leq |B|$, then $\|A\|_p \leq \|B\|_p$. We have $\|A\|_p=\|A^*\|_p$ for all $A$ from $\mathcal{A}$ ($0 < p <+\infty$) and $\|\cdot\|_p$ is a seminorm for $1 \leq p <+\infty$. We estimate the distance from any element of unital $\mathcal{A}$ to the scalar subalgebra in the seminorm $\|\cdot\|_1$. We investigate geometric properties of semiorthogonal projections from $\mathcal{A}$. If a trace $\varphi$ is finite, then the set of all finite sums of pairwise products of projections and semiorthogonal projections (in any order) of $\mathcal{A}$ with coefficients from $\mathbb{R}^+ $ is not dense in $\mathcal{A}$.