Abstract:
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 $\mathcal{A}$.
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