Geodesic
Ideal $F$-norms on $C^*$-algebras. II
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2019), pp. 90-96.

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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 $\mathcal{A}$.
Keywords: Hilbert space, linear operator, projection, semiorthogonal projection, unitary operator, inequality, $C^*$-algebra, trace, ideal $F$-norm. 
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A. M. Bikchentaev. Ideal $F$-norms on $C^*$-algebras. II. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2019), pp. 90-96. http://geodesic.mathdoc.fr/item/IVM_2019_3_a6/

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