## Abstract

Let $\mathfrak{\pi \x9d\x94\x85}\left(\mathcal{\beta \x84\x8b}\right)$ and ${\mathfrak{\pi \x9d\x94\x85}}^{s}\left(\mathcal{\beta \x84\x8b}\right)$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathcal{\beta \x84\x8b}$ and the real Jordan algebra of all self-adjoint operators on $\mathcal{\beta \x84\x8b}$, respectively. Suppose $\mathcal{\pi \x9d\x92\xb2}$ and $\mathcal{\pi \x9d\x92\pm}$ are subsets of $\mathfrak{\pi \x9d\x94\x85}\left(\mathcal{\beta \x84\x8b}\right)$ or ${\mathfrak{\pi \x9d\x94\x85}}^{s}\left(\mathcal{\beta \x84\x8b}\right)$ and $F\left(\beta \x8b\x85\right)$ is the $c$-numerical radius or the diameter of an operator. Under an assumption on $\mathcal{\pi \x9d\x92\xb2}$ and $\mathcal{\pi \x9d\x92\pm}$, characterizations are obtained for surjective maps $\mathrm{\Xi \xa6}:\mathcal{\pi \x9d\x92\xb2}\beta \x86\x92\mathcal{\pi \x9d\x92\pm}$ satisfying

$$F\left(AB\right)=F\left(\mathrm{\Xi \xa6}\left(A\right)\mathrm{\Xi \xa6}\left(B\right)\right)\phantom{\rule{1em}{0ex}}\left(A,B\beta \x88\x88\mathcal{\pi \x9d\x92\xb2}\right).$$

When $dim\mathcal{\beta \x84\x8b}\beta \x89\u20af3$, to establish the proofs, some general results are obtained for functions $F:\mathfrak{\pi \x9d\x94\x85}\left(\mathcal{\beta \x84\x8b}\right)\beta \x86\x92\left[0,+\mathrm{\beta \x88\x9e}\right)$ satisfying $\left({P}_{1}\right)$ $F\left(UA{U}^{\beta \x88\x97}\right)=F\left(A\right)$ for all $A\beta \x88\x88\mathfrak{\pi \x9d\x94\x85}\left(\mathcal{\beta \x84\x8b}\right)$ and unitary $U$ on $\mathcal{\beta \x84\x8b}$; $\left({P}_{2}\right)$ For $A\beta \x88\x88\mathfrak{\pi \x9d\x94\x85}\left(\mathcal{\beta \x84\x8b}\right)$, $F\left(A\right)=0$ if and only if $A$ is a multiple of the identity; $\left({P}_{3}\right)$ There are nonnegative real numbers $\mathrm{\Xi \pm},\mathrm{\Xi \xb2}$ with ${\mathrm{\Xi \pm}}^{2}+{\mathrm{\Xi \xb2}}^{2}\beta \x890$ such that $F\left(T\right)=\mathrm{\Xi \pm}\beta \x88\u20afT\beta \x88\u20af+\mathrm{\Xi \xb2}|tr\left(T\right)|$ for each rank-one $T\beta \x88\x88\mathfrak{\pi \x9d\x94\x85}\left(\mathcal{\beta \x84\x8b}\right)$.

## Citation

Yanfang Zhang. Xiaochun Fang. "Maps preserving the $c$-numerical radius of products for operators in $\mathfrak{B}(H)$." Rocky Mountain J. Math. 50 (6) 2265 - 2280, December 2020. https://doi.org/10.1216/rmj.2020.50.2265

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