# MAPPING PRESERVING NUMERICAL RANGE OF OPERATOR PRODUCTS ON C*-ALGEBRAS

• MABROUK, MOHAMED (Department of Mathematics College of Applied Sciences and Department of Mathematics Faculty of Sciences)
• Published : 2015.11.30

#### Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two unital $C^*$-algebras. Denote by W(a) the numerical range of an element $a{\in}\mathcal{A}$. We show that the condition W(ax) = W(bx), ${\forall}x{\in}\mathcal{A}$ implies that a = b. Using this, among other results, it is proved that if ${\phi}$ : $\mathcal{A}{\rightarrow}\mathcal{B}$ is a surjective map such that $W({\phi}(a){\phi}(b){\phi}(c))=W(abc)$ for all a, b and $c{\in}\mathcal{A}$, then ${\phi}(1){\in}Z(B)$ and the map ${\psi}={\phi}(1)^2{\phi}$ is multiplicative.

#### References

1. R. An and J. Hou, Additivity of Jordan multiplicative maps on Jordan operator algebras, Taiwanese J. Math. 10 (2006), no. 1, 45-64. https://doi.org/10.11650/twjm/1500403798
2. F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and elements of normed algebras, Cambridge Univ. Press, London, 1971.
3. M. Bresar and S. Spela, Determining elements in Banach algebras through spectral properties, J. Math. Anal. Appl. 393 (2012), no. 1, 144-150. https://doi.org/10.1016/j.jmaa.2012.03.058
4. J. T. Chan, Numerical radius preserving operators on B(H), Proc. Amer. Math. Soc. 123 (1995), no. 5, 1437-1439. https://doi.org/10.1090/S0002-9939-1995-1231293-7
5. M. A. Chebotar, W. F. Ke, P. K. Lee, and N. C. Wong, Mappings preserving zero products, Studia Math. 155 (2003), no. 1, 77-94. https://doi.org/10.4064/sm155-1-6
6. J. B. Conway, A Course in Functional Analysis, Springer, 1990.
7. H. L. Gau and C. K. Li, C*-isomorphisms, Jordan isomorphisms, and numerical range preserving maps, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2907-2914. https://doi.org/10.1090/S0002-9939-07-08807-7
8. P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982.
9. J. Hou and Q. Di, Maps preserving numerical ranges of operator products, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1435-1446. https://doi.org/10.1090/S0002-9939-05-08101-3
10. R. V. Kadisson, Isometries of operator algebras, Ann. of Math. 54 (1951), no. 2, 325- 338. https://doi.org/10.2307/1969534
11. E. C. Lance, Unitary operators on Hilbert C*-modules, Bull. London Math. Soc. 4 (1994), no. 4, 363-366.
12. C. K. Li, A survey on linear preservers of numerical ranges and radii, Taiwanese J. Math. 5 (2001), no. 3, 477-496. https://doi.org/10.11650/twjm/1500574944
13. C. K. Li and E. Poon, Maps preserving the joint numerical radius distance of operators, Linear Algebra Appl. 437 (2012), no. 5, 1194-1204. https://doi.org/10.1016/j.laa.2012.04.018
14. L. Molnar, On isomorphisms of standard operator algebras, Studia Math. 142 (2000), no. 3, 295-302. https://doi.org/10.4064/sm-142-3-295-302
15. V. Pellegrini, Numerical range preserving operators on a Banach algebra, Studia Math. 54 (1975), no. 2, 143-147. https://doi.org/10.1002/sapm1975542143
16. J. G. Stampfli and J. P. Williams, Growth conditions and the numerical range in a Banach algebras, Tohoku Math. J. 20 (1968), 417-424. https://doi.org/10.2748/tmj/1178243070

#### Cited by

1. Numerical radius characterizations of elements in -algebras 2017, https://doi.org/10.1080/03081087.2017.1380595