# Spectrum Preserving Linear Map on Liminal $C^*$-algebras

Volume 6, Issue 4, pp 311 - 314
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### Authors

F. Golfarshchi - Department of Mathematics, Sahand University of Technology, Tabriz-Iran. A. A. Khalilzadeh - Department of Mathematics, Sahand University of Technology, Tabriz-Iran.

### Abstract

Let $A$ and $B$ be unital semi-simple Banach algebras. If $B$ is a liminal $C^*$-algebra and $\varphi$ is a surjective spectrum preserving linear mapping from $A$ to $B$, then $\varphi$ is a Jordan homomorphism.

### Keywords

• Semi-simple
• liminal $C^*$-algebra
• Spectrum preserving
• Jordan homomorphism.

•  15A86
•  47A10

### References

• [1] A. Jafarian, A. R Sourour, Spectrum preserving linear maps, J. Funct. Anal., 66 (1986), 255-261

• [2] A. R. Sourour, Invertibility preserving linear maps on L(X), Trans. Amer.Math. Soc. , 348 (1996), 13-30

• [3] B. Aupetit, Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras, J. London. Math. Soc. , 62 (2000), 917-924

• [4] B. Aupetit, H. du Toit Mouton, Spectrum preserving linear mappings in Banach algebras, J.Studi.Math. , 109 (1994), 91-100

• [5] B. Aupetit, Une generaisation du theoreme de Gleason-Kahane-Zelazko pour les algebres de Banach, Pacic. J. Math., 85 (1979), 11-17

• [6] B. Blackadar, Theory of $C^*$- subalgebra and von Neumann algebras, Springer-Verlag, Berlin (2006)

• [7] G. Murphy, $C^*$-algebras and operator theory, Academic Press, Inc (1990)

• [8] I. Kaplansky, Algebraic and analytic aspects of operator algebras, CBMS Regional Conference Series in Mathematics 1(Amre . Math. Soc)., Providence (1970)