Cent. Eur. J. Math. • 8(3) • 2010 • 597-601 DOI: 10.2478/s11533-010-0025-4 Central European Journal of Mathematics Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras Research Article Osamu Hatori 1∗ , Go Hirasawa 2† , Takeshi Miura 3‡ 1 Department of Mathematics, Faculty of Science, Niigata University, Niigata, Japan 2 Faculty of Engineering, Ibaraki University, Hitachi, Japan 3 Department of Applied Mathematics and Physics, Yamagata University, Yonezawa, Japan Received 25 December 2009; accepted 12 March 2010 Abstract: Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B , respectively, and let r(a) be the spectral radius of a. We show that if T : A→B is a surjective mapping, not assumed to be linear, satisfying r(T (a)+ T (b)) = r(a + b) for all a, b ∈A, then there exist a homeomorphism φ : M B → M A and a closed and open subset K of M B such that  T (a)(y)=   T (e)(y)ˆ a(φ(y)) y ∈ K  T (e)(y) ˆ a(φ(y)) y ∈ M B \ K for all a ∈A, where e is unit element of A. If, in addition,  T (e)=1 and  T (ie)= i on M B , then T is an algebra isomorphism. MSC: 46J10 Keywords: Uniform algebra • Commutative Banach algebra • Maximal ideal space • Shilov boundary • Algebra isomorphism • Norm-additive operator • Norm-linear operator © Versita Sp. z o.o. Dedicated to Professor Sin-Ei Takahasi for his retirement from Yamagata University with respect and affection ∗ E-mail: hatori@math.sc.niigata-u.ac.jp † E-mail: gou@mx.ibaraki.ac.jp ‡ E-mail: miura@yz.yamagata-u.ac.jp 597