Peripherally Monomial-Preserving Maps between Uniform Algebras

Abstract

Let A and B be uniform algebras and p(z, w) = z m w n a twovariable monomial. We characterize maps T from certain subsets of A into B such that σπ(p(T (f ), T (g))) ⊂ σπ(p(f, g)) holds for all f and g in the domain of T ; peripherally monomial-preserving maps. Furthermore A and B are proved to be isometrical isomorphic as Banach algebras. If the greatest common divisor of m and n is 1, then T is extended to an isometrical linear isomorphism; a weighted composition operator. An example of peripherally monomial-preserving surjections between uniform algebras which is not linear, nor multiplicative, nor injective is given when the greatest common divisor is strictly greater than 1. (2000). 46J10, 47B48. Mathematics Subject Classification

Generalizations of spectrally multiplicative surjections between uniform algebras

“…Motivated by this equality, we will consider two surjections S and T that satisfy r(S( f )T (g) − α) = r( f g − α) for some α ∈ C \ {0}, or σ π (S( f )T (g)) = σ π ( f g). In this paper, we will extend the results in [3,18] for S and T under the spectral radius condition, or the peripheral spectral condition.…”

“…Luttman and Tonev [16] proved that a peripherally multiplicative surjection S : A → B with S(1) = 1 is an isometric algebra isomorphism. Most recently, Hatori, Hino, Miura and Oka [3] have generalized the theorem of Luttman and Tonev. Among other things, they proved that if natural numbers m, n and a surjection S : A → B with S(1) = 1 satisfy σ π (S( f ) m S(g) n ) ⊂ σ π ( f m g n ) for all f , g ∈ A, then there exists an isometric algebra isomorphismS :…”

“…In [3], surjections S : A → B between uniform algebras that satisfy σ π (S( f ) m S(g) n ) ⊂ σ π ( f m g n ) ( f , g ∈ A) for some natural numbers m and n were characterized. The case when m = n = 1 was proven by Lambert, Luttman and Tonev [15,Corollary 4].…”

“…The notion of absolute multiplicativity was introduced in [3]. For example, if M is a multiplicative semigroup of A, that is M ⊂ A is closed under multiplication of A, then M is absolutely multiplicative.…”

Norm-additive in modulus maps between function algebras

“…For instance, the Gleason-Kahane-Żelazko theorem and the theorem by Kowalski and S lodkowski are related to this problem. Recently in [29] Tonev and Toneva gave non-linear sufficient conditions for isometries between dense subsets of uniformly closed function algebras, not necessarily unital, to be weighted composition operators and in particular, they obtained a characterization of norm-additive in modulus maps using a technique similar to the one used in the study of weak multiplicative maps ( [28]), or one can see some related conclusions in [5,13,14,16,26]. Moreover, Miura characterized in [21] R-linear isometries between function algebras and generalized some previous results.…”