Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative

Abstract

Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ (fg) of the product of any two elements f and g in A coincides with the spectrum σ (Tf T g). Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and nonmultiplicative unital map from a commutative C*-algebra into itself such that σ (Tf T g) = σ (fg) holds for every f, g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ (Tf T g) ⊂ σ (fg) holds for every f, g.

Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

“…This result was reminiscent of the classical Banach-Stone Theorem by demonstrating a connection between the spectral structure of C (X ) and its linear and multiplicative structures, as well as to the underlying topological structure of X . A wide range of spectral preserver problems have now been studied, and a variety of spectrum-type properties have also been shown to relate to the linear and multiplicative structures of uniform algebras [4,11,14], but also to more general unital, semi-simple commutative Banach algebras [3,5,6]. See [2] for a recent survey of spectral preservers.…”

Weak multiplicative operators on function algebras without units

“…In [11] Rao and Roy extended this result for a self-map of a uniform algebra, and in [12] for a self-map of a function algebra without unit. In [2] it was proven for surjections between distinct uniform algebras, in [3] for surjections between semisimple commutative Banach algebras with units, and in [4] between completely regular commutative Banach algebras without units. Norm-multiplicative operators, for which T f T g = f g , f, g ∈ A, were introduced in [7], where sufficient conditions for a norm-multiplicative operator between uniform algebras to be a composition operator in modulus were obtained.…”

Maps between Banach function algebras satisfying certain norm conditions

“…He proved that if X is a first countable compact Hausdorff space and C (X ) is the Banach algebra of all continuous complex-valued functions on X , then any surjective map T : C (X ) → C (X ) preserving multiplicatively the spectrum of functions, i.e., σ (T ( )T ( )) = σ ( ) for all ∈ C (X ), is a weighted composition operator, or in other words a multiplication of an algebra isomorphism by a continuous function. Generalizations of Molnár's result were given in [5,18,19] for arbitrary uniform algebras rather than C (X ) and in [6,8] for commutative semisimple Banach algebras. Considering a similar multiplicativity condition for some parts of the spectrum (called peripheral spectrum) instead of the whole spectrum, the above result was extended in [9,11,14,16].…”