### Ring homomorphisms on real Banach algebras.

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Let A and B be uniformly closed function algebras on locally compact Hausdorff spaces with Choquet boundaries Ch A and ChB, respectively. We prove that if T: A → B is a surjective real-linear isometry, then there exist a continuous function κ: ChB → z ∈ ℂ: |z| = 1, a (possibly empty) closed and open subset K of ChB and a homeomorphism φ: ChB → ChA such that T(f) = κ(f ∘φ) on K and $T\left(f\right)=\kappa \overline{fo\phi}$ on ChB K for all f ∈ A. Such a representation holds for surjective real-linear isometries between (not necessarily...

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σπ (T) of T is defined by σπ (T) = z ∈ σ(T): |z| = maxw∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σπ (φ(S)ϕ(T)) = σπ (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A 1 TA 2 −1 and ϕ(T) = A 2 TA 1 −1 for some bijective bounded linear operators A 1; A 2 of X onto Y, or of the form φ(T) = B 1 T*B 2 −1 and ϕ(T) = B 2 T*B −1 for some...

We describe the general form of isometries between uniformly closed function algebras on locally compact Hausdorff spaces in a continuation of the study by Miura. We can actually obtain the form on the Shilov boundary, rather than just on the Choquet boundary. We also give an example showing that the form cannot be extended to the whole maximal ideal space.

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 $$\widehat{T\left(a\right)}\left(y\right)=\left\{\begin{array}{c}\widehat{T\left(e\right)}\left(y\right)\widehat{a}\left(\phi \left(y\right)\right)y\in K\\ \widehat{T\left(e\right)}\left(y\right)\overline{\widehat{a}\left(\phi \left(y\right)\right)}y\in {M}_{\mathcal{B}}\setminus K\end{array}\right.$$ for all a ∈ A, where e is unit element of A. If, in addition, $$\widehat{T\left(e\right)}=1$$ and $$\widehat{T\left(ie\right)}=i$$ on M B, then T is an algebra isomorphism.

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