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In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / 0 and T: A → B is a surjective map, not assumed to be linear, satisfying $$∥T\left(f\right)T\left(g\right)+\lambda ∥=∥fg+\lambda ∥\forall f,g\in A,$$ then T is an ℝ-linear isometry and there exist an...

If X and Y are Banach spaces, then subalgebras ⊂ B(X) and ⊂ B(Y), not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on X and Y respectively. The peripheral spectrum of A ∈ is the set ${\sigma }_{\pi }\left(A\right)=\lambda \in \sigma \left(A\right):\left|\lambda \right|=ma{x}_{z\in \sigma \left(A\right)}\left|z\right|$ of spectral values of A of maximum modulus, and a map φ: → is called peripherally-multiplicative if it satisfies the equation ${\sigma }_{\pi }\left(\phi \left(A\right)\circ \phi \left(B\right)\right)={\sigma }_{\pi }\left(AB\right)$ for all A,B ∈ . We show that any peripherally-multiplicative and surjective map φ: → , neither assumed to be linear nor...

It has been shown that any Banach algebra satisfying ‖f 2‖ = ‖f‖2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, $𝔽$ ) the algebra over...

Let (X, d X) and (Y,d Y) be pointed compact metric spaces with distinguished base points e X and e Y. The Banach algebra of all $𝕂$-valued Lipschitz functions on X - where $𝕂$ is either‒or ℝ - that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = f(x): |f(x)| = ‖f‖∞ of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ra{n}_{\pi }\left(T_1\left(f\right)T_2\left(g\right)\right)\cap Ra{n}_{\pi }\left(S_1\left(f\right)S_2\left(g\right)\right)\ne \varnothing$ for all f, g ∈ Lip0(X),...