On local automorphisms of group algebras of compact groups

Abstract

Abstract. We show that with few exceptions every local isometric automorphism of the group algebra L p (G) of a compact metric group G is an isometric automorphism.

On 2-local nonlinear surjective isometries on normed spaces and C$^*$-algebras

“…Note that the weaker result that every mapping in CL(A) is surjective for the same A was proved in [9,Lemma]. Example 3.2 implies that we cannot drop the assumption on first countability.…”

“…Since A is commutative, Φ is a * -endomorphism. By the assumption that A = C(K) for some first countable compact Hausdorff space K, the rest of the proof can be completed by almost the same way as in the proof of [9,Lemma]. Indeed, we see that Φ is of the form f → f • ϕ 0 , f ∈ C(K), for some continuous surjection ϕ 0 : K → K. If ϕ 0 is not injective, then there exists a point z ∈ K such that ϕ −1 0 (z) has more than one point.…”

Automorphisms on algebras of operator-valued Lipschitz maps

“…These results motivated further research on reflexivity in the setting of group algebras and function algebras. Along this line, Molnár and Zalar, [24], studied the algebraic reflexivity of the isometric automorphism group of the convolution algebra L p (G) of a compact metric group G. Concerning function algebras, Cabello Sánchez and Molnár investigated in [10] the reflexivity of the automorphism group of Banach algebras of holomorphic functions, Fréchet algebras of holomorphic functions, and algebras of continuous functions (see also [9]). In [11], Cabello Sánchez proved that the automorphism group of L ∞ is algebraically reflexive.…”