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Let \Lambda (f) = K[x][y; f\frac {d}{dx} ] be an Ore extension of a polynomial algebra K[x] over a field K of characteristic zero where f\in K[x]. For a given polynomial f, the automorphism group of the algebra \Lambda (f) is explicitly described. The polynomial case \Lambda (0) = K[x,y] and the case of the Weyl algebra A_1= K[x][y; \frac {d}{dx} ] were done by Jung [J. Reine Angew. Math. 184 (1942), pp. 161–174] and van der Kulk [Nieuw Arch. Wisk. (3) 1 (1953), pp. 33–41], and Dixmier [Bul. Soc. Math. France 96 (1968), pp. 209–242], respectively. Alev and Dumas [Comm. Algebra 25 (1997), pp. 1655–1672] proved that the algebras \Lambda (f) and \Lambda (g) are isomorphic iff g(x) = \lambda f(\alpha x+\beta) for some \lambda, \alpha \in K\backslash \{ 0\} and \beta \in K. Benkart, Lopes and Ondrus [Trans. Amer. Math. Soc. 367 (2015), pp. 1993–2021] gave a complete description of the set of automorphism groups of algebras \Lambda (f). In this paper we complete the picture, i.e. given the polynomial f we have the explicit description of the automorphism group of \Lambda (f). The key concepts in finding the automorphism groups are the eigenform, the eigenroot and the eigengroup of a polynomial (introduced in the paper; they are of independent interest). [ABSTRACT FROM AUTHOR]

For any real \beta let H^2_\beta be the Hardy-Sobolev space on the unit ball \mathbb {B}_{n}, n\geq 1. H^2_\beta is a reproducing kernel Hilbert space and its reproducing kernel is bounded when \beta >n/2. In this paper, we characterize when the composition operator C_{\varphi } on H^{2}_{\beta } is Fredholm for a non-constant analytic map \varphi :\mathbb {B}_{n}\to \mathbb {B}_{n}. [ABSTRACT FROM AUTHOR]

We call an algebra A commutator-simple if [A,A] does not contain nonzero ideals of A. After providing several examples, we show that in these algebras derivations are determined by a condition that is applicable to the study of local derivations. This enables us to prove that every continuous local derivation D\colon L^1(G)\to L^1(G), where G is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras. [ABSTRACT FROM AUTHOR]

Let X be a finite connected poset and K a field. We give a full description of the Lie automorphisms of the incidence algebra I(X,K). In particular, we show that they are in general not proper. [ABSTRACT FROM AUTHOR]