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1. CMB 2014 (vol 57 pp. 609)

Nasr-Isfahani, Alireza
 Jacobson Radicals of Skew Polynomial Rings of Derivation Type We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive, when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type. Keywords:skew polynomial rings, Jacobson radical, derivationCategories:16S36, 16N20

2. CMB 2013 (vol 57 pp. 270)

Didas, Michael; Eschmeier, Jörg
 Derivations on Toeplitz Algebras Let $H^2(\Omega)$ be the Hardy space on a strictly pseudoconvex domain $\Omega \subset \mathbb{C}^n$, and let $A \subset L^\infty(\partial \Omega)$ denote the subalgebra of all $L^\infty$-functions $f$ with compact Hankel operator $H_f$. Given any closed subalgebra $B \subset A$ containing $C(\partial \Omega)$, we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra $\mathcal(B) \subset B(H^2(\Omega))$. In particular, we show that every derivation on $\mathcal{T}(A)$ is inner. These results are new even for $n=1$, where it follows that every derivation on $\mathcal{T}(H^\infty+C)$ is inner, while there are non-inner derivations on $\mathcal{T}(H^\infty+C(\partial \mathbb{B}_n))$ over the unit ball $\mathbb{B}_n$ in dimension $n\gt 1$. Keywords:derivations, Toeplitz algebras, strictly pseudoconvex domainsCategories:47B47, 47B35, 47L80

3. CMB 2012 (vol 57 pp. 51)

Fošner, Ajda; Lee, Tsiu-Kwen
 Jordan $*$-Derivations of Finite-Dimensional Semiprime Algebras In the paper, we characterize Jordan $*$-derivations of a $2$-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the theorem: Let $deltacolon R o R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R=Uoplus V$ such that both $U$ and $V$ are invariant under $delta$. Moreover, $*$ is the identity map of $U$ and $delta,|_U$ is a derivation, and the Jordan $*$-derivation $delta,|_V$ is inner. We also prove the theorem: Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, $operatorname{char},R e 2$, and let $delta$ be a nonzero Jordan $*$-derivation of $R$. If $delta$ is an elementary operator of $R$, then $operatorname{dim}_CRlt infty$ and $delta$ is inner. Keywords:semiprime algebra, involution, (inner) Jordan $*$-derivation, elementary operatorCategories:16W10, 16N60, 16W25

4. CMB 2012 (vol 56 pp. 534)

Filali, M.; Monfared, M. Sangani
 A Cohomological Property of $\pi$-invariant Elements Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$ be a continuous representation of $A$ on a separable Hilbert space $H$ with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of $\pi$ with respect to an orthonormal basis and suppose that for each $1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m} \|\pi_{ij}\|_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$ left $\pi$-invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all $a\in A$. In this paper we prove a link between the existence of left $\pi$-invariant elements and the vanishing of certain Hochschild cohomology groups of $A$. Our results extend an earlier result by Lau on $F$-algebras and recent results of Kaniuth-Lau-Pym and the second named author in the special case that $\pi \colon A \longrightarrow \mathbf C$ is a non-zero character on $A$. Keywords:Banach algebras, $\pi$-invariance, derivations, representationsCategories:46H15, 46H25, 13N15

5. CMB 2011 (vol 56 pp. 31)

Ayuso, Fortuny P.
 Derivations and Valuation Rings A complete characterization of valuation rings closed for a holomorphic derivation is given, following an idea of Seidenberg, in dimension $2$. Keywords:singular holomorphic foliation, derivation, valuation, valuation ringCategories:32S65, 13F30, 13A18

6. CMB 2010 (vol 54 pp. 21)

 Generalized D-symmetric Operators II Let $H$ be a separable, infinite-dimensional, complex Hilbert space and let $A, B\in{\mathcal L }(H)$, where ${\mathcal L}(H)$ is the algebra of all bounded linear operators on $H$. Let $\delta_{AB}\colon {\mathcal L}(H)\rightarrow {\mathcal L}(H)$ denote the generalized derivation $\delta_{AB}(X)=AX-XB$. This note will initiate a study on the class of pairs $(A,B)$ such that $\overline{{\mathcal R}(\delta_{AB})}= \overline{{\mathcal R}(\delta_{A^{\ast}B^{\ast}})}$. Keywords:generalized derivation, adjoint, D-symmetric operator, normal operatorCategories:47B47, 47B10, 47A30
 A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$ We strengthen certain results concerning actions of $(\Comp,+)$ on $\Comp^{3}$ and embeddings of $\Comp^{2}$ in $\Comp^{3}$, and show that these results are in fact valid over any field of characteristic zero. Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine spaceCategories:14R10, 14R20, 14R25, 13N15