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Search: MSC category 13N15 ( Derivations )

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1. 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, representations
Categories:46H15, 46H25, 13N15

2. CMB 2009 (vol 53 pp. 77)

Finston, David; Maubach, Stefan
Constructing (Almost) Rigid Rings and a UFD Having Infinitely Generated Derksen and Makar-Limanov Invariants
An example is given of a UFD which has an infinitely generated Derksen invariant. The ring is "almost rigid" meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem.

Categories:14R20, 13A50, 13N15

3. CMB 2009 (vol 52 pp. 535)

Daigle, Daniel; Kaliman, Shulim
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 space
Categories:14R10, 14R20, 14R25, 13N15

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