This article has two goals: descriptive and theoretical. On the descriptive side, the article presents a grammar of gliding and epenthesis of Upper Sorbian. The descriptive goal is worthy because Upper Sorbian has a highly complex but regular and productive system of gliding and epenthesis. Upper Sorbian stands out from a typological point of view because it has ten [sic] different strategies to satisfy Onset.

On the theoretical side, the question is whether Optimality Theory that has been designed to solve conspiracies can deal with the complexities of Upper Sorbian. The answer is that it cannot unless it is modified to admit derivational levels. A point of interest is that level 1 in Upper Sorbian must be defined as the root level, not as the expected stem level that includes roots and affixes. Further, it is demonstrated that Itô and Mester’s Crisp Edge constraint makes wrong predictions for Upper Sorbian, so a new constraint, Multi, is postulated. Also, the analysis bears on the issue of positional markedness versus positional faithfulness and the question of whether Duke of York derivations should be admitted in phonology.

By using a representation of a Lie algebra on the second Hochschild cohomology group, we construct an obstruction class to extensibility of derivations and a short exact sequence of Wells type for an abelian extension of an associative algebra.

Given an infinite, compact, monothetic group $G$ we study decompositions and structure of unbounded derivations in a crossed product $\text{C}^{\ast }$-algebra $C(G)\rtimes \mathbb{Z}$ obtained from a translation on $G$ by a generator of a dense cyclic subgroup. We also study derivations in a Toeplitz extension of the crossed product and the question whether unbounded derivations can be lifted from one algebra to the other.

The contribution of morphological awareness to reading comprehension in Hebrew was tested in 100 second- and third-grade students on three types of morphology: inflections, derivations, and construct formation, controlling for vocabulary knowledge. Third graders performed better than second graders on inflectional and construct formation awareness, but only derivations and construct formation predicted success in reading comprehension. Significant differences in reading comprehension but not in orthographic word recognition and phonological decoding were found between students with low and high morphological awareness. The results highlight the importance of examining the unique contribution of different components of morphological awareness to reading comprehension.

We prove that, under rather general conditions, the 1-cohomology of a von Neumann algebra $M$ with values in a Banach $M$ -bimodule satisfying a combination of smoothness and operatorial conditions vanishes. For instance, we show that, if $M$ acts normally on a Hilbert space ${\mathcal{H}}$ and ${\mathcal{B}}_{0}\subset {\mathcal{B}}({\mathcal{H}})$ is a norm closed $M$ -bimodule such that any $T\in {\mathcal{B}}_{0}$ is smooth (i.e., the left and right multiplications of $T$ by $x\in M$ are continuous from the unit ball of $M$ with the $s^{\ast }$ -topology to ${\mathcal{B}}_{0}$ with its norm), then any derivation of $M$ into ${\mathcal{B}}_{0}$ is inner. The compact operators are smooth over any $M\subset {\mathcal{B}}({\mathcal{H}})$ , but there is a large variety of non-compact smooth elements as well.

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

It is shown that if $G$ is a finite $p$-group of coclass 2 with $p>2$, then $G$ has a noninner automorphism of order $p$.

Let $A$ be a Banach algebra and let $\pi :\,A\,\to \,\mathcal{L}\left( H \right)$ be a continuous representation of $A$ on a separable Hilbert space $H$ with dim $H\,=\,\text{m}$. Let ${{\pi }_{ij}}$ be the coordinate functions of $\pi $ with respect to an orthonormal basis and suppose that for each $1\,\le \,j\,\le \,\text{m,}\,{{C}_{j}}\,=\,\sum\nolimits_{i=1}^{\text{m}}{\left\| {{\pi }_{ij}} \right\|}{{A}^{*}}\,<\,\infty $ and ${{\sup }_{j}}\,{{C}_{j}}\,<\,\infty $. Under these conditions, we call an element $\overline{\Phi }\,\in \,{{\iota }^{\infty }}\,\left( \mathfrak{m},\,{{A}^{**}} \right)$ left $\pi $-invariant if $a\,\cdot \overline{\Phi }\,={{\,}^{^{t}\pi }}\left( a \right)\overline{\Phi }$ 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 and the second author in the special case where $\pi :\,A\,\to \text{C}$ is a non-zero character on $A$.

Let R be a prime ring, let I be a nonzero ideal of R and let n be a fixed positive integer. We prove that if the characteristic of R is either 0 or a prime p that is greater than 2n, then an additive map d that satisfies d(xn+1)=∑ nj=0xn−jd(x)xj for all x∈I must be a derivation.

In this note, we will prove that a finite-dimensional Lie algebra L over a field of characteristic zero, admitting an abelian algebra of derivations D≤Der(L), with the property for some n>1, is necessarily solvable. As a result, we show that if L has a derivation d:L→L such that Ln⊆d(L), for some n>1, then L is solvable.

A characterization is given of those unital, 2-subhomogeneous, Fell C*-algebras which have only inner derivations. This proves Sproston and Strauss's conjecture from 1992. Various examples are given of phenomena which cannot occur for separable C*-algebras. In particular, an example is given of a C*-algebra with only inner derivations which has a quotient algebra admitting outer derivations. This answers a question of Akemann, Elliott, Pedersen and Tomiyama from 1976.

A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2: n; ω2) (of dimension one less than a power of p) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2: n; ω2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).

Using Grothendieck's semicontinuity theorem for half-exact functors, we derive two semicontinuity results on Hochschild cohomology. We apply these to show that the first Hochschild cohomogy group of the mesh algebra of a translation quiver over a domain vanishes if and only if the translation quiver is simply connected. We then establish an exact sequence relating the first Hochschild cohomology group of an algebra to that of the endomorphism algebra of a module and apply it to study the first Hochschild cohomology group of an Auslander algebra. Our main result shows that for a finite-dimensional and representation-finite algebra algebra $A$ over an algebraically closed field with Auslander algebra $\Lambda$ the following conditions are equivalent:

(1) $A$ admits no outer derivation;

(2) $\Lambda$ admits no outer derivations;

(3) $A$ is simply connected;

(4) $\Lambda$ is strongly simply connected.

We prove that the separating space of an epimorphism from a Lie–Banach algebra onto the (continuous) derivation algebra $\mathfrak{Der}(A)$ of a Banach algebra $A$ consists of derivations which map into the radical of $A$.

AMS 2000 Mathematics subject classification: Primary 46H40; 46H70. Secondary 47B47; 47B48

We investigate a connection between the differential of polylogarithms (as considered by Cathelineau) and a finite variant of them. This allows to answer a question raised by Kontsevich concerning the construction of functional equations for the finite analogs, using in part the p-adic version of polylogarithms and recent work of Besser. Kontsevich's original unpublished note is supplied (with his kind permission) in an ‘Appendix’ at the end of the paper.

We prove a bounded decomposition for higher order Hankel forms and characterize the first order Hochschild cohomology groups of the disk algebra with coefficients in the space of bounded Hankel forms of some fixed order. Although these groups are non-trivial, we prove that every bounded derivation is inner and necessarily implemented by a Hankel form of order one higher. In terms of operators, this result extends the similarity result of Aleksandrov and Peller. Both of the main structural theorems here rely on estimates involving multilinear maps on the $n$-fold product of the disk algebra and we obtain several higher order analogues of the factorization results due to Aleksandrov and Peller. 2000 Mathematics Subject Classification: 47B35, 46E15, 46E25.

Let $G$ be a locally compact group. The question of whether ${\cal H}^1(L^1(G),M(G))$, the first Hochschild cohomology group of $L^1(G)$ with coefficients in $M(G)$, is zero was first studied by B. E. Johnson and initiated his development of the theory of amenable Banach algebras. He was able to show that ${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$ whenever $G$ is amenable, a $[SIN]$-group, or a matrix group satisfying certain conditions. No group such that ${\cal H}^1(L^1(G),M(G)) \neq \{ 0 \}$ is known. In this paper, we approach the problem of whether ${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$ from several angles. Using weakly almost periodic functions, we show that ${\cal H}^1(L^1(G),L^1(G))$ is always Hausdorff for unimodular $G$. We also show that for $[IN]$-groups, every derivation $D \colon L^1(G) \to L^1(G)$ is implemented, not necessarily by an element of $M(G)$, but at least by an element of $\mbox{VN}(G)$, the group von Neumann algebra of $G$. This applies, in particular, to the group $G := {\mathbb T}^2 \rtimes \mbox{SL}(2, {\mathbb Z})$, for which it is unknown whether ${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$. Finally, we analyse the structure of derivations on $L^1(G)$; an important r\^ole is played by the closed normal subgroup $N$ of $G$ generated by the elements of $G$ with relatively compact conjugacy classes. We can write an arbitrary derivation $D \colon L^1(G) \to L^1(G)$ as a sum $D = D_N + D_{N^\perp}$, where $D_N$ and $D_{N^\perp}$ can be tackled with different techniques. Under suitable conditions, all satisfied by ${\mathbb T}^2 \rtimes \mbox{SL}(2,{\mathbb Z})$, we can show that $D_N$ is implemented by an element of $\mbox{VN}(G)$ and that $D_{N^\perp}$ is implemented by a measure. 1991 Mathematics Subject Classification: 22D05, 22D25, 43A10, 43A20, 46H25, 46L10, 46M20, 47B47, 47B48.

Let R be a semiprime ring with a derivation d, λ a left ideal of R and k, n two positive integers. Suppose that [d(xn),xn]k = 0 for all x ∊ λ. Then [λ,R]d(R) = 0. That is, there exists a central idempotent e ∊ U, the left Utumi quotient ring of R, such that d vanishes identically on eU and λ(l — e) is central in (1 — e)U