We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units $V(FG)$ of the group algebra $FG$ is locally nilpotent; (ii) the set of nontrivial nilpotent elements of $FG$ is finite and nonempty, and $V(FG)$ is an Engel group.

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$, every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba)$ for some continuous linear map $\unicode[STIX]{x1D70E}$. We show that all $C^{\ast }$-algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications.

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.

Motivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra $A$. We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$. Examples include all left Arens product algebras over $A$, but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$-module action $Q$ on a space $X$, we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak$^{\ast }$-continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.

Let $F$ be a field of characteristic $p\geq 0$ and $G$ any group. In this article, the Engel property of the group of units of the group algebra $FG$ is investigated. We show that if $G$ is locally finite, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G$ is locally nilpotent and $G^{\prime }$ is a $p$-group. Suppose that the set of nilpotent elements of $FG$ is finite. It is also shown that if $G$ is torsion, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G^{\prime }$ is a finite $p$-group and $FG$ is Lie Engel, if and only if ${\mathcal{U}}(FG)$ is locally nilpotent. If $G$ is nontorsion but $FG$ is semiprime, we show that the Engel property of ${\mathcal{U}}(FG)$ implies that the set of torsion elements of $G$ forms an abelian normal subgroup of $G$.

We develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version of the homotopy category of weakly injective kG-modules and a recollement relating the stable category, the homotopy category, and the derived category of kG-modules.

The decomposability number of a von Neumann algebra $\mathcal{M}$ (denoted by $\text{dec}\left( \mathcal{M} \right)$) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in $\mathcal{M}$. In this paper, we explore the close connection between $\text{dec}\left( \mathcal{M} \right)$ and the cardinal level of the Mazur property for the predual ${{\mathcal{M}}_{*}}$ of $\mathcal{M}$, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group $G$ as the group algebra ${{L}_{1}}(G)$, the Fourier algebra $A(G)$, the measure algebra $M(G)$, the algebra $LUC{{(G)}^{*}}$, etc. We show that for any of these von Neumann algebras, say $\mathcal{M}$, the cardinal number dec$(\mathcal{M})$ and a certain cardinal level of the Mazur property of ${{\mathcal{M}}_{*}}$ are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of $G$: the compact covering number $\kappa (G)$ of $G$ and the least cardinality $\mathcal{X}(G)$ of an open basis at the identity of $G$. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra $A{{(G)}^{**}}$.

Let F be a field of characteristic p and G a group containing at least one element of order p. It is proved that the group of units of the group algebra FG is a bounded Engel group if and only if FG is a bounded Engel algebra, and that this is the case if and only if G is nilpotent and has a normal subgroup H such that both the factor group G/H and the commutator subgroup H′ are finite p–groups.

Let p be a prime, a field of pn elements, and G a finite p-group. It is shown here that if G has a quotient whose commutator subgroup is of order p and whose centre has index pk, then the group of normalized units in the group algebra has a conjugacy class of pnk elements. This was first proved by A. Bovdi and C. Polcino Milies for the case k = 2; their argument is now generalized and simplified. It remains an intriguing question whether the cardinality of the smallest noncentral conjugacy class can always be recognized from this test.

Let $G$ be a $\sigma$-compact, locally compact group and $\mathcal I$ be a closed 2-sided ideal with finite codimension in $L^1(G)$. It is shown that there are a closed left ideal ${\mathcal L}$ having a right bounded approximate identity and a closed right ideal ${\mathcal R}$ having a left bounded approximate identity such that ${\mathcal I} = {\mathcal L} + {\mathcal R}$. The proof uses ideas from the theory of boundaries of random walks on groups. 2000 Mathematics Subject Classification: primary 43A20; secondary 42A85, 43A07, 46H10, 46H40, 60B11.

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.

A group is said to be factorizable if it has a finite number of abelian subgroups, H1, H2, … Hn, such that G = H1H2 … Hn. It is shown that, if G is a factorizable or connected locally compact group, then every derivation from L1 (G) to an arbitrary L1 (G)-bimodule X is continuous.