If ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

Following the analogy with group theory, we define the Wielandt subalgebra of a finite-dimensional Lie algebra to be the intersection of the normalisers of the subnormal subalgebras. In a non-zero algebra, this is a non-zero ideal if the ground field has characteristic 0 or if the derived algebra is nilpotent, allowing the definition of the Wielandt series. For a Lie algebra with nilpotent derived algebra, we obtain a bound for the derived length in terms of the Wielandt length and show this bound to be best possible. We also characterise the Lie algebras with nilpotent derived algebra and Wielandt length 2.

Let ℳ be a semi-finite von Neumann algebra equipped with a faithful normal trace τ. We prove a Kadec-Pelczyński type dichotomy principle for subspaces of symmetric space of measurable operators of Rademacher type 2. We study subspace structures of non-commutative Lorentz spaces Lp, q, (ℳ, τ), extending some results of Carothers and Dilworth to the non-commutative settings. In particular, we show that, under natural conditions on indices, ℓp cannot be embedded into Lp, q (ℳ, τ). As applications, we prove that for 0 < p < ∞ with p ≠ 2, ℓp cannot be strongly embedded into Lp(ℳ, τ). This provides a non-commutative extension of a result of Kalton for 0 < p < 1 and a result of Rosenthal for 1 ≦ p < 2 on Lp [0, 1].

Let X be a Banach space with the analytic UMD property, and let A and B be two commuting sectorial operators on X which admit bounded H∞ functional calculi with respect to angles θ1 and θ2 satisfying θ1 + θ2 > π. It was proved by Kalton and Weis that in this case, A + B is closed. The first result of this paper is that under the same conditions, A + B actually admits a bounded H∞ functional calculus. Our second result is that given a Banach space X and a number 1 ≦ p < ∞, the derivation operator on the vector valued Hardy space Hp (R; X) admits a bounded H∞ functional calculus if and only if X has the analytic UMD property. This is an ‘analytic’ version of the well-known characterization of UMD by the boundedness of the H∞ functional calculus of the derivation operator on vector valued Lp-spaces Lp (R; X) for 1 < p < ∞ (Dore-Venni, Hieber-Prüss, Prüss).

Generalizing and strengthening some well-known results of Higman, B. Neumann, Hanna Neumann and Dark on embeddings into two-generator groups, we introduce a construction of subnormal verbal embedding of an arbitrary (soluble, fully ordered or torsion free) ordered countable group into a twogenerator ordered group with these properties. Further, we establish subnormal verbal embedding of defect two of an arbitrary (soluble, fully ordered or torsion free) ordered group G into a group with these properties and of the same cardinality as G, and show in connection with a problem of Heineken that the defect of such an embedding cannot be made smaller, that is, such verbal embeddings of ordered groups cannot in general be normal.

Consider the quasi-variety generated by a finite algebra and assume that yields a natural duality on based on which is optimal modulo endomorphisms. We shoe that, provided satisfies certain minimality conditions, we can transfer this duality to a natural duality on based on , which is also optimal modulo endormorphisms, for any finite algebra in that has a subalgebra isomorphic to .

Let M be a commutative cancellative atomic monoid. We consider the behaviour of the asymptotic length functions and on M. If M is finitely generated and reduced, then we present an algorithm for the computation of both and where x is a nonidentity element of M. We also explore the values that the functions and can attain when M is a Krull monoid with torsion divisor class group, and extend a well-known result of Zaks and Skula by showing how these values can be used to characterize when M is half-factorial.