1. Every family of subnormal operators in a Hilbert space fulfils the Halmos-Bram condition on a suitable dense subset of its domain [2], [3]. In [2] and [4] it is shown that the generalized commutation relation implies the Halmos-Bram condition for one operator. In this paper it is proved that the generalized commutation relation implies the Halmos-Bram condition for infinite families of operators (in a special case Jorgensen proved it in a different way for finite families of operators, see [2]) and as an example of the application of this property it is shown that every family of generalized creation operators in the Bargmann space of an infinite order, indexed by mutually orthogonal vectors from I2 is subnormal. See [1] for the definitions.

In this note we give the proof of the following result (previously known for homotopically trivial and free actions on infranilmanifolds [3, Theorem 5.6]).

Theorem 1. Let G be a finite group acting freely and smoothly on a closed infranilmanifold M. Assume that dim M≠3, 4. Then the action of G is topologically conjugate to an affine action.

Extending a result of Boyle and Goodearl in [1] on V-rings it was shown in Yousif [11] that a generalized V-module (GV-module) has Krull dimension if and only if it is noetherian. Our note is based on the observation that every GV-module has a maximal submodule (Lemma 1). Applying a theorem of Shock [6] we immediately obtain that a GV-module has acc on essential submodules if and only if for every essential submodule K ⊂ M the factor module M/K has finitely generated socle. Yousif's result is obtained as a corollary.

Let R be an associative ring with unity and R-Mod the category of unital left R-modules. Soc M denotes the socle of an R-module M. If K ⊂ M is an essential submodule we write K⊴M.

An R-module M is called co-semisimple or a V-module, if every simple module is M-injective ([2], [7], [9], [10]). According to Hirano [3] M is a generalized V-module or GV-module, if every singular simple R-module is M-injective. This extends the notion of a left GV-ring in Ramamurthi-Rangaswamy [5].

It is easy to see that submodules, factor modules and direct sums of co-semisimple modules (GV-modules) are again co-semisimple (GV-modules) (e.g. [10, § 23]).

Let G be a compact abelian group and let Γ be its (discrete) dual group. Denote by M(G) the space of complex regular Borel measures on G.

Let E be a subset of Γ. Then:

(i) E is called a Rajchman set if, for all μ ∈M(G) implies

(ii) E is called a set of continuity if given ε > 0 there exists δ > 0 such that if and

(iii) E is called a parallelepiped of dimension N if |E| = 2N and there are two-element sets . (The multiplication indicated here is the group operation.)

Let R be a commutative, semi-local ring. Let On,n be the group of linear automorphisms of R2n which preserve the bilinear form . The main result of this paper is the following theorem.

Theorem A. The natural inclusion of Onn into On,n into On+1,n+1induces an isomorphism on the ith homology group if only n is large enough with respect to i.

We provide corrected versions of Theorems 1 and 2 and Corollaries 3(a) and 4 of the paper mentioned in the title.