Let M be a real hypersurface of the complex projective space Pn(C). The Ricci tensor S of M is recurrent if there exists a 1-form $\alpha$ such that $\nablaS=S\otimes\alpha;$. In this paper we show that there are no real hypersurfaces with recurrent Ricci tensor of Pn(C) under the condition that $\xi$ is a principal curvature vector.

1991 Mathematics Subject Classification 53C40 (53C25).

In this paper G denotes a non-identity finite soluble group. If A is an irreducible G-module, EndGA is a division ring by Schur's Lemma, actually a field, since G finite forces A to be finite. Moreover A is a vector space over EndGA with respect to $\alphaa:=\alpha(a),\alpha\in\rm{End}_GA,a\inA$. We let $\varphi_G(A):=\rm{dim}_{\rm{End}_GA}A$. Any chief factor of G is an irreducible G-module via the conjugation action, and it is central precisely when it is a trivial G-module. By a refined version of the Theorem of Jordan-Hölder [1, p. 33] the number $\delta_G(A)$ of complemented chief factors of G, which are G-isomorphic to a given A, is constant for any chief series of G. We say that A is complemented, as aG-module, if $\delta_G(A)>0$.

It is known that the class of mils generalizes that of pramarts and martingales in the limit. Also every Banach space-valued mil (Xn) with lim infnE(‖Xn‖)<∞ can be written in a unique form: $X_n=M_n+P_n(n\in\rm{N})$, where $(M_n)$ is a uniformly integrable martingale and $(P_n)$ converges to zero a.s. in norm. We shall show that this result still holds for a class which essentially generalizes that of mils. Another class of Banach space-valued martingale-like sequences, still containing all pramarts is defined and shown to have the decomposition above under the following much weaker condition: $\rm{lim inf}_{r\inT}E(\VertX_{\tau}\Vert)<\infty$, where T denotes the set of all bounded stopping times.

1991 Mathematics Subject Classification. 60G48, 60B11.

Let the locally finite group G be the product of two locally nilpotent subgroups A and B, and assume that H is a subgroup of G belonging to a group classℱ . The question is considered whether there exists a subgroup X of G containing H which belongs to ℱ and satisfies $X=(X\capA)(X\capB)$. Under various assumptions on G and ℱ, necessary and sufficient conditions for the existence of such a subgroup X are obtained.

Let A be a UHF-algebra and K the C*-algebra of all compact operators on a countably infinite-dimensional Hilbert space. In this note we shall find all projections p in A with $pAp\congA$ and, using these projections, we shall determine the group of automorphisms of $K_0(A\otimes$K) induced by those of $A\otimes$K in some cases.

1991 Mathematics Subject Classification 46 L40.

The ‘canonical embedding approach’ was introduced by the second author and, subsequently, it has been applied several times to prove the embeddability of certain regular extensions by groups into semidirect products by groups. In the present paper this technique is generalized so that it is suitable to handle regular extensions by inverse semigroups. As an application, B. Billhardt's embedding theorem on regular extensions of semilattices by inverse semigroups is reproved.

In this paper we give a characterization of the central elements of the algebra A** for a class of weakly sequentially complete Banach algebras A and present a detailed study of several questions related to Arens regularity of the algebra A.

1991 Mathematics Subject Classification Primary: 43A10, 46B20, 46J99.

We denote by $\rm{Aut}_{sn}(G)$ the set of all automorphisms that fix every subnormal subgroup of $G$ setwise. In their paper [5], Franciosi and de Giovanni began the study of $\rm{Aut}_{sn}(G)$. Other authors have also considered the structure of $\rm{Aut}_{sn}(G)$ under various restrictions on the structure of $G$ (Robinson [11], Cossey [2], Dalle Molle [4]). The inner automorphisms in $\rm{Aut}_{sn}(G)$ are precisely the inner automorphisms induced by elements of $\omega(G)$, the Wielandt subgroup of $G$. Recall that the Wielandt subgroup of a group $G$ is the set of all elements of $G$ that normalise each subnormal subgroup of $G$ and that $\zeta(G)$, the centre of $G$, is contained in $\omega;(G)$. Thus $\rm{Aut}_{sn}(G)\cap;\rm{Inn}(G)$ is isomorphic to $\omega(G)/\zeta(G)$ and some of the results obtained indicate that the structure of $\rm{Aut}_{sn}(G)$ is controlled by the structure of $\omega(G)/\zeta(G)$; for example, Robinson [11, Corollary 3] shows that, for a finite group $G,\rm{Aut}_{sn}(G)$ is insoluble if and only if $\omega(G)$ is insoluble. We shall prove a result of a similar nature here. One of the main results (Theorem B) of Franciosi and de Giovanni [5] is that, for a polycyclic group $G$, $\rm{Aut}_{sn}(G)$ is either finite or abelian. We shall show that $\rm{Aut}_{sn}(G)$ can indeed be infinite, but only if $\omega(G)/\zeta(G)$ is infinite.

We prove an $n$-tuple analogue of the Putnam area inequality for the spectrum of a single $p$-hyponormal operator.

By a quasi-permutation matrix we mean a square matrix over the complex field [Copf ] with non-negative integral trace. Thus every permutation matrix over [Copf ] is a quasi-permutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational field ℚ, and let c(G) be the minimal degree of a faithful representation of G by complex quasi-permutation matrices. See [1].

We give the first examples of contact metric spaces which are weakly locally $\varphi$-symmetric (that is, $\nabla R=0$ on horizontal vectors), but not strongly (that is, not all reflections with respect to the characteristic lines are local isometries). These examples are three-dimensional non-unimodular Lie groups with a left-invariant contact metric structure. We exhibit additional symmetries on these spaces.

1991 Mathematics Subject Classification: 53B20, 53C15, 53C25.

The nonabelian tensor square G[otimes ] G of a group G is generated by the symbols g[otimes ] h, g,h ∈ G, subject to the relations $$gg\prime\otimesh=(^gg\prime\otimes^gh)(g\otimesh) and g\otimeshh\prime-(g\otimesh)(^hg\otimes^hh\prime),$$ for all $g,g\prime,h,h\prime \in G< / f>, where $^gg\prime=gg\primeg^{−1}$. The nonabelian tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J.-L. Loday in [4] and [5], extending ideas of J.H.C. Whitehead in [10]. The topic of this paper is the classification of 2-generator 2-groups of class two up to isomorphism and the determination of nonabelian tensor squares for these groups.

This paper gives examples to show that a polar operator is not necessarily AC and an AC operator is not necessarily polar.

Let $X$ be a Banach space and let $B(X)$ denote the space of bounded operators on $X$. Two elements $S,T\inB(X)$ are isometrically equivalent if there exists an invertible isometry $V$ such that $TV=VS$. If $X$ is a Hilbert space, then $V$ is a unitary operator and $S$ and $T$ are said to be unitarily equivalent.

Denote by $(R,\cdot)$ the multiplicative semigroup of an associative algebra $R$ over an infinite field, and let $(R,\circ)$ represent $R$ when viewed as a semigroup via the circle operation $x\circy=x+y+xy$. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of $R$. Namely, we prove that the following conditions on $R$ are equivalent: the semigroup $(R,\circ)$ satisfies an identity; the semigroup $(R,\cdot)$ satisfies a reduced identity; and, the associated Lie algebra of $R$ satisfies the Engel condition. When $R$ is finitely generated these conditions are each equivalent to $R$ being upper Lie nilpotent.

1991 Mathematics Subject Classification 16R40, 20M07, 20M25

An operator $T\in$[Lscr ]$(H)$ is called a square root of a hyponormal operator if $T^2$ is hyponormal. In this paper, we prove the following results: Let $S$ and $T$ be square roots of hyponormal operators.

(1) If $\sigma(T)\cap[-\sigma(T)]=\phi$ or {0}, then $T$ is isoloid (i.e., every isolated point of $\sigma(T)$ is an eigenvalue of $T$).

(2) If $S$ and $T$ commute, then $ST$ is Weyl if and only if $S$ and $T$ are both Weyl.

(3) If $\sigma(T)\cap[-\sigma(T)]=\phi$ or {0}, then Weyl's theorem holds for $T$.

(4) If $\sigma(T)\cap[-\sigma(T)]=\phi$, then $T$ is subscalar. As a corollary, we get that $T$ has a nontrivial invariant subspace if $\sigma(T)$ has non-empty interior. (See [3].)