Abstract. We consider the structure of the Kac modules V(Λ) for dominant integral doubly atypical weights Λ of the Lie superalgebra s1(2/2). Primitive vectors of V(Λ) are constructed and it is shown that the number of composition factors of V(Λ) for such Λ is in exact agreement with the conjectures of [HKV]. These results are used to show that the extended Kac-Weyl character formula which was proved in [VHKTl] for singly atypical simple modules of s1(m/n), and conjectured to be valid for all finite dimensional irreducible representations of sl(m/n) in [VHKT2] is in fact valid for all finite-dimensional doubly atypical simple modules of s1(2/2).

Let K ⊂ Rd be a convex body and choose points xl, x2, …, xn randomly, independently, and uniformly from K. Then Kn = conv {x1, …, xn} is a random polytope that approximates K (as n → ∞) with high probability. Answering a question of Rolf Schneider we determine, up to first order precision, the expectation of vol K – vol Kn when K is a smooth convex body. Moreover, this result is extended to quermassintegrals (instead of volume).

Abstract. Sufficient conditions are derived for all bounded solutions of general classes of integrodifferential equations of arbitrary order with variable coefficients to be either oscillatory or convergent to zero asymptotically.

Abstract. For F a field we compute, explicitly and directly, the right Krull dimension of the algebra Qop⊗FQ for certain semisimple Artinian F-algebras Q. (Here Qop denotes the opposite ring of Q.) We use our calculation to give alternative proofs of some theorems of J. T. Stafford and A. I. Lichtman. Our methods involve a detailed study of skew polynomial rings.

Abstract. Let Φ be in the disc algebra H∞ ∩ C(T) with its restriction to T in the Zygmund space of smooth functions λ*(T). If P(Φ') ⊂ T is the set of Plessner points of Φ' and if F = Φ + Ψ, where Ψ∈C1(T), it is shown that F(P(Φ')) ⊆ C is a set of zero linear Hausdorff measure. Applications are given to the spectral theory of multiplication operators.

Let X be a complex Banach space and H be a hermitian operator on X. Then in [7] Sinclair proved that r(H) = ¶H¶, where r(H) and ¶H¶ are the spectral radius and the operator norm of H, respectively.

For a commuting n-tuple T = (T1,…, Tn) of operators on X, we denote the (Taylor) joint spectrum of T by σ(T) (see [9]) and define the joint operator norm ¶T¶ and the joint spectral radius r(T) by

and

respectively.

A description of regular group rings is well known (see [12]). Various authors have considered regular semigroup rings (see [17], [8], [10], [11], [4]). These rings have been characterized for many important classes of semigroups, although the general problem turns out to be rather difficult and still has not got a complete solution. It seems natural to describe the regular radical in semigroup rings for semigroups of the classes mentioned. In [10], the regular semigroup rings of commutative semigroups were described. The aim of the present paper is to characterize the regular radical ρ(R[S]) for each associative ring R and commutative semigroup S.

A locally convex space (E, ) has the Mazur Property if and only if every linear -sequential continuous functional is -continuous (see [11]).

In the Banach space setting, a Banach space X is a Mazur space if and only if the dual space X* endowed with the w*-topology has the Mazur property. The Mazur property was introduced by S. Mazur, and, for Banach spaces, it is investigated in detail in [4], where relations with other properties and applications to measure theory are listed. T. Kappeler obtained (see [8]) certain results for the injective tensor product and showed that L1(μ, X), the space of Bochner-integrable functions over a finite and positive measure space (S, σ, μ), is a Mazur space provided X is also, and ℓ1 does not embed in X.

By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

Let E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:

(i) biprojectivity of N(E);

(ii) biflatness of N(E);

(iii) homological finite-dimensionality of N(E);

(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).

In recent years a new approach to the study of compact symmetric spaces has been taken by Nagano and Chen [10]. This approach assigned to each pair of antipodal points on a closed geodesic a pair of totally geodesic submanifolds. In this paper we will show how these totally geodesic submanifolds can be used in conjunction with a theorem of Bott to compute homotopy in compact symmetric spaces. Some of the results are already known (see [1], [5], [11] for example) but we include them here for completeness and to illustrate this unified approach. We also exhibit a connection between the second homotopy group of a compact symmetric space and the multiplicity of the highest root. Using this in conjunction with a theorem of J. H. Cheng [6] we obtain a topological characterization of quaternionic symmetric spaces with antiquaternionic involutive isometry. The author would like to thank Prof T. Nagano for all his help and his detailed descriptions of the totally geodesic submanifolds mentioned above.

For integers a, N and H > 0 write

where ϰ denotes a non-principal Dirichlet character modulo the positive integer k and e(y) denotes e2πiy. By a well-known generalisation of the Póya–Vinogradov inequality

Throughout the paper n denotes a fixed positive integer unless otherwise specified. Let B = Bn denote the open unit ball of ℂn and let S = Sn denote its boundary, the unit sphere. The unique rotation-invariant probability measure on 5 will be denoted by σ = σn. For n = l, we use more customary notations D = B1, T = S1 and dσ1= dθ/2π. The Hardy space on B, denoted by H2(B), is then the space of functions f holomorphic on B for which

Let the group G = AB be the product of two subgroups A and B. A normal subgroup K of G is said to be factorized if K = (A ∩ K)(B ∩ K) and A ∩ B ≤ K, and this is well-known to be equivalent to the fact that K = AK ∩ BK (see [1]). Easy examples show that normal subgroups of a product of two groups need not, in general, be factorized. Therefore the determination of certain special factorized subgroups is of relevant interest in the investigation concerning the structure of a factorized group. In this direction E. Pennington [5] proved that the Fitting subgroup of a finite product of two nilpotent groups is factorized. This result was extended to infinite groups by B. Amberg and theauthors, who provedin [2] that if the soluble group G = AB with finite abelian section rank isthe product of two locally nilpotent subgroups A and B, then the Hirsch-Plotkin radical (i.e. the maximum locally nilpotent normal subgroup) of G is factorized. If G is a soluble ℒI group and the factors A and B are nilpotent, it was shown in [3] that also the Fitting subgroup of G is factorized. However, Pennington's theorem becomes false for finite soluble groups which are the productof two arbitrary subgroups. For instance, the symmetric group of degree 4 is the product of a subgroup isomorphic with the symmetric group of degree 3 and a cyclic subgroup of order 4, but its Fitting subgroup is not factorized.

The present note deals with bounded endomorphisms of free p-algebras (pseudocomplemented lattices). The idea of bounded homomorphisms was introduced by R. McKenzie in [8]. T. Katriňák [5] subsequently studied the properties of bounded homomorphisms for the varieties of p-algebras. This concept is also an efficient tool for the characterization of, so-called, splitting as well as projective algebras in the varieties of all lattices or p-algebras. For details the reader is referred to [2], [5], [6], [7] and other references therein. Let us emphasize that the main results that are contained in the above mentioned references strongly depend on the boundedness of each endomorphism of any finitely generated free algebra in a given variety.

Let A be a commutative algebra, and let M be a bimodule over A. A derivation from A into M is a linear mapping D: A→M that satisfies

If M is only a left A-module, by a derivation from A into M we mean a linear mapping D: A→M such that

Each A-bimodule M is trivially a left module. However, unless it is commutative, i.e.

the two classes of linear operators from A into M characterized by (1) and (2), respectively, need not coincide.

The object of this paper is to consider two easy propositions concerning bounded approximate identities and show that they do not extend to unbounded approximate identities. The propositions are as follows.

Proposition 1.1. Every bounded left approximate identity in a normed algebra is a left approximate identity for the completion.

Proposition 1.2. Every bounded left approximate identity in a separable normed algebra has a subsequence which is a left approximate identity.

In [6], Blyth and Varlet characterize those algebras having only principal congruences in some well known classes of algebras having distributive lattice reducts. In particular, they characterize those Stone algebras having only principal congruences. In this paper we characterize those quasi-modular p-algebras having only principal congruences and show on specializing that distributive p-algebras having only principal congruences can be described in exactly the same way as Blyth and Varlet described Stone algebras having the same property. The same problem is addressed for some distributive double p-algebras.