Let A and C be m × m matrices and let B and D be n × n matrices, all with elements in a field F. Let AT denote the transpose of A. A well-known theorem states that, if every m × m matrix X for which AX = XA also satisfies CX = XC, then C = φ(A) for some polynomial φ(λ). In this note we establish the following simple generalizations.

Theorem 1. Let A and B have the same minimal polynomial m(λ). If each m × n matrix X over F for which AX = XB also satisfies CX = XD, then C = φ(A) and D = φ(B) for a polynomial φ(λ) over F.

It is well known that if α and β are commuting *-automorphisms of a von Neumann algebra M satisfying the equation α + α-1 = β + β-1 then M can be decomposed into a direct sum of subalgebras Mp and M(l − p) by a central projection p in M such that α = β on Mp and α = β-1 on M(1 − p) (see, for instance, [6], [7], [2]). Originally this equation arose in the Tomita-Takesaki theory (see, for example, [11]) in the form of one-parameter modular automorphism groups and later on it has been studied for arbitrary automorphisms and one-parameter groups of automorphisms on von Neumann algebras [7], [8], [9]. In the case of automorphism groups satisfying the above equation, one has a similar decomposition but this time without assuming the commutativity condition (cf. [7], [8]). For another relevant work on one-parameter groups of automorphisms which is close to our papers [7] and [8], we refer to Ciorănescu and Zsidó [1]. Regarding applications, this equation has been used for arbitrary automorphisms in the geometric interpretation of the Tomita-Takesaki theory [2] and in the case of automorphism groups it has been a fundamental tool in the generalization of the Tomita-Takesaki theory to Jordan algebras [3]. We may remark that the decomposition in the commuting case [6], [7] is much simpler than in the case of automorphism groups in the non-commutative situation [8]. In some cases one can obtain the decomposition for an arbitrary pair of automorphisms without assuming their commutativity but the problem in the general case has been unresolved. Recently we have shown that if α and β are *-automorphisms of a von Neumann algebra M satisfying the equation α + α-1 = β + β-1 (without assuming the commutativity of α and β) then there exists a central projection p in M such that α2= β2 on Mp and α2 = β−2 on M(l − p) [10].

The formulae to be proved are as follows.

If p ≧ q + 1, R(l + m) > 0, R(αr − l − m + n)> −1, R(αr − l − m − n)> 0, r = 1,2,…p,

If p ≧ q + 1, R(l >0, R(l+m >0, R(αr − l − m + n)> −1,

We study a class of rings which are closely related to principal ideal domains, and prove in particular that finitely-generated projective modules over such rings are free. Examples include the ring of Lipschitz quaternions; Z[a½] with d = —3 or d = —7; and any subring R of M2(Z) such that R ⊇ M2(pZ) for some prime number/? and R/M2(pZ) is a field with p2 elements.

The definition of a power-free group will be found in [1]. It is a partial algebraic system which, roughly speaking, may be thought of as a group in which (with the exception of the identity) squares and higher powers of an element are not defined.

It has been shown [1, Theorem 3.3] that the usual cancellation laws need not hold in a power-free group. When these laws do hold, the power-free group is called cancellative. In this paper we shall be solely concerned with cancellative power-free groups and the term ‘power-free group’ is to be understood to mean ‘cancellative power-free group’.

Let M be a compact connected boundaryless surface and f: M → ℝ3 a smooth immersion transverse to a straight line L. Thus there is an even number p of points xεM such that f(x)εL. Under further transversality assumptions on f (see §3) there is a finite number q of points x of M such that the plane containing f(x) and L touches f(M) at f(x). These assumptions are mild in the sense that they hold for any f in an open dense subset of the space of smooth immersions under consideration. Suppose that the Gaussian curvature of f(M) is positive at q+ of these points and negative at q−, with q = q++ q−. Then

where e(M) denotes the Euler number of M.

On the setting of the unit ball of the complex n-space, some characterizations of M-harmonic Bloch functions are obtained. As an application, Carleson measures are characterized by means of Berezin type integrals of M-harmonic Bloch functions. As one may expect, these results carry over to M-harmonic little Bloch functions and vanishing Carleson measures.

In [13], Poincaré asked the following question: Which abstract groups can appear as monodromy groups [14] of second order, linear, homogeneous differential equations with meromorphic coefficients (which might depend on one or more parameters) on ℂ? In the present paper, we initiate a classification of monodromy groups of differential equations on compact Riemann surfaces of genus 1. We proceed as follows: Let

be the general Euler equation [1] which has two regular singular points at 0 and λ in the extended complex plane C. Further, let yu(z)(v – 1,2) be an arbitrary but fixed pair of linearly independent solutions to (1) valid in a neighborhood of some ordinary point. Analytic continuation of each solution along a closed loop A in C–{0, λ}, starting and ending at some fixed base point, produces a new solution yv,A (v = 1, 2) which can be expressed as

where the constants avlx (v1 ν = l,2) in C depend on the homotopy class [A] of A. Clearly, y1,A(z)/y2,A(z) = To(y1(z)/y2(z)) where the Möbius transformation T:w→ (a11W + a12)/(a21W + a22) depends on [A]. The set of all Mobius transformations T belonging to every possible closed loop A in C–{0,λ} forms a group G, called the monodromy group of the Euler equation (1). G is generated by the Mobius transformation belonging to a simple, closed loop Ao encircling 0. Hence, G is cyclic.

Let S be a semigroup, and let ℓ1(S) denote the l1-semigroup algebra of S. Beginning with the fundamental paper of E. Hewitt and H. Zuckerman [5], there has been a considerable amount of research done concerning the Banach algebra ℓ1(S) in the case when S is abelian; see the bibliography [7]. However, until recently, there was very little information known concerning ℓ1(S) when S was nonabelian and infinite. Now for certain classes of infinite nonabelian semigroups with involution, recent progress has been made in the study of the Banach *-algebra ℓ1(S) and the *-representations of l1(S). In [2], B. Barnes and J. Duncan prove that ℓ1(S) is Jacobson semisimple, study the spectrum of elements in ℓ1(S), and construct and study *-representations of ℓ1(S) when S is the free semigroup with a finite or countably infinite set of generators (and also in some cases where the generators satisfy certain relations). In [1], the present author considered the representation theory of ℓ1(S) where S is an inverse semigroup. This paper is a sequel to [1].

Canonically associated to a real symplectic vector space are several associative algebras. The Weyl algebra (generated by the Heisenberg commutation relations) has been the subject of much study; see [1] for example. The exponential Weyl algebra (generated by the canonical commutation relations in exponential form) has been less well studied; see [8].

The Bass–Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of π1(L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass–Serre theorem as a special case and clarifies the roles played by the various generators and relations.

Recently, Levin and Saxon [5], De Wilde and Houet [2] defined the σ-barrelledness while Husain [3] defined the countable barrelledness and countable quasibarrelledness. It is well-known that barrelled spaces are countably barrelled, and countably barrelled spaces are σ-barrelled. It is natural to ask whether there is some condition for σ-barrelled (resp. countably barrelled) spaces to be countably barrelled (resp. barrelled). Using the concept of S-absorbent sequences of sets, we are able to give such conditions in Theorem 2.5 and Corollaries 2.6 and 2.7.

Let F be a field of characteristic p ≠ 2 and G a group without 2-elements having an involution ∗. Extend the involution linearly to the group ring FG, and let (FG)− denote the set of skew elements with respect to ∗. In this paper, we show that if G is finite and (FG)− is Lie metabelian, then G is nilpotent. Based on this result, we deduce that if G is torsion, p > 7 and (FG)− is Lie metabelian, then G must be abelian. Exceptions are constructed for smaller values of p.

The KO-cohomology ring of the symmetric space SU(2n)/SO(2n) is computed by using the Bott exact sequence and some facts on the real and quaternionic representation rings of SU(2n) and SO(2n).

Let $J_\mu$ denote the Bessel function of order $\mu$. The functions $x^{-\alpha-1} J_{\alpha+2n+1}(x), n=0,1,2,\ldots,$ form an orthogonal system in the space $L^2((0,\infty), x^{2\alpha+1}dx)$ when $\alpha >{-}1$. In this paper we prove that the Fourier series associated to this system is of restricted weak type for the endpoints of the interval of mean convergence, while it is not of weak type if $\alpha\,{\ge}\,0$.

In this paper, we study the homogenised algebra B of the enveloping algebra U of the Lie algebra sℓ(2,ℂ). We look first to connections between the category of graded left B-modules and the category of U-modules, then we prove B is Koszul and Artin–Schelter regular of global dimension four, hence its Yoneda algebra B! is self-injective of radical five zeros, and the structure of B! is given. We describe next the category of homogenised Verma modules, which correspond to the lifting to B of the usual Verma modules over U, and prove that such modules are Koszul of projective dimension two. It was proved in Martínez-Villa and Zacharia (Approximations with modules having linear resolutions, J. Algebra266(2) (2003), 671–697)] that all graded stable components of a self-injective Koszul algebra are of type ZA∞. Here, we characterise the graded B!-modules corresponding to the Koszul duality to homogenised Verma modules, and prove that these are located at the mouth of a regular component. In this way we obtain a family of components over a wild algebra indexed by ℂ.

H will denote a Hilbert space of infinite dimension, ℬ(H) the algebra of bounded linear operators on H, and ℛ(H) the ideal of compact operators on H. We let σ, σe and σω denote the spectrum, essential spectrum and Weyl spectrum respectively. It is well known that for arbitrary T ∈ ℬ(H) we have by [5]

and

and

A ring R is called right PCI if every proper cyclic right R-module is injective, i.e. if C is a cyclic right R-module then CR ≅ RR or CR is injective. By [2] and [3], if R is a non-artinian right PCI ring then R is a right hereditary right noetherian simple domain. Such a domain is called a right PCI domain. The existence of right PCI domains is guaranteed by an example given in [2]. As generalizations of right PCI rings, several classes of rings have been introduced and investigated, for example right CDPI rings, right CPOI rings (see [8], [6]). In Section 2 we define right PCS, right CPOS and right CPS rings and study the relationship between all these rings.

If a convergent series of real or complex numbers is rearranged, the resulting series may or may not converge. There are therefore two problems which naturally arise.

(i) What is the condition on a given series for every rearrangement to converge?

(ii) What is the condition on a given method of rearrangement for it to leave unaffected the convergence of every convergent series?

The answer to (i) is well known; by a famous theorem of Riemann, the series must be absolutely convergent. The solution of (ii) is perhaps not so familiar, although it has been given by various authors, including R. Rado [7], F. W. Levi [6] and R. P. Agnew [2]. It is also given as an exercise by N. Bourbaki ([4], Chap. III, § 4, exs. 7 and 8).

We show that the Fourier expansion in spherical $h$-harmonics (from Dunkl's theory) of a function $f$ on the sphere converges uniformly to $f$ if this function is sufficiently differentiable.