A group $G$ is called a group with boundedly finite conjugacy classes (or a BFC-group) if $G$ is finite-by-abelian. A group $G$ satisfies the maximal condition on non-BFC-subgroups if every ascending chain of non-BFC-subgroups terminates in finitely many steps. In this paper the authors obtain the structure of finitely generated soluble-by-finite groups with the maximal condition on non-BFC subgroups.

AMS 2000 Mathematics subject classification: Primary 20E15. Secondary 20F16; 20F24

Given an integer $n$, we show that $\mathcal{I}_{n}$ embeds in a 2-generated subsemigroup of $\mathcal{I}_{n+2}$, which is an inverse semigroup. An immediate consequence of this result is the following, which is analogous to the case for groups and semigroups: every finite inverse semigroup may be embedded in a finite 2-generated semigroup which is an inverse semigroup.

AMS 2000 Mathematics subject classification: Primary 20M18. Secondary 20M20

This paper provides mathematical analysis of optical tomography in a situation when the examined object, for example the human brain, is strongly scattering with non-scattering inclusions. Light propagation in biological tissue is often modelled by the diffusion approximation of the radiative transfer equation. To be justified, the diffusion approximation demands that the medium is strongly scattering. Naturally, this is not true for non-scattering inclusions, for which some other model is needed. This is found through geometrical optics. Combination of the two models leads to an elliptic partial differential equation with boundary conditions on the outer boundary as well as on the boundaries of the non-scattering regions. The well-posedness of this forward problem is the main concern of this work.

AMS 2000 Mathematics subject classification: Primary 35Q60; 35R30

The cusp density of a hyperbolic 3-manifold is the ratio of the largest possible volume in a set of cusps with disjoint interiors to the volume in the manifold. It is known that all cusp densities fall in the interval $[0,0.853\dots]$. It is shown that the cusp densities of finite-volume orientable hyperbolic 3-manifolds are dense in this interval.

AMS 2000 Mathematics subject classification: Primary 57M50

We consider a parabolic matrix–vector system in which the diffusion matrix may be time dependent. For the time-independent case we construct approximate solutions with guaranteed error bounds using spectral information from certain matrix–vector Sturm–Liouville problems. For the time-dependent case we employ an approximation procedure which reduces the problem, on each time-step, to the time-independent case. We give an algorithm which may be used a priori at each time-step in the time-dependent case to guarantee accuracy to a specified tolerance.

AMS 2000 Mathematics subject classification: Primary 35C10; 35M10. Secondary 15A

For a Banach space $\mathfrak{X}$, let $\mathcal{B}(\mathfrak{X})$ denote the Banach algebra of all continuous linear operators on $\mathfrak{X}$. First, we study the lattice of closed ideals in $\mathcal{B}(\mathfrak{J}_p)$, where $1 \lt p \t \infty$ and $\mathfrak{J}_p$ is the $p$th James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{J}_p)$. Applications of this result include the following.

(i) The Brown–McCoy radical of $\mathcal{B}(\mathfrak{X})$, which by definition is the intersection of all maximal ideals in $\mathcal{B}(\mathfrak{X})$, cannot be turned into an operator ideal. This implies that there is no ‘Brown–McCoy’ analogue of Pietsch’s construction of the operator ideal of inessential operators from the Jacobson radical of $\mathcal{B}(\mathfrak{X})/\mathcal{A}(\mathfrak{X})$.

(ii) For each natural number $n$ and each $n$-tuple $(m_1,\dots,m_n)$ in $\{k^2\mid k\in\mathbb{N}\}\cup\{\infty\}$, there is a Banach space $\mathfrak{X}$ such that $\mathcal{B}(\mathfrak{X})$ has exactly $n$ maximal ideals, and these maximal ideals have codimensions $m_1,\dots,m_n$ in $\mathcal{B}(\mathfrak{X})$, respectively; the Banach space $\mathfrak{X}$ is a finite direct sum of James spaces and $\ell_p$-spaces.

Second, building on the work of Gowers and Maurey, we obtain further examples of Banach spaces $\mathfrak{X}$ such that all the maximal ideals in $\mathcal{B}(\mathfrak{X})$ can be classified. We show that the ideal of strictly singular operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{X})$ for each hereditarily indecomposable Banach space $\mathfrak{X}$, and we prove that there are $2^{2^{\aleph_0}}$ distinct maximal ideals in $\mathcal{B}(\mathfrak{G})$, where $\mathfrak{G}$ is the Banach space constructed by Gowers to solve Banach’s hyperplane problem.

AMS 2000 Mathematics subject classification: Primary 47D30; 47D50; 46H10; 16D30

Complete positivity of ‘atomically extensible’ bounded linear operators between $C^*$-algebras is characterized in terms of positivity of a bilinear form on certain finite-rank operators. In the case of an elementary operator on a $C^*$-algebra, the approach leads us to characterize k-positivity of the operator in terms of positivity of a quadratic form on a subset of the dual space of the algebra and in terms of a certain inequality involving factorial states of finite type I.

As an application we characterize those $C^*$-algebras where every k-positive elementary operator on the algebra is completely positive. They are either k-subhomogeneous or k-subhomogeneous by antiliminal. We also give a dual approach to the metric operator space introduced by Arveson.

AMS 2000 Mathematics subject classification: Primary 46L05. Secondary 47B47; 47B65

Let $E$ be a UMD Banach space, and $L$ a positive self-adjoint operator in $\mathrm{L}^2$ of Laplace type, for which the imaginary powers $L^{-\ri t}$ form a $C_0$-group of exponential growth $0\leq\alpha\lt \pi$ on $\mathrm{L}^p(E)$, where $1\lt p\lt\lt \infty$. Suppose $G(z)$ is holomorphic inside and on the boundary of the sector $\{z:z\neq0,\ |\arg z|\leq\phi\}$, and $z^\kappa G(z)\rightarrow0$ uniformly as $z\rightarrow\infty$ for some $\kappa\gt0$ and $\phi\gt\alpha$. Then $G(tL)$ $(t \gt0)$ defines a bounded family of linear operators on $\mathrm{L}^p(E)$; and the maximal operator $f\mapsto\sup_{t \gt 0}\|G(tL)f\|_E$ is bounded on the domain of $\log L$. The proof uses transference methods. These hypotheses hold for the maximal solution operators for the heat, wave and Schrödinger equations, and for Cesàro sums.

AMS 2000 Mathematics subject classification: Primary 47D03; 42B25; 47D09

In this short note we show that $\sup\{\|M_\nu\|:\nu\text{ is a measure on }\mathbb{R}^n\}$, where $\|M_\nu\|$ denotes the centred Hardy–Littlewood maximal operator, depends exponentially on $n$.

AMS 2000 Mathematics subject classification: Primary 42B25

Let $G$ be a countable discrete group and let $M$ be a proper free $C^r$ $G$-manifold and $N$ a $C^r$ $G$-manifold, where $1\leq r\leq\omega$. We prove that if $G$ acts properly and freely also on $N$ and if $\dim(N)\geq2\dim(M)$, then equivariant immersions form an open dense subset in the space $C^r_G(M,N)$ of all equivariant $C^r$ maps from $M$ to $N$. The space $C^r_G(M,N)$ is equipped with a topology, which coincides with the Whitney $C^r$ topology if $G$ is finite and is suited to studying equivariant maps. We also prove an equivariant version of Thom’s transversality theorem and show that $C^\omega_G(M,N)$ is dense in $C^r_G(M,N)$, for $1\leq r\leq\infty$.

AMS 2000 Mathematics subject classification: Primary 57S20

For Q the variance of some centred Gaussian random vector in a separable Banach space it is shown that, necessarily, Q factors through $\ell^2$ as a product of 2-summing operators. This factorization condition is sufficient when the Banach space is of Gaussian type 2. The stochastic integral of a deterministic family of operators with respect to a Q-Wiener process is shown to exist under a continuity condition involving the 2-summing norm. A Langevin equation

$$ \rd\bm{Z}_t+\sLa\bm{Z}_t\,\rd t=\rd\bm{B}_t, $$

with values in a separable Banach space, is studied. The operator $\sLa$ is closed and densely defined. A weak solution $(\bm{Z}_t,\bm{B}_t)$, where $\bm{Z}_t$ is centred, Gaussian and stationary, while $\bm{B}_t$ is a Q-Wiener process, is given when $\ri\sLa$ and $\ri\sLa^*$ generate $C_0$ groups and the resolvent of $\sLa$ is uniformly bounded on the imaginary axis. Both $\bm{Z}_t$ and $\bm{B}_t$ are stochastic integrals with respect to a spectral Q-Wiener process.

AMS 2000 Mathematics subject classification: Primary 60G15. Secondary 46E40; 47B10; 47D03; 60H10

A weak form of faithfulness, depending on Green’s equivalence $\mathcal{D}$, is introduced for a ring $R$ graded by a semigroup $S$. Suppose that $R$ satisfies this condition. It is shown that if $e$ and $f$ are $\mathcal{D}$-equivalent idempotents of $S$ and $R_e$ is semiprime (respectively, prime, semiprimitive, right primitive), then $R_f$ is semiprime (respectively, prime, semiprimitive, right primitive). In addition, it is shown that if $G$ and $H$ are maximal subgroups of $S$ lying in the same $\mathcal{D}$-class and $R_G$ is semiprime (respectively, prime, semiprimitive, right primitive), then $R_H$ is semiprime (respectively, prime, semiprimitive, right primitive).

AMS 2000 Mathematics subject classification: Primary 16W50. Secondary 20M25

The purpose of this paper is to derive an integral representation of the Drazin inverse of an element of a Banach algebra in a more general situation than previously obtained by the second author, and to give an application to the Moore–Penrose inverse in a $C^*$-algebra.

AMS 2000 Mathematics subject classification:Primary 46H05; 46L05

It is shown that, for every sequence $(f_n)$ of stochastically independent functions defined on $[0,1]$—of mean zero and variance one, uniformly bounded by $M$—if the series $\sum_{n=1}^\infty a_nf_n$ converges to some constant on a set of positive measure, then there are only finitely many non-null coefficients $a_n$, extending similar results by Stechkin and Ul’yanov on the Rademacher system. The best constant $C_M$ is computed such that for every such sequence $(f_n)$ any set of measure strictly less than $C_M$ is a set of uniqueness for $(f_n)$.

AMS 2000 Mathematics subject classification:Primary 42C25. Secondary 60G50

It is proved in this paper that for any non-elementary subgroup $G$ of $\mathrm{PSL}(2,\sGa_n)$, which has no elliptic element, to be not strict, there is a minimal generating system of $G$ consisting of loxodromic elements, and that if $G$ is a non-elementary subgroup of $\mathrm{PSL}(2,\sGa_n)$ of which each loxodromic element is hyperbolic, then $G$ is conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$.

AMS 2000 Mathematics subject classification: Primary 30F40. Secondary 20H10

We study a system of ordinary differential equations linked by parameters and subject to boundary conditions depending on parameters. We assume certain definiteness conditions on the coefficient functions and on the boundary conditions that yield, in the corresponding abstract setting, a right-definite case. We give results on location of the eigenvalues and oscillation of the eigenfunctions.

AMS 2000 Mathematics subject classification: Primary 34B08; 34B24

In this paper we consider soft group and crossed product $C^*$-algebras. In particular we show that soft crossed product $C^*$-algebras are isomorphic to classical crossed product $C^*$-algebras. We also prove that large classes of soft $C^*$-algebras have stable rank equal to infinity.

AMS 2000 Mathematics subject classification: Primary 46L80; 46L55

The uniform stability of the zero solution and the asymptotic behaviour of all solutions of the neutral delay differential equation

$$ [x(t)-P(t)x(t-\tau)]'+Q(t)x(t-\sigma)=0,\quad t\ge t_0, $$

are investigated, where $\tau,\sigma\in(0,\infty)$, $P\in C([t_0,\infty),\mathbb{R})$, and $Q\in C([t_0,\infty), [0,\infty))$. The obtained sufficient conditions improve the existing results in the literature.

AMS 2000 Mathematics subject classification: Primary 34K20; 34K15; 34K40

We present an alternative proof of the existence of density-conserving solutions to the discrete coagulation–fragmentation equations when the coagulation rates grow at most linearly. The proof relies on the study of the propagation of some moments of the solutions to approximating equations and simplifies the previous argument of Ball and Carr which involves rather delicate estimates. The case of multiple fragmentation is also considered, and the question of uniqueness as well.

AMS 2000 Mathematics subject classification: Primary 34A34; 82C22

As a consequence of the main result of the paper we obtain that every 2-local isometry of the $C^*$-algebra $B(H)$ of all bounded linear operators on a separable infinite-dimensional Hilbert space $H$ is an isometry. We have a similar statement concerning the isometries of any extension of the algebra of all compact operators by a separable commutative $C^*$-algebra. Therefore, on those $C^*$-algebras the isometries are completely determined by their local actions on the two-point subsets of the underlying algebras.

AMS 2000 Mathematics subject classification: Primary 47B49