Richard J. Loy

Amenability and weak amenability of second conjugate Banach algebras

For a Banach algebra A, amenability of A∗∗ necessitates amenability of A, and similarly for weak amenability provided A is a left ideal in A∗∗. For G a locally compact group, indeed more generally, L1(G)∗∗ is amenable if and only if G is finite. If L1(G)∗∗ is weakly amenable, then M(G) is weakly amenable. 0. Introduction For a Banach algebra A,A∗∗ is a Banach algebra under two Arens products, of which we will always take the first, or left, product. For further details see the survey article [8]. This product can be characterized as the extension to A∗∗×A∗∗ of the bilinear map A×A→ A : (x, y) 7→ xy with the following continuity properties: for fixed y ∈ A∗∗, x 7→ xy is weak*-continuous on A∗∗; for fixed y ∈ A, x 7→ yx is weak*-continuous on A∗∗. Here, as elsewhere, we identify A with its canonical image in A∗∗. In terms of the asymmetry here, define the topological centre of A∗∗ by Zt(A ∗∗) = {y ∈ A∗∗ : x 7→ yx is weak∗-continuous}. Clearly, Zt(A ∗∗) contains the (algebraic) centre Z(A∗∗) of A∗∗; it also contains A. In the case that Zt(A ∗∗) = A∗∗, A∗∗ is said to be Arens regular . Any C∗-algebra is Arens regular [6, Theorem 7.1], but for a locally compact group G, L(G) is Arens regular if and only if G is finite [25]. A Banach algebra A is amenable if every derivation D : A → X∗ is inner, for every Banach A-bimodule X . If one only considers the bimodule X = A, one has the notion of weak amenability. There are many alternative formulations of the notion of amenability, of which we need the following two, first introduced in [18]. For further details see [3, 16, 7]. The Banach algebra A is amenable if and only if either, and hence both, of the following hold: (i) A has an approximate diagonal , that is, a bounded net (mi) ⊂ A⊗A such that for each x ∈ A, mix− xmi → 0, π(mi)x→ x; (ii) A has a virtual diagonal , that is, an element M ∈ (A⊗A)∗∗ such that for each x ∈ A, xM = Mx, (π∗∗M)x = x. Received by the editors June 27, 1994 and, in revised form, October 19, 1994. 1991 Mathematics Subject Classification. Primary 46H20; Secondary 43A20.

Centroid sampling: a variant of importance sampling for estimating the volume of sample trees of radiata pine.

Simulation of importance sampling using a database comprising detailed measurements along the bole of 114 trees of radiata pine (Pinus radiata D. Don) revealed a point on the bole at a relative height of approximately 0.3 which, if selected as the point of importance sampling, gave minimum bias in the estimate of tree volume. A theoretical examination of solids of revolution identified this point as the centroid of the tree. A variant of importance sampling, called centroid sampling, in which the point of sampling is fixed at the height of the centroid rather than being determined at random, was then applied in conjunction with 3P sampling to estimate the volume of a forest plot of 55 trees of 36-year-old radiata pine in Stromlo Forest, Australian Capital Territory. The resulting estimate of stand volume was both precise and accurate.

Rheological implications of completely monotone fading memory

In the constitutive equation modeling of a (linear) viscoelastic material, the “fading memory” of the relaxation modulus G(t) is a fundamental concept that dates back to Boltzmann [Ann. Phys. Chem. 7, 624 (1876)]. There have been various proposals that range from the experimental and pragmatic to the theoretical about how fading memory should be defined. However, if, as is common in the rheological literature, one assumes that G(t) has the following relaxation spectrum representation: G(t)=∫0∞ exp(−t/τ)[H(τ)/τ]dτ, t > 0, then it follows automatically that G(t) is a completely monotone function. Such functions have quite deep mathematical properties, that, in a rheological context, spawn interesting and novel implications. For example, because the set of completely monotone functions is closed under positive linear combinations and products, it follows that the dynamics of a linear viscoelastic material, under appropriate stress–strain stimuli, will involve a simultaneous mixture of different molecular int...

COMPLETELY MONOTONE FADING MEMORY RELAXATION MODULII

J—oo which defines how the stress a(t) at time t depends on the earlier history of the shear rate j(r) via the relaxation modulus (kernel) G(t). Physical reality is achieved by requiring that the form of the relaxation modulus G(t) gives the Boltzmann equation fading memory, so that changes in the distant past have less effect now than the same changes in the more recent past. A popular choice, though others have previously been proposed and investigated, is the assumption that G(t) be a completely monotone function. This assumption has much deeper ramifications than have been identified, discussed or exploited in the rheological literature. The purpose of this paper is to review the key mathematical properties of completely monotone functions, and to illustrate how these properties impact on the theory and application of linear viscoelasticity and polymer dynamics. A more general representation of a completely monotone function, known in the mathematical literature, but not the rheological, is formulated and discussed. This representation is used to derive new rheological relationships. In particular, explicit inversion formulas are derived for the relationships that are obtained when the relaxation spectrum model and a mixing rule are linked through a common relaxation modulus.

Banach algebras of power series

Let C [[ t ]] denote the algebra of all formal power series over the complex field C in a commutative indeterminate t with the weak topology determined by the projections p j : Σα i t i ↦α j . A subalgebra A of C [[ t ]] is a Banach algebra of power series if it contains the polynomials and is a Banach algebra under a norm such that the inclusion map A ⊂ C [[ t ]] is continuous. Such algebras were first introduced in [13] when considering algebras with one generator, and studied, in a special case, in [23]. For a partial bibliography of their subsequent study and application see the references of [9] (note that the usage of the term Banach algebra of power series in [9] differs from that here), and also [2], [3], [11]. Indeed, an examination of their use in [11], under more general topological conditions than here, led the present author to the results of [14], [15], [16], [17].

Continuity of linear operators commuting with shifts

Abstract It is shown that linear operators on a wide class of function and measure spaces on [0, ∞) which commute with the right shift operators are necessarily continuous. Indeed, commutativity with a single nonidentity shift is often sufficient, and the results are valid for more general cones. As a consequence characterizations of perfect operators and other multipliers are obtained, and applied to prove an extension of a result of Diamond for derivations on measure algebras.

On the scaling of molecular weight distribution functionals

When formulating a constitutive equation model or a mixing rule for some synthetic or biological polymer, one is essentially solving an inverse problem. However, the data will not only include the results obtained from simple step strain, oscillatory shear, elongational, and other experiments, but also information about the molecular weight scaling of key rheological parameters (i.e., molecular weight distribution functionals) such as zero-shear viscosity, steady-state compliance, and the normal stress differences. In terms of incorporating such scaling information into the formulation of models, there is a need to understand the relationship between various models and their molecular weight scaling, since such information identifies the ways in which molecular weight scaling constrains the choice of possible models. In Anderssen and Mead (1998) it was established formally that the members of a quite general class of reptation mixing rules all had the same molecular weight scaling. The purpose of this pap...

Note on amenable algebras of operators

It is shown that an amenable algebra of operators on Hibert space which is generated by its normal elements is necessarily self-adjoint, so it is a C*-algebra.

Interconversion of Prony series for relaxation and creep

Various algorithms have been proposed to solve the interconversion equation of linear viscoelasticity when Prony series are used for the relaxation and creep moduli, G(t) and J(t). With respect to a Prony series for G(t), the key step in recovering the corresponding Prony series for J(t) is the determination of the coefficients {jk} of terms in J(t). Here, the need to solve a poorly conditioned matrix equation for the {jk} is circumvented by deriving elementary and easily evaluated analytic formulae for the {jk} in terms of the derivative dG(s)/ds of the Laplace transform G(s) of G(t).

A duality proof of sampling localisation in relaxation spectrum recovery

In a recent paper, Davies and Anderssen (1997) examined the range of relaxation times, on which the linear viscoelasticity relaxation spectrum could be reconstructed, when the oscillatory shear data were only known on a fixed finite interval of frequencies. In particular, they showed that, for such oscillatory shear data, knowledge about the relaxation spectrum could only be recovered on a specific finite interval of relaxation times. They referred to this phenomenon as sampling localisation. The purpose of this note is show how their result can be proved using a duality argument, and, thereby, establish the fundamental nature of sampling localisation in relaxation spectrum recovery