Conditions implying the uniqueness of the weak*-topology on certain group algebras

We investigate possible preduals of the measure algebra M(G) of a locally compact group and the Fourier algebra A(G) of a separable compact group. Both of these algebras are canonically dual spaces and the canonical preduals make the multiplication separately weak*-continuous so that these algebras are dual Banach algebras. In this paper we find additional conditions under which the preduals C0(G) of M(G) and C*(G) of A(G) are uniquely determined. In both cases we consider a natural comultiplication and show that the canonical predual gives rise to the unique weak*-topology making both the multiplication separately weak*-continuous and the comultiplication weak*-continuous. In particular, dual cohomological properties of these algebras are well defined with this additional structure

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C 0(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak*-continuous. Given a discrete semigroup S, the convolution algebra ℓ 1(S) also carries a coproduct. In this paper we examine preduals for ℓ 1(S) making both the product and the coproduct weak*-continuous. Under certain conditions on S, we show that ℓ 1(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ 1(S) when S is either ℤ+×ℤ or (ℕ,⋅)

Quantum Eberlein compactifications and invariant means

We propose a definition of a "$C^*$-Eberlein" algebra, which is a weak form of a $C^*$-bialgebra with a sort of "unitary generator". Our definition is motivated to ensure that commutative examples arise exactly from semigroups of contractions on a Hilbert space, as extensively studied recently by Spronk and Stokke. The terminology arises as the Eberlein algebra, the uniform closure of the Fourier-Stieltjes algebra $B(G)$, has character space $G^{\mathcal E}$, which is the semigroup compactification given by considering all semigroups of contractions on a Hilbert space which contain a dense homomorphic image of $G$. We carry out a similar construction for locally compact quantum groups, leading to a maximal $C^*$-Eberlein compactification. We show that $C^*$-Eberlein algebras always admit invariant means, and we apply this to prove various "splitting" results, showing how the $C^*$-Eberlein compactification splits as the quantum Bohr compactification and elements which are annihilated by the mean. This holds for matrix coefficients, but for Kac algebras, we show it also holds at the algebra level, generalising (in a semigroup-free way) results of Godement

The approximation property implies that convolvers are pseudo-measures

This paper (not for formal publication) grew out of the authors' attempts to understand Cowling's argument that for a locally compact group $G$ with the approximation property, we have that $PM_p(G)=CV_p(G)$ ("all convolvers are pseudo-measures".) We have ended up giving a somewhat self-contained survey of Cowling's construction of a predual for $CV_p(G)$, together with a survey of old ideas of Herz relating to Herz-Schur multipliers. Thus none of the results are new, but we make some claim to originality of presentation. We hope this account may help other researchers, and in particular, that this might spur others to study this problem

Shift invariant preduals of ℓ₁(ℤ)

The Banach space ℓ<sub>1</sub>(ℤ) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ℓ<sub>1</sub>(ℤ) weak<sup>*</sup>-continuous. This is equivalent to making the natural convolution multiplication on ℓ<sub>1</sub>(ℤ) separately weak*-continuous and so turning ℓ<sub>1</sub>(ℤ) into a dual Banach algebra. We call such preduals <i>shift-invariant</i>. It is known that the only shift-invariant predual arising from the standard duality between C<sub>0</sub>(K) (for countable locally compact K) and ℓ<sub>1</sub>(ℤ) is c<sub>0</sub>(ℤ). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak<sup>*</sup>-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c<sub>0</sub>. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of ℤ. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c<sub>0</sub>

Canary in the coal mine: Lessons from the Jarrah Forest suggest long-term negative effects of phosphorus fertilizer on biodiverse restoration after surface mining

Despite nutrient enrichment having widely reported negative impacts on biodiversity, fertilizer is routinely applied in post mining restoration to enhance plant growth and establishment. Focusing on surface mine restoration (predominately bauxite and mineral sands), we outline the long-term negative impacts of fertilizer, particularly phosphorus fertilizer, on plant community composition, species richness, fire fuel loads, and belowground impacts on nutrient-cycling. We draw from extensive research in south-western Australia and further afield, noting the geographical coincidence of surface mining, phosphorus impoverished soil and high plant biodiversity. We highlight the trade-offs between rapid plant-growth under fertilisation and the longer-term effects on plant communities and diversity. We note that the initial growth benefits of fertilisation may not persist in water-limited environments: growth of unfertilised forests can eventually match that of fertilised forest, throwing doubt on the premise that fertilisation is necessary at all

Parametric LTL on Markov Chains

This paper is concerned with the verification of finite Markov chains against parametrized LTL (pLTL) formulas. In pLTL, the until-modality is equipped with a bound that contains variables; e.g., $\Diamond_{\le x}\ \varphi$ asserts that $\varphi$ holds within $x$ time steps, where $x$ is a variable on natural numbers. The central problem studied in this paper is to determine the set of parameter valuations $V_{\prec p} (\varphi)$ for which the probability to satisfy pLTL-formula $\varphi$ in a Markov chain meets a given threshold $\prec p$, where $\prec$ is a comparison on reals and $p$ a probability. As for pLTL determining the emptiness of $V_{> 0}(\varphi)$ is undecidable, we consider several logic fragments. We consider parametric reachability properties, a sub-logic of pLTL restricted to next and $\Diamond_{\le x}$, parametric B\"uchi properties and finally, a maximal subclass of pLTL for which emptiness of $V_{> 0}(\varphi)$ is decidable.Comment: TCS Track B 201

Influence of surface pretreatment in resistance spot welding of aluminum AA1050

Resistance spot welding (RSW) of aluminum alloys implies a major problem of inconsistent quality from weld to weld due to problems of varying thickness of the oxide layer. The high resistivity of oxide layer causes strong heat development, which has significant influence on electrode life and weld quality. An experimental study of the influence of pretreatment on weld quality in RSW of AA1050 sheets with three thicknesses, comparing welding of as-received sheet with pretreated sheet by either pickling in NaOH or glass-blasting were investigated. Different weld settings were applied with low-, medium-, and high-energy inputs. The as-received sheet showed higher electrical contact resistance because of thicker oxide layer. Lower values were noticed with pickled surfaces, whereas the lowest electrical contact resistance was obtained when glass blasting, resulting in the roughest surface topography, which facilitated breakdown the oxide layer. Highest strength and smaller scatter in strength were obtained by pickling in NaOH