On character amenability of Banach algebras

AbstractWe continue our work [E. Kaniuth, A.T. Lau, J. Pym, On φ-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 85–96] in the study of amenability of a Banach algebra A defined with respect to a character φ of A. Various necessary and sufficient conditions of a global and a pointwise nature are found for a Banach algebra to possess a φ-mean of norm 1. We also completely determine the size of the set of φ-means for a separable weakly sequentially complete Banach algebra A with no φ-mean in A itself. A number of illustrative examples are discussed

Climate change and the rise of the central Asian Silk Roads

The final centuries BCE (Before Common Era) saw the main focus of trade between the Far East and Europe switch from the so called Northern Route across the Asian steppes to the classical silk roads. The cities across central Asia flourished and grew in size and importance. While clearly there were political, economic and cultural drivers for these changes, there may also have been a role for changes in climate in this relatively arid region of Asia. Analysis of a new ensemble of snapshot global climate model simulations, run every 250 years over the last 6000 years, allows us to assess the long term climatological changes seen across the central Asian arid region through which the classical Silk Roads run. While the climate is comparatively stable through the Holocene, the fluctuations seen in these simulations match significant cultural developments in the region. From 1500 BCE the deterioration of climate from a transient precipitation peak, along with technological development and the immigration of Aryan nomads, drove a shift towards urbanization and probably irrigation, culminating in the founding of the major cities of Bukhara and Samarkand around 700–500 BCE. Between 1000 and 250 BCE the modelled precipitation in the central Asian arid region undergoes a transition towards wetter climates. The changes in the Western Disturbances, which is the key weather system for central Asian precipitation, provides 10% more precipitation and the increased hydrological resources may provide the climatological foundation for the golden era of Silk Road trade

Spectral synthesis and topologies on ideal spaces for Banach *-algebras

This paper continues the study of spectral synthesis and the topologies τ∞ and τr on the ideal space of a Banach algebra, concentrating on the class of Banach *-algebras, and in particular on L1-group algebras. It is shown that if a group G is a finite extension of an abelian group then τr is Hausdorff on the ideal space of L1(G) if and only if L1(G) has spectral synthesis, which in turn is equivalent to G being compact. The result is applied to nilpotent groups, [FD]−-groups, and Moore groups. An example is given of a non-compact, non-abelian group G for which L1(G) has spectral synthesis. It is also shown that if G is a non-discrete group then τr is not Hausdorff on the ideal lattice of the Fourier algebra A(G)

Charazterization of closed ideals with boundedapproximate identities in commutative banach algebras, complemented subspaces of the group Von Neumann Algebras and applications

Let G be a locally compact group and B(G) the Fourier–Stieltjes algebra of G. Pursuing our investigations of power bounded elements in B(G), we study the extension property for power bounded elements and discuss the structure of closed sets in the coset ring of G which appear as 1-sets of power bounded elements. We also show that L1-algebras of noncompact motion groups and of noncompact IN-groups with polynomial growth do not share the so-called power boundedness property. Finally, we give a characterization of power bounded elements in the reduced Fourier–Stieltjes algebra of a locally compact group containing an open subgroup which is amenable as a discrete group