Multidimensional operator multipliers

We introduce multidimensional Schur multipliers and characterise them, generalising well-known results by Grothendieck and Peller. We define a multidimensional version of the two-dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several C *-algebras satisfying certain boundedness conditions. In the case of commutative C*-algebras, the multidimensional operator multipliersreduce to continuousmul-tidimensional Schur multipliers. We show that the multiplierswith respect to some given representations of the corresponding C*-algebrasdo not change if the representations are replaced by approximately equivalent ones. We establish a non-commutative and multidimensional version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained ascertain weak limits of elements of the algebraic tensor product of the corresponding C *-algebras.

From a ‘borderless world’ to a ‘world of borders’: ‘bringing history back in’

The first part of the paper explores some of the reasons why contemporary border studies understate the full significance of state borders and their global primacy. It is argued that this failing is rooted in a much wider lack of historical reflexivity-a reluctance to acknowledge the historical positioning (the 'where and when') of contemporary border studies themselves. This reluctance encourages a form of pseudohistory, or 'epochal thinking', which disfigures perspective on the present. Among the consequences are (a) exaggerated claims of the novelty of contemporary border change, propped up by poorly substantiated benchmarks in the past; (b) an incapacity to recognise the 'past in the present' as in the various historical deposits of state formation processes; and (c) a failure to recognise the distinctiveness of contemporary state borders and how they differ from other borders in their complexity and globality. The second part of the paper argues for a recalibration of border studies aimed at balancing their spatial emphases with a much greater, and more critical, historical sensitivity. It insists that 'boundedness', and state boundedness in particular, is a variable that must be understood historically. This is illustrated by arguing that a better analysis of contemporary border change means rethinking the crude periodisation which distinguishes the age of empire from the age of nation-states and, in turn, from some putative contemporary era 'beyond nation-states' and their borders.