We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many properties formally analogous to KK-theory including a composition product. We establish basic properties, like additivity, stability and continuity, and study categorical aspects in the setting of local C *algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover, we establish identities when tensoring with strongly self-absorbing C * -algebras. Finally, we show that the bivariant Cuntz semigroup of the present work can be used to classify all unital and stably finite C * -algebras.

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A noncommutative spacetime admitting dilation symmetry was briefly mentioned in the seminal work [8] of Doplicher, Fredenhagen and Roberts. In this paper, we explicitly construct the model in detail and carry out an indepth analysis. The C * -algebra that describes this quantum spacetime is determined, and it is shown that it admits an action by * -automorphisms of the dilation group, along with the expected Poincaré covariance. In order to study the main physical properties of this scale-covariant model, a free scalar neutral field is introduced as an investigation tool. Our key results are then the loss of locality and the irreducibility, or triviality, of special field algebras associated with regions of the ordinary Minkowski spacetime. It turns out, in the conclusions, that this analysis allows also to argue on viable ways of constructing a full conformally covariant model for quantum spacetime.

We provide a new and concise proof of the existence of suprema in the Cuntz semigroup using the open projection picture of the Cuntz semigroup initiated in [11]. Our argument is based on the observation that the supremum of a countable set of open projections in the bidual of a C*-algebra A is again open and corresponds to the generated hereditary sub-C*-algebra of A.

We provide an equivariant extension of the bivariant Cuntz semigroup introduced in previous work for the case of compact group actions over C * -algebras. Its functoriality properties are explored and some well-known classification results are retrieved. Connections with crossed products are investigated, and a concrete presentation of equivariant Cuntz homology is provided. The theory that is here developed can be used to define the equivariant Cuntz semigroup. We show that the object thus obtained coincides with the recently proposed one by Gardella and Santiago, and we complement their work by providing an open projection picture of it.