We show that continuous group homomorphisms between unitary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain $K$-theoretic regularity conditions, these maps can be seen to commute with the pairing between $K_0$ and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign).

Let $G(n)={\textrm {Sp}}(n,1)$ or ${\textrm {SU}}(n,1)$ . We classify conjugation orbits of generic pairs of loxodromic elements in $G(n)$ . Such pairs, called ‘nonsingular’, were introduced by Gongopadhyay and Parsad for ${\textrm {SU}}(3,1)$ . We extend this notion and classify $G(n)$ -conjugation orbits of such elements in arbitrary dimension. For $n=3$ , they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group. We further construct twist-bend parameters to glue such representations and obtain local parametrization for generic representations of the fundamental group of a closed (genus $g \geq 2$ ) oriented surface into $G(3)$ .

We prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with respect to every quasitrace. This condition is a key ingredient in the study of the realization problem of functions on the cone of quasitraces as ranks of positive elements. In the course of our investigation, we identify additional structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of quasitraces.

Given a $2$-commutative diagram

in a symmetric monoidal $(\infty ,2)$-category $\mathscr{E}$ where $X,Y\in \mathscr{E}$ are dualizable objects and $\unicode[STIX]{x1D711}$ admits a right adjoint we construct a natural morphism $\mathsf{Tr}_{\mathscr{E}}(F_{X})\xrightarrow[{}]{~~~~~}\mathsf{Tr}_{\mathscr{E}}(F_{Y})$ between the traces of $F_{X}$ and $F_{Y}$, respectively. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty ,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah–Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

We prove that localization operators associated to ridgelet transforms withLp symbols are boundedlinear operators on L2(Rn).Operators closely related to these localization operators are shown to be in the traceclass and a trace formula for them is given.

In this paper we explore the influence of the physical and social environment (the design space) son the formation of shared understanding in multidisciplinary design teams. We concentrate on the creative design meeting as a microenvironment for studying processes of design communication. Our applied research context entails the design of mixed physical–digital interactive systems supporting design meetings. Informed by theories of embodiment that have recently gained interest in cognitive science, we focus on the role of interactive “traces,” representational artifacts both created and used by participants as scaffolds for creating shared understanding. Our research through design approach resulted in two prototypes that form two concrete proposals of how the environment may scaffold shared understanding in design meetings. In several user studies we observed users working with our systems in natural contexts. Our analysis reveals how an ensemble of ongoing social as well as physical interactions, scaffolded by the interactive environment, grounds the formation of shared understanding in teams. We discuss implications for designing collaborative tools and for design communication theory in general.

We show how to derive the uniqueness of graded or ordinary traces on some algebras of log-polyhomogeneous pseudodifferential operators from the uniqueness of their restriction to classical pseudodifferential ones.

Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a well-known theorem relating traces and Conway operators in Cartesian categories.