Mathematics

We show that if C ∗ u (X) is a uniform Roe algebra associated to a bounded geometry metric space X, then all bounded derivations on C ∗ u (X) are inner.

Let x 0 be a self-adjoint random variable and c t be a free circular Brownian motion, freely independent from x 0 . We use the Hamilton--Jacobi method to compute the Brown measure ρ t of x 0 + c t . The Brown measure is absolutely continuous with a density that is \emph{constant along the vertical direction} in the support of ρ t . The support of the Brown measure of x 0 + c t is related to the subordination function of the free additive convolution of x 0 + s t , where s t is the free semicircular Brownian motion, freely independent from x 0 . Furthermore, the push-forward of ρ t by a natural map is the law of x 0 + s t . Let u be a unitary random variable and b t is the free multiplicative Brownian motion freely independent from u , we compute the Brown measure μ t of the free multiplicative Brownian motion u b t , extending the recent work by Driver--Hall--Kemp. The measure is absolutely continuous with a density of the special form 1 r 2 w t (θ) in polar coordinates in its support. The support of μ t is related to the subordination function of the free multiplicative convolution of u u t where u t is the free unitary Brownian motion free independent from u . The push-forward of μ t by a natural map is the law of u u t . In the special case that u is Haar unitary, the Brown measure μ t follows the \emph{annulus law}. The support of the Brown measure of u b t is an annulus with inner radius e −t/2 and outer radius e t/2 . The density in polar coordinates is given by 1 2πt 1 r 2 in its support.

We show that Bunce-Deddens algebras, which are AT-algebras, are also limits of circle algebras for Rieffel's quantum Gromov-Hausdorff distance, and moreover, form a continuous family indexed by the Baire space. To this end, we endow Bunce-Deddens algebras with a quantum metric structure, a step which requires that we reconcile the constructions of the Latremoliere's Gromov-Hausdorff propinquity and Rieffel's quantum Gromov-Hausdorff distance when working on order-unit quantum metric spaces. This work thus continues the study of the connection between inductive limits and metric limits.

In this paper, we construct, for a certain class of semigroup dynamical systems, two operator algebras that are universal with respect to their corresponding covariance conditions: one being self-adjoint, and another being non-self-adjoint. We prove that the C ∗ -envelope of the non-self-adjoint operator algebra is precisely the self-adjoint one. This result leads to a number of new examples of operator algebras and their C ∗ -envelopes, with many from number fields and commutative rings. We further establish the functoriality of these operator algebras along with their applications.

This paper is a continuation of the paper entitled "Subshifts, λ -graph bisystems and C ∗ -algebras", arXiv:1904.06464. A λ -graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying certain compatibility condition on their edge labeling. For any two-sided subshift Λ , there exists a λ -graph bisystem satisfying a special property called FPCC. We will construct an AF-algebra F L with shift automorphism ρ L from a λ -graph bisystem ( L − , L + ) , and define a C ∗ -algebra R L by the crossed product F L ⋊ ρ L Z . It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If λ -graph bisystems come from two-sided subshifts, these C ∗ -algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We will present a simplicity condition of the C ∗ -algebra R L and the K-theory formulas of the C ∗ -algebras F L and R L . The K-group for the AF-algebra F L is regarded as a two-sided extension of the dimension group of subshifts.

We study the quantum ( C ∗ ) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, C ∗ -extreme points of normalized POVMs on countable spaces (in particular for finite sets) are always spectral measures (normalized projection valued measures). More generally it is shown that atomic C ∗ -extreme points are spectral. A Krein-Milman type theorem for POVMs has also been proved. As an application it is shown that a map on any commutative unital C ∗ -algebra with countable spectrum (in particular C n ) is C ∗ -extreme in the set of unital completely positive maps if and only if it is a unital ∗ -homomorphism.

A systematic theory of product and diagonal states is developed for tensor products of Z 2 -graded ∗ -algebras, as well as Z 2 -graded C ∗ -algebras. As a preliminary step to achieve this goal, we provide the construction of a {\it fermionic C ∗ -tensor product} of Z 2 -graded C ∗ -algebras. Twisted duals of positive linear maps between von Neumann algebras are then studied, and applied to solve a positivity problem on the infinite Fermi lattice. Lastly, these results are used to define fermionic detailed balance (which includes the definition for the usual tensor product as a particular case) in general C ∗ -systems with gradation of type Z 2 , by viewing such a system as part of a compound system and making use of a diagonal state.

For a second-countable locally compact Hausdorff étale groupoid G with a continuous 2 -cocycle σ we find conditions that guarantee that ℓ 1 (G,σ) has a unique C ∗ -norm.

Given a normal subgroup bundle A of the isotropy bundle of a groupoid Σ , we obtain a twisted action of the quotient groupoid Σ/A on the bundle of group C ∗ -algebras determined by A whose twisted crossed product recovers the groupoid C ∗ -algebra C ∗ (Σ) . Restricting to the case where A is abelian, we describe C ∗ (Σ) as the C ∗ -algebra associated to a T -groupoid over the tranformation groupoid obtained from the canonical action of Σ/A on the Pontryagin dual space of A . We give some illustrative examples of this result.

We give a new construction of a C*-algebra from a cancellative semigroup P via partial isometric representations, generalising the construction from the second named author's thesis. We then study our construction in detail for the special case when P is an LCM semigroup. In this case we realize our algebras as inverse semigroup algebras and groupoid algebras, and apply our construction to free semigroups and Zappa-Szép products associated to self-similar groups.