算子代数

We investigate when discrete, amenable groups have $C^*$-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse in an open problem. We show that if $C^*(G)$ has real rank zero, then…

We identify and characterize unital completely positive (UCP) maps on finite dimensional $C^*$-algebras for which the Choi-Effros product extended to the space generated by peripheral eigenvectors matches with the original product. We…

The Calkin algebra is not isomorphic to the corona of the stabilization of the Cuntz algebra~${\mathcal O}_\infty$, any other Kirchberg algebra, or even the corona of the stabilization of any unital, ${\mathcal Z}$-stable ${\mathrm…

We prove a version of Pedersen's outer conjugacy theorem for coactions of compact groups, which characterizes outer conjugate coactions of a compact group in terms of properties of the dual actions. In fact, we show that every isomorphism…

Generalizing the case of the Toeplitz algebra by Brake and Winter, we prove that the nuclear dimension of a C*-algebra extension of C(X) by the compact operators is equal to the dimension of X.

We establish the functional relations between generating series of higher-order free cumulants and moments in higher-order free probability, solving an open problem posed fifteen years ago by Collins, Mingo, \'Sniady and Speicher. We…

We show that conditional expectations, optimal hypotheses, disintegrations, and adjoints of unital completely positive maps, are all instances of Bayesian inverses. We study the existence of the latter by means of the Tomita-Takesaki…

We describe a construction for the full C$^*$-algebra of a possibly unsaturated Fell bundle over a possibly non-Hausdorff locally compact \'etale groupoid without appealing to Renault's disintegration theorem. This construction generalises…

For a unital spectral triple $(\mathcal{A}, H,D)$, we study when its truncation converges to itself. The spectral truncation is obtained by using the spectral projection $P_{\Lambda}$ of $D$ onto $[-\Lambda,\Lambda]$ to deal with the case…

If the symmetry (fixed invertible self adjoint map) of Krein spaces is replaced by a fixed unitary, then we obtain the notion of S-spaces which was introduced by Szafraniec. Assume $\alpha$ to be an automorphism on a $C^*$-algebra. In this…

We construct compact parametrix implementations of covariant derivations on the quantum annulus.

Let $p\colon \mathcal{A} \to G$ be a saturated Fell bundle over a locally compact, Hausdorff, second countable, {\'e}tale groupoid~$G$, and let $\mathrm{C}^*(G;\mathcal{A})$ denote its full $\mathrm{C}^*$-algebra. We prove an…

We introduce the notion of biexactness for general von Neumann algebras, naturally extending the notion from group theory. We show that biexactness implies solidity for von Neumann algebras, and that many of the examples of solid von…

We describe a weak tracial analog of approximate representability under the name "weak tracial approximate representability" for finite group actions. Let $G$ be a finite abelian group, let $A$ be an infinite-dimensional simple unital…

Let X be a compact Hausdorff space. Then the radius of comparison rc ( C (X)) is related to the covering dimension dim (X) by rc ( C (X)) \geq [ dim (X) - 7 ] / 2. Except for the additive constant, this improves a result of Elliott and Niu,…

We compare actions on C*-algebras of two constructions of locally compact quantum groups, the bicrossed product and the double crossed product. We give a duality between them as a generalization of Baaj-Skandalis duality. In the case of…

We establish factoriality and non-injectivity in full generality for the mixed $q$-Araki-Woods von Neumann algebra associated to a separable real Hilbert space $\mathsf{H}_{\mathbf{R}}$ with $\dim\mathsf{H}_{\mathbf{R}}\geq 2$, a strongly…

This paper explores the tension between multiple models and rigidity for groupoid $C^*$-algebras. We begin by identifying $\Gamma$-Cartan subalgebras $D$ inside twisted groupoid $C^*$-algebras $C^*_r(G, \omega)$, using similar techniques to…

In 1988, Haagerup and St{\o}rmer conjectured that any pointwise inner automorphism of a type $\rm III_1$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that any type $\rm III_1$ factor…

We treat the notion of principal symbol mapping on a compact smooth manifold as a $\ast$-homomorphism of $C^{\ast}$-algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the…