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For an element in an algebra-valued *-noncommutative probability space, equivalent conditions for algebra-valued R-diagonality (a notion introduced by Sniady and Speicher) are proved. Formal power series relations involving the moments and…

We prove analogues of the Riemann-Roch Theorem and the Hodge Theorem for noncommutative tori (of any dimension) equipped with complex structures, and discuss implications for the question of how to distinguish "noncommutative abelian…

We prove a version of the Cuntz--Krieger Uniqueness Theorem for $C^*$-algebras of arbitrary relative generalized Boolean dynamical systems. We then describe properties of a $C^*$-algebra of a relative generalized Boolean dynamical system…

In this paper, we study the strong extension groups of Cuntz--Krieger algebras, and present a formula to compute the groups. We also detect the position of the Toeplitz extension of a Cuntz--Krieger algebra in the strong extension group and…

Results about angles between Haagerup--Schultz projections for DT-operators whose measures have atoms are proved, which in some cases imply that such operators are non-spectral. Several examples are considered.

Klep and Schweighofer asked whether the Nirgendsnegativsemide-finitheitsstellensatz holds for a symmetric noncommutative polynomial whose evaluations at bounded self-adjoint operators on any nontrivial Hilbert space are not negative…

Let $X$ be a path connected, locally path connected and semilocally simply connected space; let $\tilde{X}$ be its universal cover. We discuss the existence and description of a Haar system on the fundamental groupoid $\Pi_1(X)$ of $X$. The…

We associate mirror games with the universal game algebra and use the *-representation to describe quantum commuting operator strategies. We provide an algebraic characterization of whether or not a mirror game has perfect commuting…

In this paper, we obtain a new proof result of Shlyakhtenko which states that if $G$ is a sofic, finitely presented group with vanishing first $\ell^2$-Betti number, then $L(G)$ is strongly 1-bounded. Our proof of this result adapts and…

For each odd integer $n \geq 3$, we construct a rank-3 graph $\Lambda_n$ with involution $\gamma_n$ whose real C*-algebra $C^*_\mathbb{R}(\Lambda_n, \gamma_n)$ is stably isomorphic to the exotic Cuntz algebra $\mathcal E_n^\mathbb{R}$. This…

We introduce a notion of teleportation scheme between subalgebras of semi-finite von Neumann algebras in the commuting operator model of locality. Using techniques from subfactor theory, we present unbiased teleportation schemes for…

In this paper, we establish noncommutative Burkholder inequalities with asymmetric diagonals in symmetric operator spaces. Our proof mainly relies on a new complex interpolation result on asymmetric vector valued spaces and a duality…

This note treats a simple minded question: what does a typical random matrix range look like? We study the relationship between various modes of convergence for tuples of operators, on the one hand, and continuity of matrix ranges with…

In this paper we study various von Neumann algebraic rigidity aspects for the property (T) groups that arise via the Rips construction developed by Belegradek and Osin in geometric group theory \cite{BO06}. Specifically, developing a new…

Let $q = e^{i \theta} \in \mathbb{T}$ (where $\theta \in \mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, i.e., $u$ and $v$ are unitaries such that $vu = quv$. In this paper we find the optimal constant $c = c_\theta$ such that $u,v$…

We prove many new results about interacting Fock spaces. We pose many open problems; for most of them we prove that their solutions have no choice but being nontrivial. We ask the kind reader to consult the extended abstract in the paper.

We characterize Cuntz-Nica-Pimsner algebras for compactly aligned product systems over quasi-lattice ordered groupoids. We show that the full cross sectional $C^*$-algebras of Fell bundles of Morita equivalence bimodules are isomorphic to…

Let $\Omega$ be a class of unital ${\rm C^*}$-algebras which have the second type tracial nuclear dimensional at moat $n$ (or have tracial nuclear dimensional at most $n$). Let $A$ be an infinite dimensional unital simple ${\rm…

We study the space of all $d$-tuples of unitaries $u=(u_1,\ldots, u_d)$ using dilation theory and matrix ranges. Given two $d$-tuples $u$ and $v$ generating C*-algebras $\mathcal A$ and $\mathcal B$, we seek the minimal dilation constant…

Given a locally compact group $G=Q\ltimes V$ such that $V$ is Abelian and such that the action of $Q$ on the Pontryagin dual $\hat V$ has a free orbit of full measure, we construct a family of unitary dual $2$-cocycles $\Omega_\omega$ (aka…