Related papers: Unitaries Permuting Two Orthogonal Projections
It is known that, under strong combinatorial axioms, $O_N\subset O_N^*\subset O_N^+$ are the only orthogonal quantum groups. We prove here similar results for the noncommutative spheres $S^{N-1}_\mathbb R\subset S^{N-1}_{\mathbb R,*}\subset…
We give multidimensional generalizations of several transformation formulae for basic hypergeometric series of a specific type. Most of the upper parameters of the series differ multiplicatively from corresponding lower parameters by a…
Given a bounded operator $Q$ on a Hilbert space $\mathcal{H}$, a pair of bounded operators $(T_1, T_2)$ on $\mathcal{H}$ is said to be $Q$-commuting if one of the following holds: \[ T_1T_2=QT_2T_1 \text{ or }T_1T_2=T_2QT_1 \text{ or…
We describe a particular class of pairs of quantum observables which are extremal in the convex set of all pairs of compatible quantum observables. The pairs in this class are constructed as uniformly noisy versions of two mutually unbiased…
We show that if $\mu$ is a self-affine measure on the plane defined by an iterated function system of contractions with diagonal linear parts, then under an irrationality assumption on the entries of the linear parts, $$ \dim_{\rm H} \mu…
We relate extensions of completely unitary VOAs and (commutative) Q-systems. As an application, we show that any unitary extension of a completely unitary VOA is completely unitary.
Two rational polygons $P$ and $Q$ are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection (equivalently, a piecewise affine-linear bijection that preserves the integer lattice $\mathbb{Z} \times…
A finite shift plane can be equivalently defined via abelian relative difference sets as well as planar functions. In this paper, we present a generic way to construct unitals in finite shift planes of odd orders $q^2$. We investigate…
Given two orthogonal projections $\{P,Q\}$ in a non commutative tracial probability space, we prove relations between the moments of $P+Q$, of $\sqrt{-1}(PQ-QP)$ and of $P+QPQ$ and those of the angle operator $PQP$. Our proofs are purely…
DeTurck and Yang have shown that in the neighbourhood of every point of a $3$-dimensional Riemannian manifold, there exists a system of orthogonal coordinates (that is, whith respect to which the metric has diagonal form). We show that this…
We consider pseudo-unitary quantum systems and discuss various properties of pseudo-unitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal…
For an arbitrary separated scheme $X$ of finite type over a finite field $\mathbb F_q$ and an integer $j=-1,-2,$ we prove under the assumption of resolution of singularities, that the two groups $H_{-1}(X,\mathbb Z(j))$ and…
We present a complete characterization of diagonal unitary covariant (DU-covariant) superchannels, i.e. higher-order transformations transforming quantum channels into themselves. Necessary and sufficient conditions for complete positivity…
We show how localization and smoothing techniques can be used to establish universality in the bulk of the spectrum for a fixed positive measure mu on [-1,1]. Assume that mu is a regular measure, and is absolutely continuous in an open…
The two-parametric quantum superalgebra $U_{p,q}[gl(2/1)]$ is consistently defined. A construction procedure for induced representations of $U_{p,q}[gl(2/1)]$ is described and allows us to construct explicitly all (typical and nontypical)…
Let $P$ and $Q$ be idempotents on a Hilbert space $\mathcal{H}.$ The minus order $P\preceq Q$ is defined by the equation $PQ=QP=P.$ In this note, we first present some necessary and sufficient conditions for which the supremum and infimum…
Relying on results due to Shmerkin and Solomyak, we show that outside a $0$-dimensional set of parameters, for every planar homogeneous self-similar measure $\nu$, with strong separation, dense rotations and dimension greater than $1$,…
Constructing the Semi - Unitary Transformation (SUT) to obtain the supersymmetric partner Hamiltonians for a one dimensional harmonic oscillator, it has been shown that under this transformation the supersymmetric partner loses its ground…
Generalizing the case of an infinite discrete metric space of finite diameter, we say that a discrete metric space $(X,d)$ is a Kuiper space, if the group of invertible elements of its uniform Roe algebra is norm-contractible. Various…
We study the operator space UMD property, introduced by Pisier in the context of noncommutative vector-valued Lp-spaces. It is unknown whether the property is independent of p in this setting. We prove that for 1<p,q<\infty, the Schatten…