arXiv subjects
Publications and source records attributed to Lyudmila Turowska.
We introduce new properties of Haagerup-type for Fell bundles over discrete groups and discuss their relationship with Haagerup properties for the associated $C^*$-algebras. In particular, our results on Fell bundles associated with twisted unital $C^*$-dynamical systems extend previous results known in the untwisted case.
For a locally compact group G we introduce and study the reduced Beurling-Fourier-Stieltjes algebra, a weighted analogue of the reduced Fourier-Stieltjes algebra, together with the algebra of completely bounded multipliers of the associated weighted Fourier algebra. We show, in particular, that these two algebras coincide when G is amenable. For a general locally compact group G, we identify them as subspaces of the reduced Fourier-Stieltjes algebra and of the space of functions that locally belong to the Fourier algebra, respectively. Furthermore, we establish sufficient conditions on the group and the weight under which the algebra of completely bounded multipliers of the weighted Fourier algebra embeds into its unweighted counterpart.
Let $G$ be a simple finite graph, and let $\mathcal U_G$ be the related quantum graph. We study the game algebra $C(\mathrm{Qut}(\mathcal U_G))$ of quantum automorphism of $\mathcal U_G$. Moreover, we prove that for any graph $G$ with $|V(G)|\geq 3$, the quantum graph $\mathcal U_G$ admits nonlocal symmetry, meaning that there exists a perfect quantum no-signaling correlation
Motivated by a recent result on finite-dimensional Hilbert spaces, we prove a Jensen's inequality for partial traces in semifinite von Neumann algebras. We also prove a similar inequality in the framework of general (non-tracial) von Neumann algebras.
We study no-signalling correlations over Cantor spaces, placing the product of infinitely many copies of a finite non-local game in a unified general setup. We define the subclasses of local, quantum spatial, approximately quantum and quantum commuting Cantor correlations and describe them in terms of states on tensor products of inductive limits of operator systems. We provide a correspondence between no-signalling (resp. approximately quantum, quantum commuting) Cantor correlations and sequences of correlations of the same type over the projections onto increasing number of finitely many coordinates. We introduce Cantor games, and associate canonically such a game to a sequence of finite input/output games, showing that the numerical sequence of the values of the games in the sequence converges to the corresponding value of the compound Cantor game.
We develop a general framework for self-testing, in which bipartite correlations are described by states on the commuting tensor product of a pair of operator systems. We propose a definition of a local isometry between bipartite quantum systems in the commuting operator model, and define self-testing and abstract self-testing in the latter generality. We show that self-tests are in the general case always abstract self-tests and that, in some cases, the converse is also true. We apply our framework in a variety of instances, including to correlations with quantum inputs and outputs, quantum commuting correlations for the CHSH game, synchronous correlations, contextuality scenarios, quantum colourings and Schur quantum channels.
We discuss convergence in the Fourier algebra A(G) of a locally compact group G and provide a new characterisation of the local spectral sets of G.
We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum.
We study no-signalling correlations, defined over a quadruple of second countable compact Hausdorff spaces. Using operator-valued information channels over abstract alphabets, we define the subclasses of local, quantum spatial and quantum commuting measurable no-signalling correlations. En route, we establish measurable versions of the Stinespring's Dilation Theorem. We define values of measurable non-local games of local, quantum spatial and quantum commuting type, as well as inner versions thereof, and show how the asymptotic values of a finite non-local game can be viewed as special cases of the corresponding inner values of a measurable game, canonically associated with the given finite game.
We develop a resource-theoretical approach that allows us to quantify values of two-player, one-round cooperative games with quantum inputs and outputs, as well as values of quantum probabilistic hypergraphs. We analyse the quantum game values arising from the type hierarchy of quantum no-signalling correlations, establishing tensor norm expressions for each of the correlation types. As a consequence, we provide metric characterisations of state convertibility via LOSR and LOCC.En route, we obtain an alternative description of the maximal tensor products of ternary rings of operators.
Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras our parameters are related to quantitative versions of Arveson's Null Set Theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalise quantum versions of Strassen's Theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.
We introduce classical and quantum no-signalling bicorrelations and characterise the different types thereof in terms of states on operator system tensor products, exhibiting connections with bistochastic operator matrices and with dilations of quantum magic squares. We define concurrent bicorrelations as a quantum input-output generalisation of bisynchronous correlations. We show that concurrent bicorrelations of quantum commuting type correspond to tracial states on the universal C*-algebra of the projective free unitary quantum group, showing that in the quantum input-output setup, quantum permutations of finite sets must be replaced by quantum automorphisms of matrix algebras. We apply our results to study the quantum graph isomorphism game, describing the game C*-algebra in this case, and make precise connections with the algebraic notions of quantum graph isomorphism, existing presently in the literature.
In this paper, we develop a new approach that allows to identify the Gelfand spectrum of weighted Fourier algebras as a subset of an abstract complexification of the corresponding group for a wide class of groups and weights. This generalizes recent related results of Ghandehari-Lee-Ludwig-Spronk-Turowska (Adv. Math. 2021) about the spectrum of Beurling-Fourier algebras on some Lie groups. In the case of discrete groups we show that the spectrum of Beurling-Fourier algebra is homeomorphic to $G$.
We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely $SU(n)$, the Heisenberg group $\mathbb{H}$, the reduced Heisenberg group $\mathbb{H}_r$, the Euclidean motion group $E(2)$ and its simply connected cover $\widetilde{E}(2)$. We will determine the spectrum of Beurling-Fourier algebras on each of the aforementioned groups emphasizing its connection to the complexification of underlying Lie groups. We also demonstrate "polynomially growing" weights does not change the spectrum and show the associated regularity of the resulting Beurling-Fourier algebras.
We introduce concurrent quantum non-local games, quantum output mirror games and concurrent classical-to-quantum non-local games, as quantum versions of synchronous non-local games, and provide tracial characterisations of their perfect strategies belonging to various correlation classes. We define *-algebras and C*-algebras of concurrent classical-to-quantum and concurrent quantum non-local games, and algebraic versions of the orthogonal rank of a graph. We show that quantum homomorphisms of quantum graphs can be viewed as entanglement assisted classical homomorphisms of the graphs, and give descriptions of the perfect quantum commuting and the perfect approximately quantum strategies for the quantum graph homomorphism game. We specialise the latter results to the case where the inputs of the game are based on a classical graph.
Given a skew-symmetric real $n\times n$ matrix $Θ$ we consider the universal enveloping $C^*$-algebra $\mathsf{CAR}_Θ$ of the $*$-algebra generated by $a_1, \ldots, a_n$ subject to the relations \[ a_i^* a_i + a_i a_i^* = 1, \ \] \[ a_i^* a_j = e^{2 πi Θ_{i,j}} a_j a_i^*, \] \[ a_i a_j = e^{-2 πi Θ_{i,j}} a_j a_i. \] We prove that $\mathsf{CAR}_Θ$ has a $C(K_n)$-structure, where $K_n = \left[ 0,\frac{1}{2} \right]^n$ is the hypercube and describe the fibers. We classify irreducible representations of $\mathsf{CAR}_Θ$ in terms of irreducible representations of a higher-dimensional noncommutative torus. We prove that for a given irrational skew-symmetric $Θ_1$ there are only finitely many $Θ_2$ such that $\mathsf{CAR}_{Θ_1} \simeq \mathsf{CAR}_{Θ_2}$. Namely, $\mathsf{CAR}_{Θ_1} \simeq \mathsf{CAR}_{Θ_2}$ implies $(Θ_1)_{ij} = \pm (Θ_2)_{σ(i,j)} \mod \mathbb{Z}$ for a bijection $σ$ of the set $\{(i,j) : i < j, \ i, j = 1, \ldots, n\}$. For $n = 2$ we give a full classification: $\mathsf{CAR}_{θ_1} \simeq \mathsf{CAR}_{θ_2}$ iff $θ_1 = \pm θ_2 \mod \mathbb{Z}$.
We obtain descriptions of central operator-valued Schur and Herz-Schur multipliers, akin to a classical characterisation due to Grothendieck, that reveals a close link between central (linear) multipliers and bilinear multipliers into the trace class. Restricting to dynamical systems where a locally compact group acts on itself by translation, we identify their convolution multipliers as the right completely bounded multipliers, in the sense of Junge-Neufang-Ruan, of a canonical quantum group associated with the underlying group. We provide characterisations of contractive idempotent operator-valued Schur and Herz-Schur multipliers. Exploiting the link between Herz-Schur multipliers and multipliers on transformation groupoids, we provide a combinatorial characterisation of groupoid multipliers that are contractive and idempotent.
Given a $C^*$-dynamical system $(A,G,α)$, with $G$ a discrete group, Schur $A$-multipliers and Herz--Schur $(A,G,α)$-multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of $A \rtimes_{α, r} G$. The resulting characterisations of exactness and SOAP of $A \rtimes_{α, r} G$ generalise the corresponding statements for the reduced group $C^*$-algebra.