DrRupert Levene

We study unique games and estimate some of their values. We prove that if a unique game has a quantum-assisted value close to 1, then it must have a perfect deterministic strategy. We introduce a family of unique games based on groups that generalize XOR games, and show that when the group is the cyclic group of order 3, then these games correspond to a 3-labelling problem for directed graphs.

A graph is said to be orthogonalisable if the set of real symmetric matrices whose off-diagonal pattern is prescribed by its edges contains an orthogonal matrix. We determine some necessary and some sufficient conditions on the sizes of the connected components of two graphs for their join to be orthogonalisable. In some cases, those conditions coincide, and we present several families of joins of graphs that are orthogonalisable.

Given a graph G, consider the family of real symmetric matrices with the property that the pattern of their nonzero off-diagonal entries corresponds to the edges of G. For the past 30 years a central problem has been to determine which spectra are realizable in this matrix class. Using combinatorial methods, we identify a family of graphs and multiplicity lists whose realizable spectra are highly restricted. In particular, we construct trees with multiplicity lists that require a unique spectrum, up to shifting and scaling. This represents the most extreme possible failure of spectral arbitrariness for a multiplicity list, and greatly extends all previously known instances of this phenomenon, in which only single linear constraints on the eigenvalues were observed.

Motivated by applications in matrix constructions used in the inverse eigenvalue problem for graphs, we study a concept of genericity for eigenvectors associated with a given spectrum and a connected graph. This concept generalizes the established notion of a nowhere-zero eigenbasis. Given any simple graph G on n vertices and any spectrum with no multiple eigenvalues, we show that the family of eigenbases for symmetric matrices with this graph and spectrum is generic, strengthening a result of Monfared and Shader. We illustrate applications of this result by constructing new achievable ordered multiplicity lists for partial joins of graphs and providing several families of joins of graphs that are realizable by a matrix with only two distinct eigenvalues.

An important facet of the inverse eigenvalue problem for graphs is to determine the minimum number of distinct eigenvalues of a particular graph. We resolve this question for the join of a connected graph with a path. We then focus on bordering a matrix and attempt to control the change in the number of distinct eigenvalues induced by this operation. By applying bordering techniques to the join of graphs, we obtain numerous results on the nature of the minimum number of distinct eigenvalues as vertices are joined to a fixed graph.

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 relative commutants N′∩M$$N'\cap M$$ of a large class of finite-index inclusions N⊆M$$N\subseteq M$$ of tracial von Neumann algebras, where the unbiased condition means that no information about the teleported observables is contained in the classical communication sent between the parties. For a large class of subalgebras N of matrix algebras Mn(C)$$M_n({\mathbb {C}})$$, including those relevant to hybrid classical/quantum codes, we show that any tight teleportation scheme for N necessarily arises from an orthonormal unitary Pimsner–Popa basis of Mn(C)$$M_n({\mathbb {C}})$$ over N′$$N'$$, generalising work of Werner (J Phys A 34(35):7081–7094, 2001). Combining our techniques with those of Brannan–Ganesan–Harris (J Math Phys 63(11): 112204, 2022) we compute quantum chromatic numbers for a variety of quantum graphs arising from finite-dimensional inclusions N⊆M$$N\subseteq M$$.

We introduce a notion of compatibility for multiplicity matrices. This gives rise to a necessary condition for the join of two (possibly disconnected) graphs G and H to be the pattern of an orthogonal symmetric matrix, or equivalently, for the minimum number of distinct eigenvalues q of G ∨ H to be equal to two. Under additional hypotheses, we show that this necessary condition is also sufficient. As an application, we prove that q ( G ∨ H ) is either two or three when G and H are unions of complete graphs, and we characterise when each case occurs.

A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant matrix norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the matroidal class of ( k , l ) -sparse graphs for suitable k and l. An edge-colouring technique is developed to characterise infinitesimal rigidity for product norms and then applied to show that the graph of a minimally rigid bar-joint framework in the space of 2 × 2 symmetric (respectively, hermitian) matrices with the trace norm admits an edge-disjoint packing consisting of a (Euclidean) rigid graph and a spanning tree.

We establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi-finite von Neumann algebras. Namely, we establish a generalisation to this setting of Nielsen’s theorem on the convertibility of quantum states under local operations and classical communication (LOCC) schemes. Along the way, we introduce an appropriate generalisation of LOCC operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von Neumann algebras. As an application of our result in the setting of II1$$\mathrm {II}_1$$-factors, we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of Vidal, thus yielding a new way to quantify entanglement in that context. Building on previous work in the infinite-dimensional setting, we show that trace vectors play the role of maximally entangled states for general II1$$\mathrm {II}_1$$-factors. Examples are drawn from infinite spin chains, quasi-free representations of the CAR, and discretised versions of the CCR.

We propose a Nordhaus–Gaddum conjecture for q ( G ) , the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph G: for every graph G excluding four exceptions, we conjecture that q ( G ) + q ( G c ) ≤ | G | + 2 , where G c is the complement of G. We compute q ( G c ) for all trees and all graphs G with q ( G ) = | G | − 1 , and hence we verify the conjecture for trees, unicyclic graphs, graphs with q ( G ) ≤ 4 , and for graphs with | G | ≤ 7 .

We consider the tensor product of the completely depolarising channel on d × d matrices with the map of Schur multiplication by a k × k correlation matrix and characterise, via matrix theory methods, when such a map is a mixed (random) unitary channel. When d = 1, this recovers a result of O’Meara and Pereira and for larger d, is equivalent to a result of Haagerup and Musat that was originally obtained via the theory of factorisation through von Neumann algebras. We obtain a bound on the distance between a given correlation matrix for which this tensor product is nearly mixed unitary and a correlation matrix for which such a map is exactly mixed unitary. This bound allows us to give an elementary proof of another result of Haagerup and Musat about the closure of such correlation matrices without appealing to the theory of von Neumann algebras.

We generalize some well-known graph parameters to operator systems by considering their underlying quantum channels. In particular, we introduce the quantum complexity as the dimension of the smallest co-domain Hilbert space a quantum channel requires to realize a given operator system as its non-commutative confusability graph. We describe quantum complexity as a generalized minimum semidefinite rank and, in the case of a graph operator system, as a quantum intersection number. The quantum complexity and a closely related quantum version of orthogonal rank turn out to be upper bounds for the Shannon zero-error capacity of a quantum channel, and we construct examples for which these bounds beat the best previously known general upper bound for the capacity of quantum channels, given by the quantum Lovász theta number.

Abstract We define the Schur multipliers of a separable von Neumann algebra with Cartan maximal abelian self-adjoint algebra , generalizing the classical Schur multipliers of ( ℓ 2 ). We characterize these as the normal -bimodule maps on . If contains a direct summand isomorphic to the hyperfinite II 1 factor, then we show that the Schur multipliers arising from the extended Haagerup tensor product ⊗ eh are strictly contained in the algebra of all Schur multipliers.

<p> We consider non-selfadjoint operator algebras <math alttext="script upper L left-parenthesis upper G comma lamda right-parenthesis"> <semantics> <mrow> <mrow> <mi mathvariant="script">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi> λ </mi> <mo stretchy="false">)</mo> </mrow> <annotation>\mathcal {L} (G,\lambda )</annotation> </semantics> </math> generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs  <math alttext="upper G"> <semantics> <mi>G</mi> <annotation>G</annotation> </semantics> </math> . These algebras may be viewed as non-commutative generalizations of weighted Bergman space algebras or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for <math alttext="German upper L left-parenthesis upper G comma lamda right-parenthesis"> <semantics> <mrow> <mrow> <mi mathvariant="fraktur">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi> λ </mi> <mo stretchy="false">)</mo> </mrow> <annotation>\mathfrak {L} (G,\lambda )</annotation> </semantics> </math> in the case of the single vertex graph with two edges and a suitable choice of left weight function  <math alttext="lamda"> <semantics> <mi> λ </mi> <annotation>\lambda</annotation> </semantics> </math> . </p>

We show that the set of Schur idempotents with hyperreflexive range is a Boolean lattice which contains all contractions. We establish a preservation result for sums which implies that the weak* closed span of a hyperreflexive and a ternary masa-bimodule is hyperreflexive, and prove that the weak* closed span of finitely many tensor products of a hyperreflexive space and a hyperreflexive range of a Schur idempotent (respectively, a ternary masa-bimodule) is hyperreflexive.

We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.

It is well-known that if T is a Dm-Dn bimodule map on the m×n complex matrices, then T is a Schur multiplier and ‖T‖cb=‖T‖. If n=2 and T is merely assumed to be a right D2-module map, then we show that ‖T‖cb=‖T‖. However, this property fails if m⩾2 and n⩾3. For m⩾2 and n=3, 4 or n⩾m2 we give examples of maps T attaining the supremumC(m,n)=supT‖cb:Ta rightDn-module map onMm,nwith‖T‖≤1},we show that C(m,m2)=m and succeed in finding sharp results for C(m,n) in certain other cases. As a consequence, if H is an infinite-dimensional Hilbert space and D is a masa in B(H), then there is a bounded right D-module map on K(H) which is not completely bounded.

The Hardy space H2 (ℝ) for the upper half‐plane together with a multiplicative group of unimodular functions u(λ) = exp(i(λ1ψ1 + … +λnψn)), with λ ∈ ℝn, gives rise to a manifold ℳ of orthogonal projections for the subspaces u(λ) H2 (ℝ) of L2 (ℝ). For classes of admissible functions ψi the strong operator topology closures of ℳ and ℳ ∪ ℳ⊥ are determined explicitly as various n‐balls and n‐spheres. The arguments used are direct and rely on the analysis of oscillatory integrals (E. M. Stein, Harmonic analysis: real‐variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43 (Princeton University Press, Princeton, NJ, 1993)) and Hilbert space geometry. Some classes of these closed projection manifolds are classified up to unitary equivalence. In particular, the Fourier–Plancherel 2‐sphere and the hyperbolic 3‐sphere of Katavolos and Power (A. Katavolos and S. C. Power, Translation and dilation invariant subspaces of L2(ℝ), J. reine angew. Math. 552 (2002) 101–129) appear as distinguished special cases admitting non‐trivial unitary automorphism groups, which are explicitly described.

We continue the study of multidimensional operator multipliers initiated in [K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc., in press]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C∗-algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C∗-algebra of compact operators in terms of tensor products, generalising results of Saar [H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C∗-Algebra, Diplomarbeit, Universität des Saarlandes, Saarbrücken, 1982].

We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu’s non-commutative disc algebras and to free semigroup algebras as well.

The subspaces and subalgebras of B(H) which are hyperreflexive with constant 1 are completely classified. It is shown that there are 1-hyperreflexive subspaces for which the complete hyperreflexivity constant is strictly greater than 1. The constants for CT⊗B(H) are analyzed in detail.

Abstract We consider the w*-closed operator algebra generated by the image of the semigroup SL 2 (ℝ+) under a unitary representation ρ of SL 2 (ℝ) on the Hilbert space L 2 (ℝ). We show that is a reflexive operator algebra and = Alg where is a double triangle subspace lattice. Surprisingly, is also generated as a w*-closed algebra by the image under ρ of a strict subsemigroup of SL 2 (ℝ + ).

The hyperbolic algebra [Formula: see text], studied recently by Katavolos and Power [5], is the weak star closed operator algebra on L 2 (ℝ) generated by H ∞ (ℝ), as multiplication operators, and by the dilation operators V t , t≥0, given by V t f(x)=e t/2 f(e t x). We show that [Formula: see text] is a reflexive operator algebra and that the four-dimensional manifold [Formula: see text] (with the natural topology) is the reflexive hull of a natural two-dimensional subspace.