Ivan G. Todorov

Operator system structures on ordered spaces

Given an Archimedean order unit space (V,V^+,e), we construct a minimal operator system OMIN(V) and a maximal operator system OMAX(V), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system S is completely boundedly isomorphic to either OMIN(S) or to OMAX(S). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(M_n) to OMAX(M_m) if and only if it is entanglement breaking.

Estimating quantum chromatic numbers

Abstract We develop further the new versions of quantum chromatic numbers of graphs introduced by the first and fourth authors. We prove that the problem of computation of the commuting quantum chromatic number of a graph is solvable by an SDP algorithm and describe an hierarchy of variants of the commuting quantum chromatic number which converge to it. We introduce the tracial rank of a graph, a parameter that gives a lower bound for the commuting quantum chromatic number and parallels the projective rank, and prove that it is multiplicative. We describe the tracial rank, the projective rank and the fractional chromatic numbers in a unified manner that clarifies their connection with the commuting quantum chromatic number, the quantum chromatic number and the classical chromatic number, respectively. Finally, we present a new SDP algorithm that yields a parameter larger than the Lovasz number and is yet a lower bound for the tracial rank of the graph. We determine the precise value of the tracial rank of an odd cycle.

Multidimensional operator multipliers

We introduce multidimensional Schur multipliers and characterise them, generalising well-known results by Grothendieck and Peller. We define a multidimensional version of the two-dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several C*-algebras satisfying certain boundedness conditions. In the case of commutative C * algebras, the multidimensional operator multipliers reduce to continuous multidimensional Schur multipliers. We show that the multipliers with respect to some given representations of the corresponding C * -algebras do not change if the representations are replaced by approximately equivalent ones. We establish a non-commutative and multidimensional version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained as certain weak limits of elements of the algebraic tensor product of the corresponding C * -algebras.

Compact operators on Hilbert modules

We prove that an adjointable contraction acting on a countably generated Hilbert module over a separable unital C*-algebra is compact if and only if the set of its second contractive perturbations is separable.

Spectral Synthesis and Masa-Bimodules

Generalizing a result of Arveson on finite width CSL algebras, we prove that finite width masa-bimodules satisfy spectral synthesis. Introducing a new class of masa-bimodules, we show that there exists a non-synthetic masa-bimodule, such that the maximal algebras over which it is a bimodule, are synthetic.

Private algebras in quantum information and infinite-dimensional complementarity

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.

Compactness properties of operator multipliers

Abstract 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, vollstandig beschrankte Abbildungen mit Werten in einer nuklearen C ∗ -Algebra, Diplomarbeit, Universitat des Saarlandes, Saarbrucken, 1982].

Ideals of the Fourier algebra, supports and harmonic operators

We examine the common null spaces of families of Herz-Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in J. Funct. Anal. 266 (2014), 6473-6500 can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.

Ideals of A(G) and bimodules over maximal abelian selfadjoint algebras

This paper is concerned with weak⁎ closed masa-bimodules generated by A(G)-invariant subspaces of VN(G). An annihilator formula is established, which is used to characterise the weak⁎ closed subspaces of B(L2(G)) which are invariant under both Schur multipliers and a canonical action of M(G) on B(L2(G)) via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis.

Normalisers of operator algebras and tensor product formulas

We establish a tensor product formula for bimodules over maximal abelian selfadjoint algebras and their supports. We use this formula to show that if A is the tensor product of finitely many continuous nest algebras, B is a CSL algebra and A and B have the same normaliser semi-group then either A = B or A∗ = B. We show that the result does not hold without the assumption that the nests be continuous, answering in the negative a question raised in [28].