Theory Of Operators In Spaces Research Articles

Unique matrix factorizations associated to bilinear forms and Schur multipliers

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m×n matrices. The theory of operator spaces provides a set up which describes 4 norm optimal factorizations of Grothendieck's sort. It is shown that 3 of the optimal factorizations are uniquely determined and the remaining one is unique in some cases.

ANY DUAL OPERATOR SPACE IS WEAKLY LOCALLY REFLEXIVE

Abstract We introduce the notion of weakly local reflexivity in operator space theory and prove that any dual operator space is weakly locally reflexive.

Bilinear forms, Schur multipliers, complete boundedness and duality

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex $m \times n$ matrices. Based on the theory of operator spaces and completely bounded maps we present norm optimal versions of these results and two norm optimal factorization results related to the Schur product. We show that the spaces of bilinear forms and of Schur multipliers are conjugate duals to each other with respect to their completely bounded norms.

Thomson decompositions of measures in the disk

We study the classical problem of identifying the structure of P 2 ( μ ) \mathcal {P}^2(\mu ) , the closure of analytic polynomials in the Lebesgue space L 2 ( μ ) L^2(\mu ) of a compactly supported Borel measure μ \mu living in the complex plane. In his influential work, Thomson [Ann. of Math. (2) 133 (1991), pp. 477–507] showed that the space decomposes into a full L 2 L^2 -space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures μ \mu supported on the closed unit disk D ¯ \overline {\mathbb {D}} which have a part on the open disk D \mathbb {D} which is similar to the Lebesgue area measure, and a part on the unit circle T \mathbb {T} which is the restriction of the Lebesgue linear measure to a general measurable subset E E of T \mathbb {T} , we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space P 2 ( μ ) \mathcal {P}^2(\mu ) . It turns out that the space splits according to a certain natural decomposition of measurable subsets of T \mathbb {T} which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.

The 3-D vector hysteresis model of SMC materials based on the SAE and play operator

The hysteresis modeling of ferromagnetic materials in electrical equipment is one of the basic theoretical studies in the field of electrical engineering. In this paper, a 3-D vector hysteresis model is proposed based on Stacked Auto-Encoder (SAE) model and hysteresis operator space theory. Multiple 3-D vector Play hysteresis operators are used to construct an hysteresis operator space to create the high-dimensional operator data. By the operator space, the nonlinear hysteresis relationship in the hysteresis data is transformed into a nonlinear mapping between the operator data and the output of the model. And the SAE model is used to characterize this nonlinear mapping. In the training of the SAE model, training set is composed of operator data and magnetic field strength data. The structure of the vector hysteresis model is determined by taking the output of the operator space as the input of the SAE model. The simulation results show that the model can effectively describe the nonlinear characteristics and anisotropic of the Soft Magnetic Composite under 3-D vector excitation. In addition, this model has certain generalization ability.

Almost synchronous quantum correlations

The study of quantum correlation sets initiated by Tsirelson in the 1980s and originally motivated by questions in the foundations of quantum mechanics has more recently been tied to questions in quantum cryptography, complexity theory, operator space theory, group theory, and more. Synchronous correlation sets introduced by Paulsen et al. [J. Funct. Anal. 270, 2188–2222 (2016)] are a subclass of correlations that has proven particularly useful to study and arises naturally in applications. We show that any correlation that is almost synchronous, in a natural ℓ1 sense, arises from a state and measurement operators that are well-approximated by a convex combination of projective measurements on a maximally entangled state. This extends a result of Paulsen et al. [J. Funct. Anal. 270, 2188–2222 (2016)] that applies to exactly synchronous correlations. Crucially, the quality of approximation is independent of the dimension of the Hilbert spaces or of the size of the correlation. Our result allows one to reduce the analysis of many classes of nonlocal games, including rigidity properties, to the case of strategies using maximally entangled states that are generally easier to manipulate.

Hybrid intelligent hysteresis model based on DBN-DNN algorithm and fusion Preisach operator

The hysteresis modeling of ferromagnetic materials in electrical equipment is one of the basic theoretical studies in the field of electrical engineering. Long training time and poor accuracy are the main challenges of the traditional hysteresis models. In this paper, a Hybrid Intelligent Hysteresis Model (HIHM) is proposed by combining the hysteresis operator space theory and Deep Belief Networks-Deep Neural Network (DBN-DNN) algorithm. The Preisach Fusion Operator (PFO) is constructed by fusing multiple traditional Preisach operators with coefficients satisfying the Gaussian distribution. And multiple PFOs are used to construct an operator space to create the high-dimensional operator data. By the operator space, the nonlinear hysteresis relationship in the hysteresis data is transformed into a nonlinear mapping between the operator data and the output of the model. The DBN-DNN model is used to characterize this nonlinear mapping. The structure of the HIHM is determined by taking the output of the operator space as the input of the DBN model. The generation of first-order reversal curve proves that the model has certain practicality. In addition, the validity and generalization of this model is verified through the simulation with experimental data of non-orientation (NO) silicon steel B35A210 at quasi-static frequency.

The Ramsey property for operator spaces and noncommutative Choquet simplices

The noncommutative Gurarij space NG, initially defined by Oikhberg, is a canonical object in the theory of operator spaces. As the Fraïssé limit of the class of finite-dimensional nuclear operator spaces, it can be seen as the noncommutative analogue of the classical Gurarij Banach space. In this paper, we prove that the automorphism group of NG is extremely amenable, i.e. any of its actions on compact spaces has a fixed point. The proof relies on the Dual Ramsey Theorem, and a version of the Kechris–Pestov–Todorcevic correspondence in the setting of operator spaces.Recent work of Davidson and Kennedy, building on previous work of Arveson, Effros, Farenick, Webster, and Winkler, among others, shows that nuclear operator systems can be seen as the noncommutative analogue of Choquet simplices. The analogue of the Poulsen simplex in this context is the matrix state space NP of the Fraïssé limit A(NP) of the class of finite-dimensional nuclear operator systems. We show that the canonical action of the automorphism group of NP on the compact set NP1 of unital linear functionals on A(NP) is minimal and it factors onto any minimal action, whence providing a description of the universal minimal flow of Aut(NP).

Mapping ideals of quantum group multipliers

We study the dual relationship between quantum group convolution maps L1(G)→L∞(G) and completely bounded multipliers of Gˆ. For a large class of locally compact quantum groups G, we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with Mcb(L1(Gˆ)), yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter coinciding with ℓ1(bGˆ), where bG is the quantum Bohr compactification of G. For quantum groups whose dual has bounded degree, we show that the completely compact convolution maps coincide with C(bG). Our techniques comprise a mixture of operator space theory and abstract harmonic analysis, including Fubini tensor products, the non-commutative Grothendieck inequality, quantum Eberlein compactifications, and a suitable notion of inner co-amenable quantum group, that we introduce and for which we exhibit examples from the bicrossed product construction. Our main results are new even in the setting of group von Neumann algebras VN(G) for quasi-SIN locally compact groups G.

Correspondence theory on p-Fock spaces with applications to Toeplitz algebras

We prove several results concerning the theory of Toeplitz algebras over p-Fock spaces using a correspondence theory of translation invariant symbol and operator spaces. The most notable results are: The full Toeplitz algebra is the norm closure of all Toeplitz operators with bounded uniformly continuous symbols. This generalizes a result obtained by J. Xia [25] in the case p=2, which was proven by different methods. Further, we prove that every Toeplitz algebra which has a translation invariant C⁎ subalgebra of the bounded uniformly continuous functions as its set of symbols is linearly generated by Toeplitz operators with the same space of symbols.

Local theory of free noncommutative functions: germs, meromorphic functions, and Hermite interpolation

Free analysis is a quantization of the usual function theory much like operator space theory is a quantization of classical functional analysis. Basic objects of free analysis are noncommutative functions. These are maps on tuples of matrices of all sizes that preserve direct sums and similarities. This paper investigates the local theory of noncommutative functions. The first main result shows that for a scalar point Y Y , the ring O Y ua \mathcal {O}^{\operatorname {ua}}_Y of uniformly analytic noncommutative germs about Y Y is an integral domain and admits a universal skew field of fractions, whose elements are called meromorphic germs. A corollary is a local-global rank principle that connects ranks of matrix evaluations of a matrix A A over O Y ua \mathcal {O}^{\operatorname {ua}}_Y with the factorization of A A over O Y ua \mathcal {O}^{\operatorname {ua}}_Y . Different phenomena occur for a semisimple tuple of nonscalar matrices Y Y . Here it is shown that O Y ua \mathcal {O}^{\operatorname {ua}}_Y contains copies of the matrix algebra generated by Y Y . In particular, there exist nonzero nilpotent uniformly analytic functions defined in a neighborhood of Y Y , and O Y ua \mathcal {O}^{\operatorname {ua}}_Y does not embed into a skew field. Nevertheless, the ring O Y ua \mathcal {O}^{\operatorname {ua}}_Y is described as the completion of a free algebra with respect to the vanishing ideal at Y Y . This is a consequence of the second main result, a free Hermite interpolation theorem: if f f is a noncommutative function, then for any finite set of semisimple points and a natural number L L there exists a noncommutative polynomial that agrees with f f at the chosen points up to differentials of order L L . All the obtained results also have analogs for (nonuniformly) analytic germs and formal germs.

Interactions between Operator Space Theory and Quantum Probability with Applications to Quantum Information

Interactions between Operator Space Theory and Quantum Probability with Applications to Quantum Information

Polynomials in operator space theory: matrix ordering and algebraic aspects

We extend the $\lambda$-theory of operator spaces given by Defant and Wiesner (2014), that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach $*$-algebras. Given matrix regular operator spaces and operator systems, we introduce cones related to $\lambda$ for the algebraic tensor product that respect the matricial structure of matrix regular operator spaces and operator systems, respectively. The ideal structure of $\lambda$-tensor product of $C^*$-algebras has also been discussed.

SINGULAR INTEGRALS SUPPORTED BY SUBVARIETIES FOR VECTOR-VALUED FUNCTIONS

In this paper, we show that singular integrals supported by subvarieties are bounded on $L^{p}(\mathbb{R}^{n};\mathbf{X})$ for $1<p<\infty$ and some UMD space $\mathbf{X}$. In the terminology from operator space theory, we prove that singular integrals supported by subvarieties are completely $L^{p}$-bounded.

Quantum-Proof Randomness Extractors via Operator Space Theory

Quantum-proof randomness extractors are an important building block for classical and quantum cryptography as well as device independent randomness amplification and expansion. Furthermore they are also a useful tool in quantum Shannon theory. It is known that some extractor constructions are quantum-proof whereas others are provably not [Gavinsky et al., STOC'07]. We argue that the theory of operator spaces offers a natural framework for studying to what extent extractors are secure against quantum adversaries: we first phrase the definition of extractors as a bounded norm condition between normed spaces, and then show that the presence of quantum adversaries corresponds to a completely bounded norm condition between operator spaces. From this we show that very high min-entropy extractors as well as extractors with small output are always (approximately) quantum-proof. We also study a generalization of extractors called randomness condensers. We phrase the definition of condensers as a bounded norm condition and the definition of quantum-proof condensers as a completely bounded norm condition. Seeing condensers as bipartite graphs, we then find that the bounded norm condition corresponds to an instance of a well studied combinatorial problem, called bipartite densest subgraph. Furthermore, using the characterization in terms of operator spaces, we can associate to any condenser a Bell inequality (two-player game) such that classical and quantum strategies are in one-to-one correspondence with classical and quantum attacks on the condenser. Hence, we get for every quantum-proof condenser (which includes in particular quantum-proof extractors) a Bell inequality that can not be violated by quantum mechanics.

Mini-Workshop: Operator Spaces and Noncommutative Geometry in Interaction

In recent years, operator space theory has made remarkable appearances in noncommutative geometry, notably in the study of $C^\*$-algebras of real reductive groups and the unbounded picture of Kasparov theory. In both these developments, a central rôle is played by operator modules and the Haagerup tensor product. This workshop brought together experts in the aforementioned fields to deepen this interaction.

A Modular Framework for Quantum-Proof Randomness Extractors

A quantum-proof extractor is a function that is used to extract randomness from any weakly random source X in the presence of prior quantum information about X. It is known that some constructions are quantum-proof, such as Trevisan’s construction. However, these extractors are generally restrictive for applications on the one-bit output construction and the weak design. Here, we give a modular framework to combine multi-bit output extractors (not only one-bit) with pseudorandom transform, and show that it is sound in the presence of quantum side information. Then combined with the theory of operator spaces, we improve previous theoretical proofs, and discuss the security of two-bit output extractor by giving a tighter bound for it.

Vector-valued Hilbert transforms along curves

In this paper, we show that Hilbert transforms along some curves are bounded on $L^p({\mathbb R}^n;X)$ for some $1<p<\infty$ and some UMD spaces $X$. In particular, we prove that the Hilbert transform along some curves are completely $L^p$-bounded in the terminology from operator space theory. Moreover, we obtain the $L^p(\mathbb{R}^n;X)$-boundedness of anisotropic singular integrals by using the "method of rotations" of Calder\'{o}n-Zygmund. All these results extend the existing related ones.

Survey on nonlocal games and operator space theory

This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states.

Unbounded Violation of Quantum Steering Inequalities.

We construct steering inequalities that exhibit unbounded violation. The concept was to exploit the relationship between steering violation and the uncertainty relation. To this end, we apply mutually unbiased bases and anticommuting observables, known to exhibit the strongest uncertainty. In both cases, we are able to procure unbounded violations. Our approach is much more constructive and transparent than the operator space theory approach employed to obtain large violation of Bell inequalities. Importantly, using anticommuting observables we are able to obtain a dichotomic steering inequality with unbounded violation. Thus far, there is no analogous result for Bell inequalities. Interestingly, both the dichotomic inequality and one of our inequalities cannot be directly obtained from existing uncertainty relations, which strongly suggest the existence of an unknown kind of uncertainty relation.