Noncommutative Analogue Research Articles

Topological Grading of Semigroup C*-Algebras

The paper deals with the abelian cancellative semigroups and the reduced semigroup C*-algebras. It is supposed that there exist epimorphisms from the semigroups onto the group of integers modulo n. For these semigroups we study the structure of the reduced semigroup C*-algebras which are also called the Toeplitz algebras. Such a C*-algebra can be defined for any non-abelian left cancellative semigroup. It is a very natural object in the category of C*-algebras because this algebra is generated by the left regular representation of a semigroup. In the paper, by a given epimorphism σ we construct the grading of a semigroup C*-algebra. To this aim the notion of the σ-index of a monomial is introduced. This notion is the main tool in the construction of the grading. We make use of the σ-index to define the linear independent closed subspaces in the semigroup C*-algebra. These subspaces constitute the C*-algebraic bundle, or the Fell bundle, over the group of integers modulo n. Moreover, it is shown that this grading of the reduced semigroup C*-algebra is topological. As a corollary, we obtain the existence of the contractive linear operators that are non-commutative analogs of the Fourier coefficients. Using these operators, we prove the result on the geometry of the underlying Banach space of the semigroup C*-algebra

On quasisymmetric power sums

In the 1995 paper entitled “Noncommutative symmetric functions”, Gelfand et al. defined two noncommutative symmetric function analogues for the power sum basis of the symmetric functions. This paper explores the combinatorial properties of their duals, two distinct quasisymmetric power sum bases. In contrast to the symmetric power sums, the quasisymmetric power sums have a more complex combinatorial description. This paper offers a first detailed exploration of these two relatively unstudied quasisymmetric bases, in which we show that they refine the classical symmetric power sum basis, we give transition matrices to other well-understood bases, and we provide explicit formulas for products of quasisymmetric power sums.

Constrained characteristic functions, multivariable interpolation, and invariant subspaces

In this paper, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety mathcal{V}_{f,varphi,mathcal{I}}(mathcal{H}) in terms of constrained characteristic functions. As an application, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements, which can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions. On the other hand, we provide a Sarason-type commutant lifting theorem. Applying this result, we solve the Nevanlinna–Pick-type interpolation problem in our setting. Moreover, we also obtain a Beurling-type characterization of the joint invariant subspaces under the operators B_{1},ldots,B_{n}, where the n-tuple (B_{1},ldots,B_{n}) is the universal model associated with the abstract noncommutative variety mathcal{V}_{f,varphi,mathcal{I}}.

The kernel of chromatic quasisymmetric functions on graphs and hypergraphic polytopes

The chromatic symmetric function on graphs is a celebrated graph invariant. Analogous chromatic maps can be defined on other objects, as presented by Aguiar, Bergeron and Sottile. The problem of identifying the kernel of some of these maps was addressed by Féray, for the Gessel quasisymmetric function on posets.On graphs, we show that the modular relations and isomorphism relations span the kernel of the chromatic symmetric function. This helps us to construct a new invariant on graphs, which may be helpful in the context of the tree conjecture. We also address the kernel problem in the Hopf algebra of generalized permutahedra, introduced by Aguiar and Ardila. We present a solution to the kernel problem on the Hopf algebra spanned by hypergraphic polytopes, which is a subfamily of generalized permutahedra that contains a number of polytope families.Finally, we consider the non-commutative analogues of these quasisymmetric invariants, and establish that the word quasisymmetric functions, also called non-commutative quasisymmetric functions, form the terminal object in the category of combinatorial Hopf monoids. As a corollary, we show that there is no combinatorial Hopf monoid morphism between the combinatorial Hopf monoid of posets and that of hypergraphic polytopes.

Schatten Class Operators in a Representation Space of Canonical Commutation Relations

The Schatten classes $${\mathfrak{S}_p}$$ of operators in a Hilbert space—a noncommutative analog of the classical lp spaces—play a significant role in recent studies on quantum information theory. In the present paper we give sufficient conditions for an operator to belong to $${\mathfrak{S}_p}$$ in the case where the Hilbert space is the space of an irreducible representation of the canonical commutation relations.

Quantum groups with projection and extensions of locally compact quantum groups

The main result of the paper is the characterization of those locally compact quantum groups with projection, i.e. non-commutative analogs of semidirect products, which are extensions as defined by L. Vainerman and S. Vaes. It turns out that quantum groups with projection are usually not extensions.We discuss several examples including the quantum \mathrm{U}_q(2) . The major tool used to obtain these results is the co-duality for coideals in algebras of functions on locally compact quantum groups and the concept of a normal coideal.

Some noncommutative minimal surfaces

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class.In this paper we show that the generic noncommutative projective plane (corresponding to the three dimensional Sklyanin algebra) as well as noncommutative analogues of P1×P1 and the more general Van den Bergh quadrics satisfy very strong minimality conditions. Translated into an algebraic question, where one is interested in a maximality condition, we prove the following result.Theorem A:Let R be a Sklyanin algebra or a Van den Bergh quadric that is infinite dimensional over its centre and letA⊇Rbe any connected graded noetherian maximal order, with the same graded quotient ring as R. Then, up to taking Veronese rings, A is isomorphic to R.Let T be an elliptic algebra (that is, the coordinate ring of a noncommutative surface containing an elliptic curve). Then, under an appropriate homological condition, we prove that every connected graded noetherian overring of T is obtained by blowing down finitely many lines (line modules) of self-intersection(−1).

Cyclic inclusion-exclusion and the kernel of P -partitions

Following the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing functions on the graph (or of P -partitions of the poset).We describe the kernel of this linear map, using a simple combinatorial operation that we call cyclic inclusion- exclusion. Our result also holds for the natural non-commutative analog and for the commutative and non-commutative restrictions to bipartite graphs.

Notes on derivations of Murray–von Neumann algebras

Let M be a type II1 von Neumann factor and let S(M) be the associated Murray-von Neumann algebra of all measurable operators affiliated to M. We extend a result of Kadison and Liu [30] by showing that any derivation from S(M) into an M-bimodule B⊊S(M) is trivial. In the special case, when M is the hyperfinite type II1-factor R, we introduce the algebra AD(R), a noncommutative analogue of the algebra of all almost everywhere approximately differentiable functions on [0,1] and show that it is a proper subalgebra of S(R). This algebra is strictly larger than the corresponding ring of continuous geometry introduced by von Neumann. Further, we establish that the classical approximate derivative on (classes of) Lebesgue measurable functions on [0,1] admits an extension to a derivation from AD(R) into S(R), which fails to be spatial. Finally, we show that for a Cartan masa A in a hyperfinite II1-factor R there exists a derivation δ from A into S(A) which does not admit an extension up to a derivation from R to S(R).

Derivations and reflection positivity on the quantum cylinder

We describe the general structure of unbounded derivations in the quantum cylinder. We prove a noncommutative analog of reflection positivity for Laplace-type operators in a noncommutative cylinder following the ideas of the proof by Jaffe and Ritter (2008) of reflection positivity for Laplace operators on manifolds equipped with a reflection.

Standard conjecture D for matrix factorizations

We prove the non-commutative analogue of Grothendieck's Standard Conjecture D for the dg-category of matrix factorizations of an isolated hypersurface singularity in characteristic 0. Along the way, we show the Euler pairing for such dg-categories of matrix factorizations is positive semi-definite.

The Haemers Bound of Noncommutative Graphs

We continue the study of the quantum channel version of Shannon’s zero-error capacity problem. We generalize the celebrated Haemers bound to noncommutative graphs (obtained from quantum channels). We prove basic properties of this bound, such as additivity under the direct sum and submultiplicativity under the tensor product. The Haemers bound upper bounds the Shannon capacity of noncommutative graphs, and we show that it can outperform other known upper bounds, including noncommutative analogues of the Lovasz theta function (Duan–Severini–Winter, IEEE Trans. Inform. Theory , 2013 and Boreland–Todorov–Winter, arXiv preprint , 2019).

Noncommutative strong maximals and almost uniform convergence in several directions

AbstractOur first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the$L_p$-norm of the$\limsup $of a sequence of operators as a localized version of a$\ell _\infty /c_0$-valued$L_p$-space. In particular, our main result gives a strong$L_1$-estimate for the$\limsup $—as opposed to the usual weak$L_{1,\infty }$-estimate for the$\mathop {\mathrm {sup}}\limits $—with interesting consequences for the free group algebra.Let$\mathcal{L} \mathbf{F} _2$denote the free group algebra with$2$generators, and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside$L_1(\mathcal{L} \mathbf{F} _2)$for which the free Poisson semigroup converges to the initial data. Currently, the best known result is$L \log ^2 L(\mathcal{L} \mathbf{F} _2)$. We improve this result by adding to it the operators in$L_1(\mathcal{L} \mathbf{F} _2)$spanned by words without signs changes. Contrary to other related results in the literature, this set grows exponentially with length. The proof relies on our estimates for the noncommutative$\limsup $together with new transference techniques.We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak$(\Phi ,\Phi )$inequality—as opposed to weak$(\Phi ,1)$—for noncommutative multiparametric martingales and$\Phi (s) = s (1 + \log _+ s)^{2 + \varepsilon }$. This logarithmic power is an$\varepsilon $-perturbation of the expected optimal one. The proof combines a refinement of Cuculescu’s construction with a quantum probabilistic interpretation of M. de Guzmán’s original argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu’s projections.

Uniformity in $C^*$-algebras

We introduce a notion of a uniform structure on the set of all representations of a given separable, not necessarilly commutative $C^*$-algebra $\mathfrak{A}$ by introducing a suitable family of metrics on the set of representations of $\mathfrak{A}$ and investigate its properties. We define the noncommutative analogue of the notion of the modulus of continuity of an element in $C^*$-algebra and we establish its basic properties. We also deal with morphisms of $C^*$-algebras by defining two notions of uniform continuity and show their equivalence.

Hilbert’s 17th problem in free skew fields

Abstract This paper solves the rational noncommutative analogue of Hilbert’s 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of Hermitian matrices in its domain, then it is a sum of Hermitian squares of noncommutative rational functions. This result is a generalisation and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without Hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every matricial solution of a linear matrix inequality $L\succeq 0$ if and only if it belongs to the rational quadratic module generated by L. The essential intermediate step toward this Positivstellensatz for functions with singularities is an extension theorem for invertible evaluations of linear matrix pencils.

Howe duality for quantum queer superalgebras

We establish a new Howe duality between a pair of quantum queer superalgebras (Uq−1(qn), Uq(qm)). The key ingredient is the construction of a non-commutative analogue Aq(qn,qm) of the symmetric superalgebra S(Cmn|mn) with the use of quantum coordinate queer superalgebra. It turns out that this superalgebra is equipped with a Uq−1(qn)⊗Uq(qm)-supermodule structure that admits a multiplicity-free decomposition. We also show that the (Uq−1(qn),Uq(qm))-Howe duality implies the Sergeev-Olshanski duality.

Derivations on Murray–von Neumann algebras

Let $\mathcal{M}$ be a type II$_1$ von Neumann factor and let $S(\mathcal{M})$ be the associated Murray-von Neumann algebra of all measurable operators affiliated to $\mathcal{M}.$ We extend a result of Kadison and Liu \cite{KL} by showing that any derivation from $S(\mathcal{M})$ into an $\mathcal{M}$-bimodule $\mathcal{B}\subsetneq S(\mathcal{M})$ is trivial. In the special case, when $\mathcal{M}$ is the hyperfinite type II$_1-$factor $\mathcal{R}$, we introduce the algebra $AD(\mathcal{R})$, a noncommutative analogue of the algebra of all almost everywhere approximately differentiable functions on $[0,1]$ and show that it is a proper subalgebra of $S(\mathcal{R})$. This algebra is strictly larger than the corresponding ring of continuous geometry introduced by von Neumann. Further, we establish that the classical approximate derivative on (classes of) Lebesgue measurable functions on $[0,1]$ admits an extension to a derivation from $AD(\mathcal{R})$ into $S(\mathcal{R})$, which fails to be spatial. Finally, we show that for a Cartan masa $\mathcal{A}$ in a hyperfinite II$_1-$factor $\mathcal{R}$ there exists a derivation $\delta$ from $\mathcal{A}$ into $S(\mathcal{A})$ which does not admit an extension up to a derivation from $\mathcal{R}$ to $S(\mathcal{R}).$

Noncommutative Knörrer periodicity and noncommutative Kleinian singularities

We establish a version of Knörrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let A be a left noetherian AS-regular algebra, let f be a normal and regular element of A of positive degree, and take B=A/(f). Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over B and those over (a noncommutative analog of) its second double branched cover (B#)#. Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with Cassidy. These results are applied to the noncommutative Kleinian singularities studied by the second and fourth authors with Chan and Zhang.

Strong differential subordinates for noncommutative submartingales

We introduce a notion of strong differential subordination of noncommutative semimartingales, extending Burkholder’s definition from the classical case (Ann. Probab. 22 (1994) 995–1025). Then we establish the maximal weak-type $(1,1)$ inequality under the additional assumption that the dominating process is a submartingale. The proof rests on a significant extension of the maximal weak-type estimate of Cuculescu and a Gundy-type decomposition of an arbitrary noncommutative submartingale. We also show the corresponding strong-type $(p,p)$ estimate for $1<p<\infty $ under the assumption that the dominating process is a nonnegative submartingale. This is accomplished by combining several techniques, including interpolation-flavor method, Doob–Meyer decomposition and noncommutative analogue of good-$\lambda$ inequalities.

Column extreme multipliers of the Free Hardy Space

The full Fock space over $\mathbb C ^d$ can be identified with the free Hardy space, $H^2 (\mathbb B ^d _\mathbb N)$ - the unique non-commutative reproducing kernel Hilbert space corresponding to a non-commutative Szeg\"{o} kernel on the non-commutative, multi-variable open unit ball $\mathbb B ^d _\mathbb N := \bigsqcup _{n=1} ^\infty \left( \mathbb C^{n\times n} \otimes \mathbb C ^d \right) _1$. Elements of this space are free or non-commutative functions on $\mathbb B ^d _\mathbb N$. Under this identification, the full Fock space is the canonical non-commutative and several-variable analogue of the classical Hardy space of the disk, and many classical function theory results have faithful extensions to this setting. In particular to each contractive (free) multiplier $B$ of the free Hardy space, we associate a Hilbert space $\mathcal H(B)$ analogous to the deBranges-Rovnyak spaces in the unit disk, and consider the ways in which various properties of the free function $B$ are reflected in the Hilbert space $\mathcal H(B)$ and the operators which act on it. In the classical setting, the $\mathcal H(b)$ spaces of analytic functions on the disk display strikingly different behavior depending on whether or not the function $b$ is an extreme point in the unit ball of $H^\infty(\mathbb D)$. We show that such a dichotomy persists in the free case, where the split depends on whtether or not $B$ is what we call {\it column extreme}.