Noncommutative Functions Research Articles

Noncommutative Schur functions for posets

The machinery of noncommutative Schur functions is a general approach to Schur positivity of symmetric functions initiated by Fomin-Greene. Hwang recently adapted this theory to posets to give a new approach to the Stanley-Stembridge conjecture. We further develop this theory to prove that the symmetric function associated to any P-Knuth equivalence graph is Schur positive. This settles a conjecture of Kim and the third author, and refines results of Gasharov, Shareshian-Wachs, and Hwang on the Schur positivity of chromatic symmetric functions.

Non-commutative Function Theory and Free Probability

The workshop brought together researchers in two fields, non-commutative function theory and free probability. There was a mini-course in each of these areas, and speakers in all the talks were encouraged to give expository talks that illuminated the broader reaches of their fields.

Quantitative mean ergodic inequalities: Power bounded operators acting on one single noncommutative Lp space

In this paper, we establish the quantitative mean ergodic theorems for two subclasses of power bounded operators on a fixed noncommutative Lp space with 1<p<∞, which mainly concerns power bounded invertible operators and Lamperti contractions. Our approach to the quantitative ergodic theorems relies on noncommutative square function inequalities. The establishment of the latter involves several new ingredients such as the almost orthogonality and Calderón-Zygmund arguments for non-smooth kernels from semi-commutative harmonic analysis, the extension properties of the operators under consideration from operator theory, and a noncommutative version of the classical transference method due to Coifman and Weiss.

Tensor algebras of subproduct systems and noncommutative function theory

AbstractWe revisit tensor algebras of subproduct systems with Hilbert space fibers, resolving some open questions in the case of infinite-dimensional fibers. We characterize when a tensor algebra can be identified as the algebra of uniformly continuous noncommutative functions on a noncommutative homogeneous variety or, equivalently, when it is residually finite-dimensional: this happens precisely when the closed homogeneous ideal associated with the subproduct system satisfies a Nullstellensatz with respect to the algebra of uniformly continuous noncommutative functions on the noncommutative closed unit ball. We show that – in contrast to the finite-dimensional case – in the case of infinite-dimensional fibers, this Nullstellensatz may fail. Finally, we also resolve the isomorphism problem for tensor algebras of subproduct systems: two such tensor algebras are (isometrically) isomorphic if and only if their subproduct systems are isomorphic in an appropriate sense.

ADRA: Extending Digital Computing-In-Memory With Asymmetric Dual-Row-Activation

Computing in-memory (CiM) has emerged as an attractive technique to mitigate the von-Neumann bottleneck. Current digital CiM approaches for in-memory operands are based on multi-wordline assertion for computing bit-wise Boolean functions and arithmetic functions such as addition. However, most of these techniques, due to the many-to-one mapping of input vectors to bitline voltages, are limited to CiM of commutative functions, leaving out an important class of computations such as subtraction. In this paper, we propose a CiM approach, which solves the mapping problem through an asymmetric wordline biasing scheme, enabling (a) simultaneous single-cycle memory read and CiM of primitive Boolean functions (b) computation of any Boolean function and (c) CiM of non-commutative functions such as subtraction and comparison. While the proposed technique is technology-agnostic, we show its utility for ferroelectric transistor (FeFET)-based non-volatile memory. Compared to the standard near-memory methods (which require two full memory accesses per operation), we show that our method can achieve a full scale two-operand digital CiM using just one memory access, leading to a 23.2% -72.6% decrease in energy-delay product (EDP).

Sheaves of Noncommutative Smooth and Holomorphic Functions Associated with the Non-Abelian Two-Dimensional Lie Algebra

It was shown in papers of Dosi and in a recent article of the author that there is a sheaf of Frechet-Arens-Michael algebras (locally solvable in the complex case and of polynomial growth in the real) on the character space of a nilpotent Lie algebra. For the Lie algebra of the group of affine transformations of the line (the simplest non-nilpotent solvable Lie algebra) we construct analogous sheaves of non-commutative smooth and non-commutative holomorphic functions on a special set of representations.

On the analytic structure of second-order non-commutative probability spaces and functions of bounded Fréchet variation

In this paper, we propose a new approach to the central limit theorem (CLT) based on functions of bounded Fréchet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on a weaker form of a large deviation principle for the operator norm; a Poincaré-type inequality for the linear statistics; and the existence of a second-order limit distribution. This approach frames into a single setting many known random matrix ensembles and as a consequence, classical central limit theorems for linear statistics are recovered and new ones are established, e.g. the CLT for the continuously differentiable linear statistics of block Gaussian matrices. In addition, our main results contribute to the understanding of the analytical structure of second-order non-commutative probability spaces. On the one hand, they pinpoint the source of the unbounded nature of the bilinear functional associated to these spaces; on the other hand, they lead to a general archetype for the integral representation of the second-order Cauchy transform, [Formula: see text]. Furthermore, we establish that the covariance of resolvents converges to this transform and that the limiting covariance of analytic linear statistics can be expressed as a contour integral in [Formula: see text].

Schur functions in noncommuting variables

In 2004 Rosas and Sagan asked whether there was a way to define a basis in the algebra of symmetric functions in noncommuting variables, NCSym, having properties analogous to the classical Schur functions. This was because they had constructed a partial such set that was not a basis. We answer their question by defining Schur functions in noncommuting variables using a noncommutative analogue of the Jacobi-Trudi determinant. Our Schur functions in NCSym map to classical Schur functions under commutation, and a subset of them indexed by set partitions forms a basis for NCSym. Amongst other properties, Schur functions in NCSym also satisfy a noncommutative analogue of the product rule for classical Schur functions in terms of skew Schur functions.We also show how Schur functions in NCSym are related to Specht modules, and naturally refine the Rosas-Sagan Schur functions. Moreover, by generalizing Rosas-Sagan Schur functions to skew Schur functions in the natural way, we prove noncommutative analogues of the Littlewood-Richardson rule and coproduct rule for them. Finally, we relate our functions to noncommutative symmetric functions by proving a subset of our functions are natural extensions of noncommutative ribbon Schur functions, and immaculate functions indexed by integer partitions.

Operator noncommutative functions

AbstractWe establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g., $B({\mathcal H})$ . In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the $k{\mathrm {th}}$ directional derivative of any NC function at a scalar point is a k-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.

Noncommutative Weil conjecture

In this article, following an insight of Kontsevich, we extend the Weil conjecture, as well as the strong form of the Tate conjecture, from the realm of algebraic geometry to the broad noncommutative setting of dg categories. Moreover, we establish a functional equation for the noncommutative Hasse-Weil zeta functions, compute the l-adic and p-adic absolute values of the eigenvalues of the cyclotomic Frobenius, and provide a complete description of the category of noncommutative numerical motives in terms of Weil q-numbers. As a first application, we prove the noncommutative Weil conjecture and the noncommutative strong form of the Tate conjecture in several cases: twisted schemes, Calabi-Yau dg categories associated to hypersurfaces, noncommutative gluings of schemes, root stacks, (twisted) global orbifolds, and finite-dimensional dg algebras. As a second application, we provide an alternative noncommutative proof of the Weil conjecture and of the strong form of the Tate conjecture in the particular cases of intersections of two quadrics and linear sections of determinantal varieties.

Free-fermions and skew stable Grothendieck polynomials

We present a free-fermionic presentation of the skew (dual) stable Grothendieck polynomials. A direct proof of their determinantal formulas is given from this presentation. We also introduce a combinatorial method to describe the multiplication map and its adjoint over the space of skew (dual) stable Grothendieck polynomials. This calculation requires the use of noncommutative supersymmetric Schur functions.

Tracial smooth functions of non-commuting variables and the free Wasserstein manifold

Using new spaces of tracial non-commutative smooth functions, we formulate a free probabilistic analog of the Wasserstein manifold on $\mathbb{R}^d$ (the formal Riemannian manifold of smooth probability densities on $\mathbb{R}^d$), and we use it to s

Functions of class $C^\infty$ in non-commuting variables in the context of triangular Lie algebras

We construct a certain completion $C^\infty_\mathfrak{g}$ of the universal enveloping algebra of a triangular real Lie algebra $\mathfrak{g}$. It is a Fréchet-Arens-Michael algebra that consists of elements of polynomial growth and satisfies to the following universal property: every Lie algebra homomorphism from $\mathfrak{g}$ to a real Banach algebra all of whose elements are of polynomial growth has an extension to a continuous homomorphism with domain $C^\infty_\mathfrak{g}$. Elements of this algebra can be called functions of class $C^\infty$ in non-commuting variables. The proof is based on representation theory and employs an ordered $C^\infty$-functional calculus. Beyond the general case, we analyze two simple examples. As an auxiliary material, the basics of the general theory of algebras of polynomial growth are developed. We also consider local variants of the completion and obtain a sheaf of non-commutative functions on the Gelfand spectrum of $C^\infty_\mathfrak{g}$ in the case when $\mathfrak{g}$ is nilpotent. In addition, we discuss the theory of holomorphic functions in non-commuting variables introduced by Dosi and apply our methods to prove theorems strengthening some his results.

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a ‘noncommutative Hoffman-Rossi theorem’ giving the existence of weak* continuous ‘noncommutative representing measures’ for so-called D-characters. These results may also be viewed as ‘module’ Hahn-Banach extension theorems for weak* continuous ‘characters’ into possibly noninjective von Neumann algebras. In closing we introduce the notion of ‘noncommutative Jensen measures’, and show that as in the classical case representing measures of logmodular algebras are Jensen measures. The proofs of the two main cycles of results rely on the delicate interplay of Tomita-Takesaki theory, noncommutative Radon-Nikodym derivatives, Connes cocycles, Haagerup noncommutative Lp-spaces, Haagerup's reduction theorem, etc.

Noncommutative partially convex rational functions

Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. A strengthening of partial convexity arising in connection with BMIs – xy -convexity – is also considered. A characterization of xy -convex polynomials is given.

Jensen’s Inequality for Separately Convex Noncommutative Functions

Abstract Classically, Jensen’s Inequality asserts that if $X$ is a compact convex set, and $f:K\to {\mathbb {R}}$ is a convex function, then for any probability measure $\mu $ on $K$, that $f(\text {bar}(\mu ))\le \int f\; \text {d}\mu $, where $\text {bar}(\mu )$ is the barycenter of $\mu $. Recently, Davidson and Kennedy proved a noncommutative (“nc”) version of Jensen’s inequality that applies to nc convex functions, which take matrix values, with probability measures replaced by ucp maps. In the classical case, if $f$ is only a separately convex function, then $f$ still satisfies the Jensen inequality for any probability measure that is a product measure. We prove a noncommutative Jensen inequality for functions that are separately nc convex in each variable. The inequality holds for a large class of ucp maps that satisfy a noncommutative analogue of Fubini’s theorem. This class of ucp maps includes any free product of ucp maps built from Boca’s theorem, or any ucp map that is conditionally free in the free-probabilistic sense of Młotkowski. As an application to free probability, we obtain some operator inequalities for conditionally free ucp maps applied to free semicircular families.

Two-Dimensional Pauli Equation in Noncommutative Phase-Space

We study the Pauli equation in a two-dimensional noncommutative phase-space by considering a constant magnetic field perpendicular to the plane. The noncommutative problem is related to the equivalent commutative one through a set of two-dimensional Bopp-shift transformations. The energy spectrum and the wave function of the two-dimensional noncommutative Pauli equation are found, where the problem in question has been mapped to the Landau problem. In the classical limit, we have derived the noncommutative semiclassical partition function for one- and N- particle systems. The thermodynamic properties such as the Helmholtz free energy, mean energy, specific heat and entropy in noncommutative and commutative phasespaces are determined. The impact of the phase-space noncommutativity on the Pauli system is successfully examined.

Sharp Estimates of Noncommutative Bochner–Riesz Means on Two-Dimensional Quantum Tori

In this paper, we establish the full $$L_p$$ boundedness of noncommutative Bochner–Riesz means on two-dimensional quantum tori, which completely resolves an open problem raised in Chen et al. (Commun Math Phys 322(3):755–805, 2013) in the sense of the $$L_p$$ convergence for two dimensions. The main ingredients are sharp estimates of noncommutative Kakeya maximal functions and geometric estimates in the plane. We make the most of noncommutative theories of maximal/square functions, together with microlocal decompositions in both proofs of sharper estimates of Kakeya maximal functions and Bochner–Riesz means.

Submodules in Polydomains and Noncommutative Varieties

Tensor product of Fock spaces is analogous to the Hardy space over the unit polydisc. This plays an important role in the development of noncommutative operator theory and function theory in the sense of noncommutative polydomains and noncommutative varieties. In this paper we study joint invariant subspaces of tensor product of full Fock spaces and noncommutative varieties. We also obtain, in particular, by using techniques of noncommutative varieties, a classification of joint invariant subspaces of n-fold tensor products of Drury–Arveson spaces.

Blaschke–singular–outer factorization of free non-commutative functions

By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner–outer factorization. Here, a bounded analytic function is called inner or outer if multiplication by this function defines an isometry or has dense range, respectively, as a linear operator on the Hardy Space, H2, of analytic functions in the complex unit disk with square-summable Taylor series. This factorization can be further refined; any inner function θ decomposes uniquely as the product of a Blaschke inner function and a singular inner function, where the Blaschke inner contains all the vanishing information of θ, and the singular inner factor has no zeroes in the unit disk.We prove an exact analogue of this factorization in the context of the full Fock space, identified as the Non-commutative Hardy Space of analytic functions defined in a certain multi-variable non-commutative open unit ball.