Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P².The generic noncommutative plane corresponds to the Sklyanin algebra S=Skl(E,σ) constructed from an automorphism σ of infinite order on an elliptic curve E⊂P². In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1−n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P²?E.     

Measures from Dixmier traces and zeta functions

For L^∞-functions on a (closed) compact Riemannian manifold, the noncommutative residue and the Dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the Lebesgue integral. The identifications are shown to continue to, and be sharp at, L²-functions. For functions strictly in Lp, 1?p<2, symmetrised noncommutative residue and Dixmier trace formulas must be introduced, for which the identification is shown to continue for the noncommutative residue. However, a failure is shown for the Dixmier trace formulation at L¹-functions. It is shown the noncommutative residue remains finite and recovers the Lebesgue integral for any integrable function while the Dixmier trace expression can diverge. The results show that a claim in the monograph [J.M. Gracia-Bondía, J.C. Várilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser Adv. Texts, Birkhäuser, Boston, 2001], that the equality on C^∞-functions between the Lebesgue integral and an operator-theoretic expression involving a Dixmier trace (obtained from Connes' Trace Theorem) can be extended to any integrable function, is false. The results of this paper include a general presentation for finitely generated von Neumann algebras of commuting bounded operators, including a bounded Borel or L^∞ functional calculus version of C^∞ results in IV.2.δ of [A. Connes, Noncommutative Geometry, Academic Press, New York, 1994].     

Noncommutative correction to the Cornell potential in heavy-quarkonium atoms

We investigate the effect of space–time noncommutativity on the Cornell potential in heavy-quarkonium systems. It is known that the space–time noncommutativity can create bound states, and we therefore consider the noncommutative geometry of the space–time as a correction in quarkonium models. Furthermore, we take the experimental hyperfine measurements of the bottomium ground state as an upper limit on the noncommutative energy correction and derive the maximum possible value of the noncommutative parameter θ, obtaining θ ≤ 37.94 · 10^?34 m². Finally, we use our model to calculate the maximum value of the noncommutative energy correction for energy levels of charmonium and bottomium in 1S and 2S levels. The energy correction as a binding effect in quarkonium system is smaller for charmonium than for bottomium, as expected.     

Quantum extensions of semigroups generated by Bessel processes

We construct a quantum extension of the Markov semigroup of the classical Bessel process of orderv≥1 to the noncommutative von Neumann algebra ß(L ²(0, +∞)) of bounded operators onL ²(0, +∞).     

Noncommutative cyclic covers and maximal orders on surfaces

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi-Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer-Severi varieties and, in the quaternion case, we show how to obtain some Brauer-Severi varieties from G-Hilbert schemes of P¹-bundles.     

Nonseparable Banach algebras whose squares are pathological

There exists a nonseparable noncommutative Banach algebra A such that A² is nonclosed and of finite codimension in A. There exists a similar A with A² = A such that the natural mapping of the algebraic tensor product A ? A into A is not open, and there is no bound on the number of summands needed to express an element as an element of A².     

Noncommutative classical invariant theory

In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspace? _d ^m of the noncommutative invariant algebra? _d consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S _m . We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(? _d,t) whose coefficients are the number of irreducibleS _m -modules in the decomposition of? _d ^m , and show that it is rational. Finally, we develop some analogues of all this for covariants.     

Classification of Pointed Hopf Algebras of Dimension p 2 over any Algebraically Closed Field

Let p be a prime. We complete the classification of pointed Hopf algebras of dimension p ² over an algebraically closed field k. When char k?≠?p, our result is the same as the well-known result for char k?=?0. When char k?=?p, we obtain 14 types of pointed Hopf algebras of dimension p ², including a unique noncommutative and noncocommutative type.     

Sufficiency and strong commutants in quantum probability theory

A probability algebra (A, *, ω) consisting of a^*algebraA with a faithful state ω provides a framework for an unbounded noncommutative probability theory. A characterization of symmetric probability algebra is obtained in terms of an unbounded strong commutant of the left regular representation ofA. Existence of coarse-graining is established for states that are absolutely continuous or continuous in the induced topology. Sufficiency of a^*subalgebra relative to a family of states is discussed in terms of noncommutative Radon-Nikodym derivatives (a form of Halmos-Savage theorem), and is applied to couple of examples (including the canonical algebra of one degree of freedom for Heisenberg commutation relation) to obtain unbounded analogues of sufficiency results known in probability theory over a von Neumann algebra.     

The Eilenberg-Watts theorem over schemes

We study obstructions to a direct limit preserving right exact functor F between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, or if F is exact, all obstructions vanish and we recover the Eilenberg-Watts Theorem. This result is crucial to the proof that the noncommutative Hirzebruch surfaces constructed in C. Ingalls, D. Patrick (2002) [6] are noncommutative P¹-bundles in the sense of M. Van den Bergh [10].     

Local and 2-local derivations on noncommutative Arens algebras

The paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra L ^ω(M,τ) associated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every 2-local derivation on L ^ω(M,τ) is a spatial derivation, and if M is a finite von Neumann algebra, then each local derivation on L ^ω(M,τ) is also a spatial derivation and every 2-local derivation on M is in fact an inner derivation.     

Local index theory over foliation groupoids

We give a local proof of an index theorem for a Dirac-type operator that is invariant with respect to the action of a foliation groupoid G. If M denotes the space of units of G then the input is a G-equivariant fiber bundle P→M along with a G-invariant fiberwise Dirac-type operator D on P. The index theorem is a formula for the pairing of the index of D, as an element of a certain K-theory group, with a closed graded trace on a certain noncommutative de Rham algebra Ω^*B associated to G. The proof is by means of superconnections in the framework of noncommutative geometry.     

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let M be aσ-finite von Neumann algebra and let AM be a maximal subdiagonal algebra with respect to a faithful normal conditional expectationΦ.Based on the Haagerup’s noncommutative Lpspace Lp(M)associated with M,we consider Toeplitz operators and the Hilbert transform associated with A.We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space H2(M)is just the right analytic Toeplitz algebra.Furthermore,the Hilbert transform on noncommutative Lp(M)is shown to be bounded for 1p∞.As an application,we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative H1(M)as a concrete space of operators.     

A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming

A method for computing global minima of real multivariate polynomials based on semidefinite programming was developed by N.Z. Shor, J.B. Lasserre and P.A. Parrilo. The aim of this article is to extend a variant of their method to noncommutative symmetric polynomials in variables X and Y satisfying YX−XY=1 and X^*=X, Y^*=−Y. Global minima of such polynomials are defined and showed to be equal to minima of the spectra of the corresponding differential operators. We also discuss how to exploit sparsity and symmetry. Several numerical experiments are included. The last section explains how our theory fits into the framework of noncommutative real algebraic geometry.     

Generalized Weyl correspondence and Moyal multiplier algebras

We show that the Weyl correspondence and the concept of a Moyal multiplier can be naturally extended to generalized function classes that are larger than the class of tempered distributions. This generalization is motivated by possible applications to noncommutative quantum field theory. We prove that under reasonable restrictions on the test function space E ? L², any operator in L² with a domain E and continuous in the topologies of E and L² has a Weyl symbol, which is defined as a generalized function on the Wigner-Moyal transform of the projective tensor square of E. We also give an exact characterization of the Weyl transforms of the Moyal multipliers for the Gel??fand-Shilov spaces S _?? ^?? .     

Freeness of linear and quadratic forms in von Neumann algebras

We characterize the semicircular distribution by freeness of linear and quadratic forms in noncommutative random variables from tracial W^?-probability spaces with relaxed moment conditions.     

Natural examples of Valdivia compact spaces

We collect examples of Valdivia compact spaces, their continuous images and associated classes of Banach spaces which appear naturally in various branches of mathematics. We focus on topological constructions generating Valdivia compact spaces, linearly ordered compact spaces, compact groups, L¹ spaces, Banach lattices and noncommutative L¹ spaces.     

Calcul fonctionnel et fonctions carrées dans les espaces Lp non commutatifs

We introduce suitable square functions for sectorial operators on noncommutative L^p-spaces, and we investigate their relationships with H^∞ functional calculus. To cite this article: M. Junge et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).