Operator algebras of functions

We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite-dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.     

Weight-dependent commutation relations and combinatorial identities

We derive combinatorial identities for variables satisfying specific systems of commutation relations, in particular elliptic commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and $y$ satisfying $yx=qxy$. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we derive results for an elliptic derivative of elliptic commuting variables, and finally study weight-dependent extensions of the Weyl algebra which we connect to rook theory.     

Operator algebras with contractive approximate identities

We give several applications of a recent theorem of the second author, which solved a conjecture of the first author with Hay and Neal, concerning contractive approximate identities; and another of Hay from the theory of noncommutative peak sets, thereby putting the latter theory on a much firmer foundation. From this theorem it emerges there is a surprising amount of positivity present in any operator algebras with contractive approximate identity. We exploit this to generalize several results previously available only for C^?-algebras, and we give many other applications.     

Generic algebras with quadratic commutation relations

Generic algebras with m3 generators and quadratic relations are completely described. A conjecture of A. M. Vershik about their finite-dimensionality is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 176–180, 1986.     

Operator synthesis. I. Synthetic sets, bilattices and tensor algebras

The interplay between the invariant subspace theory and spectral synthesis for locally compact abelian group discovered by Arveson (Ann. of Math. (2) 100 (1974) 433) is extended to include other topics as harmonic analysis for Varopoulos algebras and approximation by projection-valued measures. We propose a “coordinate” approach which nevertheless does not use the technique of pseudo-integral operators, as well as a coordinate free one which allows to extend to non-separable spaces some important results and constructions of Arveson. We solve some problems posed in Arveson (1974).     

Nichols algebras of group type with many quadratic relations

We classify Nichols algebras of irreducible Yetter–Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known finite-dimensional Nichols algebras of nonabelian group type appear in the result of our classification. We find a new finite-dimensional Nichols algebra over fields of characteristic two.     

Operator algebras associated with polymorphisms

A short survey of the problems and developments in the theory of operator algebras associated with semigroup dynamical systems is presented. The main part is an exposition of results concerning the algebras generated by multivalued transformations (polymorphisms). Bibliography: 14 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 23–27.     

Jordan zero-product preserving additive maps on operator algebras

Let Φ:A→B be an additive surjective map between some operator algebras such that AB+BA=0 implies Φ(A)Φ(B)+Φ(B)Φ(A)=0. We show that, under some mild conditions, Φ is a Jordan homomorphism multiplied by a central element. Such operator algebras include von Neumann algebras, C^∗-algebras and standard operator algebras, etc. Particularly, if H and K are infinite-dimensional (real or complex) Hilbert spaces and A=B(H) and B=B(K), then there exists a nonzero scalar c and an invertible linear or conjugate-linear operator U:H→K such that either Φ(A)=cUAU⁻¹ for all A∈B(H), or Φ(A)=cUA^∗U⁻¹ for all A∈B(H).     

Operator algebras with contractive approximate identities: Weak compactness and the spectrum

We continue our study of operator algebras with contractive approximate identities (cais) by presenting a couple of interesting examples of operator algebras with cais, which in particular answer questions raised in previous papers in this series, for example about whether, roughly speaking, ‘weak compactness’ of an operator algebra, or the lack of it, can be seen in the spectra of its elements.     

2-Local automorphisms of operator algebras

A not necessarily continuous, linear or multiplicative function θ from an algebra A into itself is called a 2-local automorphism if θ agrees with an automorphism of A at each pair of points in A. In this paper, we study when a 2-local automorphism of a C^∗-algebra, or a standard operator algebra on a locally convex space, is an automorphism.     

Non-commutative Arens algebras and their derivations

Given a von Neumann algebra M with a faithful normal semi-finite trace τ, we consider the non-commutative Arens algebra Lω(M,τ)=_?p?1Lp(M,τ) and the related algebras and which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra is inner and all derivations of the algebras Lω(M,τ) and are spatial and implemented by elements of . In particular we obtain that if the trace τ is finite then any derivation on the non-commutative Arens algebra Lω(M,τ) is inner.     

Vertex operator algebras associated to modified regular representations of affine Lie algebras

Let G be a simply-connected complex Lie group with simple Lie algebra g and let be its affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of N-graded vertex operator algebras (VOAs) associated to g. These vertex operator algebras contain the algebra of regular functions on G as the conformal weight 0 subspaces and are -modules of dual levels in the sense that , where h^∨ is the dual Coxeter number of g. This family of VOAs was previously studied by Arkhipov-Gaitsgory and Gorbounov-Malikov-Schechtman from different points of view. We show that when k is irrational, the vertex envelope of the vertex algebroid associated to G and the level k is isomorphic to the vertex operator algebra we constructed above. The case of rational levels is also discussed.     

Reflexivity of Operator Weighted Shifts

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Operator algebras associated with multiplicative convolutions of arithmetic functions

The action of N on l~2(N) is studied in association with the multiplicative structure of N. Then the maximal ideal space of the Banach algebra generated by N is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The C*-algebra generated by N does not contain any non-zero projection of finite rank. This assertion is equivalent to the existence of infinitely many primes. The von Neumann algebra generated by N is B(l~2(N)), the set of all bounded operators on l~2(N).Moreover, the differential operator on l~2(N,1/n(n+1)) defined by ▽f = μ * f is considered, where μ is the Mbius function. It is shown that the spectrum σ(▽) contains the closure of {ζ(s)-1: Re(s) 1}. Interesting problems concerning are discussed.     

Algebra with polynomial commutation relations for the Zeeman-Stark effect in the hydrogen atom

We study the Zeeman-Stark effect for the hydrogen atom in crossed homogeneous electric and magnetic fields. A nonhomogeneous perturbing potential can also be present. If the crossed fields satisfy some resonance relation, then the degeneration in the resonance spectral cluster is removed only in the second-order term of the perturbation theory. The averaged Hamiltonian in this cluster is expressed in terms of generators of some dynamical algebra with polynomial commutation relations; the structure of these relations is determined by a pair of coprime integers contained in the resonance ratio. We construct the irreducible hypergeometric representations of this algebra. The averaged spectral problem in the irreducible representation is reduced to a second-or third-order ordinary differential equation whose solutions are model polynomials. The asymptotic behavior of the solution of the original problem concerning the Zeeman-Stark effect in the resonance cluster is constructed using the coherent states of the dynamical algebra. We also describe the asymptotic behavior of the spectrum in nonresonance clusters, where the degeneration is already removed in the first-order term of the perturbation theory.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 3, pp. 530–555, March, 2005.     

An Example on Operator Ideals

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Tubular algebras and affine Kac-Moody algebras

The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.     

Noncommutative algebras associated to complexes and graphs

A new class of noncommutative algebras associated to complexes and graphs is introduced. Algebras associated to one-dimensional complexes are studied.