Dixmier's Problem Research Articles

Fixed points for bounded orbits in Hilbert spaces

Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact a-compact groups (e.g., countable groups). Along the way, we introduce a moderate variant of the classical induction of representations and we generalize the Gaboriau-Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the measure-theoretic solution to the von Neumann problem for locally compact groups. We illustrate the latter result by giving a partial answer to the Dixmier problem for locally compact groups.

Nonseparable UHF algebras I: Dixmier's problem

There are three natural ways to define UHF (uniformly hyperfinite) C ∗ -algebras, and all three definitions are equivalent for separable algebras. In 1967 Dixmier asked whether the three definitions remain equivalent for not necessarily separable algebras. We give a complete answer to this question. More precisely, we show that in small cardinality two definitions remain equivalent, and give counterexamples in other cases. Our results do not use any additional set-theoretic axioms beyond the usual axioms, namely ZFC.

The Dixmier problem, lamplighters and Burnside groups

J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier.

The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

Let A n be the n th Weyl algebra and P m be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras { A n ⊗ P m } is proved: an algebra A admits a finite set δ 1 , … , δ s of commuting locally nilpotent derivations with generic kernels and ∩ i = 1 s ker ( δ i ) = K iff A ≃ A n ⊗ P m for some n and m with 2 n + m = s , and vice versa. The inversion formula for automorphisms of the algebra A n ⊗ P m (and for P ̂ m ≔ K 〚 x 1 , … , x m 〛 ) has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287–330]) given σ ∈ Aut K ( P m ) , then deg σ − 1 ≤ ( deg σ ) m − 1 (the proof is algebro-geometric). We extend this result (using [non-holonomic] D - modules): given σ ∈ Aut K ( A n ⊗ P m ) , then deg σ − 1 ≤ ( deg σ ) 2 n + m − 1 . Any automorphism σ ∈ Aut K ( P m ) is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102–119], a similar result is proved for σ ∈ Aut K ( A n ⊗ P m ) . One can amalgamate two old open problems ( the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209–242. [6]] problem 1) into a single question, ( JD): is a K - algebra endomorphism σ : A n ⊗ P m → A n ⊗ P m an algebra automorphism provided σ ( P m ) ⊆ P m and det ( ∂ σ ( x i ) ∂ x j ) ∈ K ∗ ≔ K ∖ { 0 } ? ( P m = K [ x 1 , … , x m ] ) . It follows immediately from the inversion formula that this question has an affirmative answer iff both conjectures have (see below) [ iff one of the conjectures has a positive answer (as follows from the recent papers [Y. Tsuchimoto, Endomorphisms of Weyl algebra and p -curvatures, Osaka J. Math. 42(2) (2005) 435–452. [10]] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. ArXiv:math.RA/0512171. [5]])].

Dixmier's Problem 6 for the Weyl Algebra (The Generic Type Problem)

ABSTRACT In Dixmier (1968), the author posed six problems for the Weyl algebra A 1 over a field K of characteristic zero. Problems 3, 6, and 5 were solved respectively by Joseph (1975) and Bavula (2005a). Problems 1, 2, and 4 are still open. In this article a short proof is given to Dixmier's problem 6 for the ring of differential operators 𝒟 (X) on a smooth irreducible algebraic curve X. It is proven that, for a given maximal commutative subalgebra C of 𝒟 (X), (almost) all noncentral elements of it have the same type, more precisely, have exactly one of the following types: (i) strongly nilpotent; (ii) weakly nilpotent; (iii) generic; (iv) generic, except for a subset K*a + K of strongly semi-simple elements; (iv) generic, except for a subset K*a + K of weakly semi-simple elements, where K* := K\\{0}. The same results are true for other popular algebras.

DIXMIER'S PROBLEM 6 FOR SOMEWHAT COMMUTATIVE ALGEBRAS AND DIXMIER'S PROBLEM 3 FOR THE RING OF DIFFERENTIAL OPERATORS ON A SMOOTH IRREDUCIBLE AFFINE CURVE

In [6], J. Dixmier posed six problems for the Weyl algebra A1 over a field K of characteristic zero. Problems 3, 5,and 6 were solved respectively by Joseph and Stein [7]; the author [1]; and Joseph [7]. Problems 1, 2, and 4 are still open. For an arbitrary algebra A, Dixmier's problem 6 is essentially aquestion: whether an inner derivation of the algebra A of the type ad f(a), a ∈ A, f(t) ∈ K[t], deg t(f(t)) > 1, has a nonzero eigenvalue. We prove that the answer is negative for many classes of algebras (e.g., rings of differential operators [Formula: see text] on smooth irreducible algebraic varieties, all prime factor algebras of the universal enveloping algebra [Formula: see text] of a completely solvable algebraic Lie algebra [Formula: see text]). This gives an affirmative answer (with a short proof) to an analogue of Dixmier's Problem 6 for certain algebras of small Gelfand–Kirillov dimension, e.g. the ring of differential operators [Formula: see text] on a smooth irreducible affine curve X, Usl(2), etc. (see [3] for details). In this paper an affirmative answer is given to an analogue of Dixmier's Problem 3 but for the ring [Formula: see text].

Dixmier's Problem 5 for the Weyl algebra

The purpose of this paper is to give a negative answer to Problem 5 and a positive answer (in the case of homogeneous elements) to Problem 4 of the article of J. Dixmier [Bull. Soc. Math. France 96 (1968) 209–242] on the first Weyl algebra.

A Question of Rentschler and the Dixmier Problem

The following theorem is the main result of the paper. THEOREM. Let R and T be somewhat commutative algebras with the same holonomic number and let φ: R → T be an algebra homomorphism. Then every holonomic T-module M is (via φ) a holonomic R-module and has finite length as an R-module. When applied to the Weyl algebra this result gives a positive answer to a question of Rentschler. In the important case where R = D(X) and T = D(Y) are rings of differential operators on smooth irreducible algebraic affine varieties X and Y of the same dimension the result means that holonomicity is preserved by twisting (by an arbitrary algebra homomorphism). A short proof is given of the well-known fact that an affirmative solution to the Dixmier Problem (whether every algebra endomorphism φ of the Weyl algebra An is an automorphism) implies the Jacobian Conjecture. The Dixmier Problem has a positive answer if and only if the (twisted from both sides) A n -bimodule φAnφ is simple for each φ. (To begin with, it is not even clear whether it is finitely generated). The theorem implies that it has finite length (moreover, the bimodule is holonomic, thus simple subfactors of it have the least possible Gelfand-Kirillov dimension).

Dixmier's Problem for Verma and Principal Series Submodules

The starting point of this paper is Dixmier's problem relating Verma modules to principal series submodules [5; Probleme 30]. Not only is this problem completely solved, but this relationship is given a very precise form. This allows, for example, one to use the results of Jantzen to determine the multiplicities in the Jordan-Hölder series for the principal series of a complex semisimple Lie group.