Addendum to “Connesʼ embedding conjecture and sums of hermitian squares” [Adv. Math. 217 (4) (2008) 1816–1837]

We show that Connes? embedding conjecture (CEC) is equivalent to a real version of the same (RCEC). Moreover, we show that RCEC is equivalent to a real, purely algebraic statement concerning trace positive polynomials. This purely algebraic reformulation of CEC had previously been given in both a real and a complex version in a paper of the last two authors. The second author discovered a gap in this earlier proof of the equivalence of CEC to the real algebraic reformulation (the proof of the complex algebraic reformulation being correct). In this note, we show that this gap can be filled with help of the theory of real von Neumann algebras.     

Noncommutative simplicial complexes and the Baum—Connes conjecture

We associate a noncommutative C ^*-algebra with every locally finite simplicial complex.We determine the K-theory of these algebras and show that they can be used to obtain a conceptual explanation for the Baum—Connes conjecture. Our main technical result determines the algebra generated by the coefficients of a universal projection in an algebraic crossed product by a discrete group , in terms of such a noncommutative algebra associated with a simplicial complex defined by . Submitted: May 2001, Revised: October 2001, Revised: April 2002.     

On simplicial commutative algebras with vanishing André-Quillen homology

In this paper, we study the André-Quillen homology of simplicial commutative ℓ-algebras, ℓ a field, having certain vanishing properties. When ℓ has non-zero characteristic, we obtain an algebraic version of a theorem of J.-P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an application, indicate how the algebraic Serre theorem can be used to resolve a conjecture of D. Quillen for algebras of finite type over Noetherian rings, having non-zero characteristic. Oblatum 3-III-1999 & 3-V-2000?Published online: 11 October 2000     

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.     

Co-amenability and Connes’s embedding problem

A longstanding open question of Connes asks whether any finite von Neumann algebra embeds into an ultraproduct of finite-dimensional matrix algebras.As of yet,algebras verified to satisfy the Connes’s embedding property belong to just a few special classes (e.g.,amenable algebras and free group factors).In this article,we prove that von Neumann algebras satisfying Popa’s co-amenability have Connes’s embedding property.     

Lie algebras with an algebraic adjoint representation revisited

A well-known theorem due to E. Zelmanov proves that PI-Lie algebras with an algebraic adjoint representation over a field of characteristic zero are locally finite-dimensional. In particular, a Lie algebra (over a field of characteristic zero) whose adjoint representation is algebraic of bounded degree is locally finite-dimensional. In this paper it is proved that a prime nondegenerate PI-Lie algebra with an algebraic adjoint representation over a field of characteristic zero is simple and finite-dimensional over its centroid, which is an algebraic field extension of the base field. We also give a new and shorter proof of the local finiteness of Lie algebras with an algebraic adjoint representation of bounded degree.     

Donovan’s conjecture,crossed products and algebraic group actions

Donovan’s conjecture, on blocks of finite group algebras over an algebraically closed field of prime characteristicp, asserts that for any finitep-groupD, there are only finitely many Morita equivalence classes of blocks with defect groupD. The main result of this paper is a reduction theorem: It suffices to prove the conjecture for groups generated by conjugates ofD. A number of other finiteness results are proved along the way. The main tool is a result on actions of algebraic groups.     

Character Connes amenability of dual Banach algebras

We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A^** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.     

Clusters and seeds in acyclic cluster algebras

Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky in order to model the dual canonical basis of a quantum group and total positivity in algebraic groups. Cluster categories were introduced as a representation-theoretic model for cluster algebras. In this article we use this representation-theoretic approach to prove a conjecture of Fomin and Zelevinsky, that for cluster algebras with no coefficients associated to quivers with no oriented cycles, a seed is determined by its cluster. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.

     

Algebraic isomorphisms of limit algebras

We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider the consequences of this result. In particular, we give partial solutions to a conjecture and an open problem by Power. As a further consequence, we describe epimorphisms between various classes of limit algebras.

     

A note on Poisson derivations

Free Poisson algebras are very closely connected with polynomial algebras, and the Poisson brackets are used to solve many problems in affine algebraic geometry. In this note, we study Poisson derivations on the symplectic Poisson algebra, and give a connection between the Jacobian conjecture with derivations on the symplectic Poisson algebra.     

Divisible groups that are brauer groups ∗

Throughour this paper G denotes an abelian divisible torsion group. It is not unreasonable to conjecture that such a G must occur as the Brauer group B(K) of some field K. Some evidence to support this conjecture is provided in [3]; it is proved there that if G is countable then G ? B(K) for some K algebraic over the rational field Q [3, Theorem 2]. In this note we provide still more evidence in support of htis conjecture.     

Universal cycles and homological invariants of locally convex algebras

Using an appropriate notion of locally convex Kasparov modules, we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras; examples are obtained from spectral triples. Our considerations are based on the action of algebraic K-theory on these functors, and involve compatibility properties of the induction process with this action, and with Kasparov-type products. This is based on an appropriate interpretation of the Connes–Skandalis connection formalism. As an application, we prove Bott periodicity and a Thom isomorphism for algebras of Schwartz functions. As a special case, this applies to the theories kk for locally convex algebras considered by Cuntz.     

On spineless cacti, Deligne’s conjecture and Connes-Kreimer’s Hopf algebra

Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne’s conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology of Chas and Sullivan.     

On the Hopf Algebraic Structure of Lie Group Integrators

A commutative but not cocommutative graded Hopf algebra H_N, based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure of unordered rooted trees H_C, developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that H_N is naturally obtained from a universal object in a category of noncommutative derivations and, in particular, it forms a foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and nonrecursive definitions of the coproduct and the antipode are derived. The relationship between H_N and four other Hopf algebras is discussed. The dual of H_N is a Hopf algebra of Grossman and Larson based on ordered rooted trees. The Hopf algebra H_C of Butcher, Connes, and Kreimer is identified as a proper Hopf subalgebra of H_N using the image of a tree symmetrization operator. The Hopf algebraic structure of the shuffle algebra H_Sh is obtained from H_N by a quotient construction. The Hopf algebra H_P of ordered trees by Foissy differs from H_N in the definition of the product (noncommutative concatenation for H_P and shuffle for H_N) and the definitions of the coproduct and the antipode, however, these are related through the tree symmetrization operator.     

Twisted Equivariant KK-Theory and the Baum–Connes Conjecture for Group Extensions

In this paper we study the stability of the Baum–Connes conjecture with coefficients undertaking group extensions. For this, it is necessary to extend Kasparov's equivariant KK-theory to an equivariant theory for twisted actions of groups on C ^*-algebras. As a consequence of our stability results, we are able to reduce the problem of whether closed subgroups of connected groups satisfy the Baum–Connes conjecture, with coefficients to the special case of center-free semi-simple Lie groups.     

Fibred coarse embeddings,a-T-menability and the coarse analogue of the Novikov conjecture

The connection between the coarse geometry of metric spaces and analytic properties of topological groupoids is well known. One of the main results of Skandalis, Tu and Yu is that a space admits a coarse embedding into Hilbert space if and only if a certain associated topological groupoid is a-T-menable. This groupoid characterisation then reduces the proof that the coarse Baum–Connes conjecture holds for a coarsely embeddable space to known results for a-T-menable groupoids. The property of admitting a fibred coarse embedding into Hilbert space was introduced by Chen, Wang and Yu to provide a property that is sufficient for the maximal analogue to the coarse Baum–Connes conjecture and in this paper we connect this property to the traditional coarse Baum–Connes conjecture using a restriction of the coarse groupoid and homological algebra. Additionally we use this results to give a characterisation of the a-T-menability for residually finite discrete groups.     

The No Gap Conjecture for tame hereditary algebras

The “No Gap Conjecture” of Brüstle–Dupont–Pérotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two maximal green sequences can be “polygonally deformed” into each other. We prove this stronger conjecture for all tame hereditary algebras over any field, equivalently, for any acyclic tame skew-symmetrizable exchange matrix.     

A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space

In 1975 A. Connes proved the fundamental result that injective factors on a separable Hilbert space are hyperfinite. In this paper a new proof of this result is presented in which the most technical parts of Connes proof are avoided. Particularly the proof does not rely on automorphism group theory. The starting point in this approach is Wassermann's simple proof of injective ? semidiscrete together with Choi and Effros' characterization of semidiscrete von Neumann algebras as those von Neumann algebras N for which the identity map on N has an approximate completely positive factorization through n × n-matrices.