In this article we study cocycles of discrete countable groups with values in 2 G and the ring of affiliated operators UG. We clarify properties of the first cohomology of a group G with coefficients in 2 G and answer several questions from De Cornulier et al. Moreover, we obtain strong results about the existence of free subgroups and the subgroup… (More)
We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the viewpoint of convex optimization. We relate the question to sums of squares decompositions of a certain Her-mite matrix. If… (More)
We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford algebras for a new approach to the problem. Our main result is that if −1 is not a sum of hermitian squares in the Clifford… (More)
In this paper we study characters on special linear groups SL n (R), where R is either an infinite field or the localization of an order in a number field. We give several applications to the theory of measure-preserving actions, operator-algebraic superrigidity, and almost homomorphisms.
In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant. We provide a natural decomposition of the entropy into summands contributed by… (More)
We show that for any amenable group Γ and any ZΓ-module M of type FL with vanishing Euler characteristic, the entropy of the natural Γ-action on the Pontryagin dual of M is equal to the L 2-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module ZΓ/ZΓf is equal to the logarithm of the Fuglede-Kadison… (More)
In this note we address various algorithmic problems that arise in the computation of the operator norm in unitary representations of a group on Hilbert space. We show that the operator norm in the universal unitary representation is computable if the group is residually finite-dimensional or amenable with decidable word problem. Moreover, we relate the… (More)