On the representation of simple Riesz groups

In this paper we answer Open Problem 2 of Goodearl's book on partially ordered abelian groups in the case of partially ordered sim- ple groups. As a consequence, we obtain a version of the Theorem of structure of dimension groups in the case of simple Riesz groups. Also, we give a method for constructing torsion-free strictly perforated simple Riesz groups of rank one, and we see that every dense additive subgroup of Q can be obtained using this method

Operator *-correspondences in analysis and geometry

An operator *-algebra is a non-selfadjoint operator algebra with completely isometric involution. We show that any operator *-algebra admits a faithful representation on a Hilbert space in such a way that the involution coincides with the operator adjoint up to conjugation by a symmetry. We introduce operator *-correspondences as a general class of inner product modules over operator *-algebras and prove a similar representation theorem for them. From this we derive the existence of linking operator *-algebras for operator *-correspondences. We illustrate the relevance of this class of inner product modules by providing numerous examples arising from noncommutative geometry.Comment: 31 pages. This work originated from the MFO workshop "Operator spaces and noncommutative geometry in interaction

Full regularity for a C*-algebra of the Canonical Commutation Relations. (Erratum added)

The Weyl algebra,- the usual C*-algebra employed to model the canonical commutation relations (CCRs), has a well-known defect in that it has a large number of representations which are not regular and these cannot model physical fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs of a countably dimensional symplectic space (S,B) and such that its representation set is exactly the full set of regular representations of the CCRs. This construction uses Blackadar's version of infinite tensor products of nonunital C*-algebras, and it produces a "host algebra" (i.e. a generalised group algebra, explained below) for the \sigma-representation theory of the abelian group S where \sigma(.,.):=e^{iB(.,.)/2}. As an easy application, it then follows that for every regular representation of the Weyl algebra of (S,B) on a separable Hilbert space, there is a direct integral decomposition of it into irreducible regular representations (a known result). An Erratum for this paper is added at the end.Comment: An erratum was added to the original pape

On a counterexample to a conjecture by Blackadar

Blackadar conjectured that if we have a split short-exact sequence 0 -> I -> A -> A/I -> 0 where I is semiprojective and A/I is isomorphic to the complex numbers, then A must be semiprojective. Eilers and Katsura have found a counterexample to this conjecture. Presumably Blackadar asked that the extension be split to make it more likely that semiprojectivity of I would imply semiprojectivity of A. But oddly enough, in all the counterexamples of Eilers and Katsura the quotient map from A to A/I is split. We will show how to modify their examples to find a non-semiprojective C*-algebra B with a semiprojective ideal J such that B/J is the complex numbers and the quotient map does not split.Comment: 6 page

Localisation and colocalisation of KK-theory at sets of primes

Given a set of prime numbers S, we localise equivariant bivariant Kasparov theory at S and compare this localisation with Kasparov theory by an exact sequence. More precisely, we define the localisation at S to be KK^G(A,B) tensored with the ring of S-integers Z[S^-1]. We study the properties of the resulting variants of Kasparov theory.Comment: 16 page

The strong Novikov conjecture for low degree cohomology

We show that for each discrete group G, the rational assembly map K_*(BG) \otimes Q \to K_*(C*_{max} G) \otimes \Q is injective on classes dual to the subring generated by cohomology classes of degree at most 2 (identifying rational K-homology and homology via the Chern character). Our result implies homotopy invariance of higher signatures associated to these cohomology classes. This consequence was first established by Connes-Gromov-Moscovici and Mathai. Our approach is based on the construction of flat twisting bundles out of sequences of almost flat bundles as first described in our previous work. In contrast to the argument of Mathai, our approach is independent of (and indeed gives a new proof of) the result of Hilsum-Skandalis on the homotopy invariance of the index of the signature operator twisted with bundles of small curvature.Comment: 11 page

A Simple Separable Exact C*-Algebra not Anti-isomorphic to Itself

We give an example of an exact, stably finite, simple. separable C*-algebra D which is not isomorphic to its opposite algebra. Moreover, D has the following additional properties. It is stably finite, approximately divisible, has real rank zero and stable rank one, has a unique tracial state, and the order on projections over D is determined by traces. It also absorbs the Jiang-Su algebra Z, and in fact absorbs the 3^{\infty} UHF algebra. We can also explicitly compute the K-theory of D, namely K_0 (D) = Z[1/3] with the standard order, and K_1 (D) = 0, as well as the Cuntz semigroup of D.Comment: 16 pages; AMSLaTeX. The material on other possible K-groups for such an algebra has been moved to a separate paper (1309.4142 [math.OA]

Property (RD) for Hecke pairs

As the first step towards developing noncommutative geometry over Hecke C*-algebras, we study property (RD) (Rapid Decay) for Hecke pairs. When the subgroup H in a Hecke pair (G,H) is finite, we show that the Hecke pair (G,H) has (RD) if and only if G has (RD). This provides us with a family of examples of Hecke pairs with property (RD). We also adapt Paul Jolissant's works in 1989 to the setting of Hecke C*-algebras and show that when a Hecke pair (G,H) has property (RD), the algebra of rapidly decreasing functions on the set of double cosets is closed under holomorphic functional calculus of the associated (reduced) Hecke C*-algebra. Hence they have the same K_0-groups.Comment: A short note added explaining other methods to prove that the subalgebra of rapidly decreasing functions is smooth. This is the final version as published. The published version is available at: springer.co

The embedding structure and the shift operator of the U(1) lattice current algebra

The structure of block-spin embeddings of the U(1) lattice current algebra is described. For an odd number of lattice sites, the inner realizations of the shift automorphism areclassified. We present a particular inner shift operator which admits a factorization involving quantum dilogarithms analogous to the results of Faddeev and Volkov.Comment: 14 pages, Plain TeX; typos and a terminological mishap corrected; version to appear in Lett.Math.Phy

On globally non-trivial almost-commutative manifolds

Within the framework of Connes' noncommutative geometry, we define and study globally non-trivial (or topologically non-trivial) almost-commutative manifolds. In particular, we focus on those almost-commutative manifolds that lead to a description of a (classical) gauge theory on the underlying base manifold. Such an almost-commutative manifold is described in terms of a 'principal module', which we build from a principal fibre bundle and a finite spectral triple. We also define the purely algebraic notion of 'gauge modules', and show that this yields a proper subclass of the principal modules. We describe how a principal module leads to the description of a gauge theory, and we provide two basic yet illustrative examples.Comment: 34 pages, minor revision