147 research outputs found
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
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
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
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
Scattering theory for lattice operators in dimension
This paper analyzes the scattering theory for periodic tight-binding
Hamiltonians perturbed by a finite range impurity. The classical energy
gradient flow is used to construct a conjugate (or dilation) operator to the
unperturbed Hamiltonian. For dimension the wave operator is given by
an explicit formula in terms of this dilation operator, the free resolvent and
the perturbation. From this formula the scattering and time delay operators can
be read off. Using the index theorem approach, a Levinson theorem is proved
which also holds in presence of embedded eigenvalues and threshold
singularities.Comment: Minor errors and misprints corrected; new result on absense of
embedded eigenvalues for potential scattering; to appear in RM
Relative commutants of strongly self-absorbing C*-algebras
The relative commutant of a strongly self-absorbing
algebra is indistinguishable from its ultrapower . This
applies both to the case when is the hyperfinite II factor and to the
case when it is a strongly self-absorbing C*-algebra. In the latter case we
prove analogous results for and reduced powers
corresponding to other filters on . Examples of algebras with
approximately inner flip and approximately inner half-flip are provided,
showing the optimality of our results. We also prove that strongly
self-absorbing algebras are smoothly classifiable, unlike the algebras with
approximately inner half-flip.Comment: Some minor correction
Loop groups and noncommutative geometry
We describe the representation theory of loop groups in terms of K-theory and
noncommutative geometry. This is done by constructing suitable spectral triples
associated with the level l projective unitary positive-energy representations
of any given loop group . The construction is based on certain
supersymmetric conformal field theory models associated with LG in the setting
of conformal nets. We then generalize the construction to many other rational
chiral conformal field theory models including coset models and the moonshine
conformal net.Comment: Revised versio
Tangling clustering of inertial particles in stably stratified turbulence
We have predicted theoretically and detected in laboratory experiments a new
type of particle clustering (tangling clustering of inertial particles) in a
stably stratified turbulence with imposed mean vertical temperature gradient.
In this stratified turbulence a spatial distribution of the mean particle
number density is nonuniform due to the phenomenon of turbulent thermal
diffusion, that results in formation of a gradient of the mean particle number
density, \nabla N, and generation of fluctuations of the particle number
density by tangling of the gradient, \nabla N, by velocity fluctuations. The
mean temperature gradient, \nabla T, produces the temperature fluctuations by
tangling of the gradient, \nabla T, by velocity fluctuations. These
fluctuations increase the rate of formation of the particle clusters in small
scales. In the laboratory stratified turbulence this tangling clustering is
much more effective than a pure inertial clustering that has been observed in
isothermal turbulence. In particular, in our experiments in oscillating grid
isothermal turbulence in air without imposed mean temperature gradient, the
inertial clustering is very weak for solid particles with the diameter 10
microns and Reynolds numbers Re =250. Our theoretical predictions are in a good
agreement with the obtained experimental results.Comment: 16 pages, 4 figures, REVTEX4, revised versio
The ordered K-theory of a full extension
Let A be a C*-algebra with real rank zero which has the stable weak
cancellation property. Let I be an ideal of A such that I is stable and
satisfies the corona factorization property. We prove that 0->I->A->A/I->0 is a
full extension if and only if the extension is stenotic and K-lexicographic. As
an immediate application, we extend the classification result for graph
C*-algebras obtained by Tomforde and the first named author to the general
non-unital case. In combination with recent results by Katsura, Tomforde, West
and the first author, our result may also be used to give a purely
K-theoretical description of when an essential extension of two simple and
stable graph C*-algebras is again a graph C*-algebra.Comment: Version IV: No changes to the text. We only report that Theorem 4.9
is not correct as stated. See arXiv:1505.05951 for more details. Since
Theorem 4.9 is an application to the main results of the paper, the main
results of this paper are not affected by the error. Version III comments:
Some typos and errors corrected. Some references adde
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