140 research outputs found
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
Leibniz Seminorms and Best Approximation from C*-subalgebras
We show that if B is a C*-subalgebra of a C*-algebra A such that B contains a
bounded approximate identity for A, and if L is the pull-back to A of the
quotient norm on A/B, then L is strongly Leibniz. In connection with this
situation we study certain aspects of best approximation of elements of a
unital C*-algebra by elements of a unital C*-subalgebra.Comment: 24 pages. Intended for the proceedings of the conference "Operator
Algebras and Related Topics". v2: added a corollary to the main theorem, plus
several minor improvements v3: much simplified proof of a key lemma,
corollary to main theorem added v4: Many minor improvements. Section numbers
increased by
Representations of Conformal Nets, Universal C*-Algebras and K-Theory
We study the representation theory of a conformal net A on the circle from a
K-theoretical point of view using its universal C*-algebra C*(A). We prove that
if A satisfies the split property then, for every representation \pi of A with
finite statistical dimension, \pi(C*(A)) is weakly closed and hence a finite
direct sum of type I_\infty factors. We define the more manageable locally
normal universal C*-algebra C*_ln(A) as the quotient of C*(A) by its largest
ideal vanishing in all locally normal representations and we investigate its
structure. In particular, if A is completely rational with n sectors, then
C*_ln(A) is a direct sum of n type I_\infty factors. Its ideal K_A of compact
operators has nontrivial K-theory, and we prove that the DHR endomorphisms of
C*(A) with finite statistical dimension act on K_A, giving rise to an action of
the fusion semiring of DHR sectors on K_0(K_A)$. Moreover, we show that this
action corresponds to the regular representation of the associated fusion
algebra.Comment: v2: we added some comments in the introduction and new references.
v3: new authors' addresses, minor corrections. To appear in Commun. Math.
Phys. v4: minor corrections, updated reference
Monoids of intervals of simple refinement monoids and non-stable K-Theory of multiplier algebras
We show that the representation of the monoid of intervals of a simple refinement monoid in terms of affine semicontinuous functions, given by Perera in 2001, fails to be faithful in the case of strictly perforated monoids. We give some potential applications of this result in the context of monoids of intervals and K-Theory of multiplier rings
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
KO-Homology and Type I String Theory
We study the classification of D-branes and Ramond-Ramond fields in Type I
string theory by developing a geometric description of KO-homology. We define
an analytic version of KO-homology using KK-theory of real C*-algebras, and
construct explicitly the isomorphism between geometric and analytic
KO-homology. The construction involves recasting the Cl(n)-index theorem and a
certain geometric invariant into a homological framework which is used, along
with a definition of the real Chern character in KO-homology, to derive
cohomological index formulas. We show that this invariant also naturally
assigns torsion charges to non-BPS states in Type I string theory, in the
construction of classes of D-branes in terms of topological KO-cycles. The
formalism naturally captures the coupling of Ramond-Ramond fields to background
D-branes which cancel global anomalies in the string theory path integral. We
show that this is related to a physical interpretation of bivariant KK-theory
in terms of decay processes on spacetime-filling branes. We also provide a
construction of the holonomies of Ramond-Ramond fields in Type II string theory
in terms of topological K-chains.Comment: 40 pages; v4: Clarifying comments added, more detailed proof of main
isomorphism theorem given; Final version to be published in Reviews in
Mathematical Physic
Almost commuting self-adjoint matrices --- the real and self-dual cases
We show that a pair of almost commuting self-adjoint, symmetric matrices is
close to a pair of commuting self-adjoint, symmetric matrices (in a uniform
way). Moreover we prove that the same holds with self-dual in place of
symmetric. The notion of self-dual Hermitian matrices is important in physics
when studying fermionic systems that have time reversal symmetry. Since a
symmetric, self-adjoint matrix is real, we get a real version of Huaxin Lin's
famous theorem on almost commuting matrices. Similarly the self-dual case gives
a version for matrices over the quaternions.
We prove analogous results for element of real C^*-algebras of "low rank." In
particular, these stronger results apply to paths of almost commuting Hermitian
matrices that are real or self-dual. Along the way we develop a theory of
semiprojectivity for real C^*-algebras.Comment: Expanded references. 33 page
Cartan subalgebras and the UCT problem, II
We show that outer approximately represenbtable actions of a finite cyclic
group on UCT Kirchberg algebras satisfy a certain quasi-freeness type property
if the corresponding crossed products satisfy the UCT and absorb a suitable UHF
algebra tensorially. More concretely, we prove that for such an action there
exists an inverse semigroup of homogeneous partial isometries that generates
the ambient C*-algebra and whose idempotent semilattice generates a Cartan
subalgebra. We prove a similar result for actions of finite cyclic groups with
the Rokhlin property on UCT Kirchberg algebras absorbing a suitable UHF
algebra. These results rely on a new construction of Cartan subalgebras in
certain inductive limits of Cartan pairs. We also provide a characterisation of
the UCT problem in terms of finite order automorphisms, Cartan subalgebras and
inverse semigroups of partial isometries of the Cuntz algebra .
This generalizes earlier work of the authors.Comment: minor revisions; final version, accepted for publication in Math.
Ann.; 26 page
D-branes, Matrix Theory and K-homology
In this paper, we study a new matrix theory based on non-BPS D-instantons in
type IIA string theory and D-instanton - anti D-instanton system in type IIB
string theory, which we call K-matrix theory. The theory correctly incorporates
the creation and annihilation processes of D-branes. The configurations of the
theory are identified with spectral triples, which are the noncommutative
generalization of Riemannian geometry a la Connes, and they represent the
geometry on the world-volume of higher dimensional D-branes. Remarkably, the
configurations of D-branes in the K-matrix theory are naturally classified by a
K-theoretical version of homology group, called K-homology. Furthermore, we
argue that the K-homology correctly classifies the D-brane configurations from
a geometrical point of view. We also construct the boundary states
corresponding to the configurations of the K-matrix theory, and explicitly show
that they represent the higher dimensional D-branes.Comment: 53 pages, corrected a few typos, version published in JHE
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