1,463 research outputs found

### Automorphisms of the UHF algebra that do not extend to the Cuntz algebra

Automorphisms of the canonical core UHF-subalgebra F_n of the Cuntz algebra
O_n do not necessarily extend to automorphisms of O_n. Simple examples are
discussed within the family of infinite tensor products of (inner)
automorphisms of the matrix algebras M_n. In that case, necessary and
sufficient conditions for the extension property are presented. It is also
addressed the problem of extending to O_n the automorphisms of the diagonal
D_n, which is a regular MASA with Cantor spectrum. In particular, it is shown
the existence of product-type automorphisms of D_n that are not extensible to
(possibly proper) endomorphisms of O_n

### Covariant Sectors with Infinite Dimension and Positivity of the Energy

We consider a Moebius covariant sector, possibly with infinite dimension, of
a local conformal net of von Neumann algebras on the circle. If the sector has
finite index, it has automatically positive energy. In the infinite index case,
we show the spectrum of the energy always to contain the positive real line,
but, as seen by an example, it may contain negative values. We then consider
nets with Haag duality on the real line, or equivalently sectors with
non-solitonic extension to the dual net; we give a criterion for irreducible
sectors to have positive energy, namely this is the case iff there exists an
unbounded Moebius covariant left inverse. As a consequence the class of sectors
with positive energy is stable under composition, conjugation and direct
integral decomposition.Comment: 25 pages, Latex2

### On discrete twisted C*-dynamical systems, Hilbert C*-modules and regularity

We first give an overview of the basic theory for discrete unital twisted
C*-dynamical systems and their covariant representations on Hilbert C*-modules.
After introducing the notion of equivariant representations of such systems and
their product with covariant representations, we prove a kind of Fell
absorption principle saying that the product of an induced regular equivariant
representation with a covariant faithful representation is weakly equivalent to
an induced regular covariant representation. This principle is the key to our
main result, namely that a certain property, formally weaker than Exel's
approximation property, ensures that the system is regular, i.e., the
associated full and reduced C*-crossed products are canonically isomorphic.Comment: Final version, to appear in Muenster J. Math. A permanence result for
the weak approximation property, some corollaries of it and two examples have
been added to Section 5. Some side results in Section 4 have been removed and
will be included in a subsequent paper. The Introduction has also been partly
rewritte

### Classification of Subsystems for Local Nets with Trivial Superselection Structure

Let F be a local net of von Neumann algebras in four spacetime dimensions
satisfying certain natural structural assumptions. We prove that if F has
trivial superselection structure then every covariant, Haag-dual subsystem B is
the fixed point net under a compact group action on one component in a suitable
tensor product decomposition of F. Then we discuss some application of our
result, including free field models and certain theories with at most countably
many sectors.Comment: 31 pages, LaTe

### Labeled Trees and Localized Automorphisms of the Cuntz Algebras

We initiate a detailed and systematic study of automorphisms of the Cuntz
algebras \O_n which preserve both the diagonal and the core $UHF$-subalgebra.
A general criterion of invertibility of endomorphisms yielding such
automorphisms is given. Combinatorial investigations of endomorphisms related
to permutation matrices are presented. Key objects entering this analysis are
labeled rooted trees equipped with additional data. Our analysis provides
insight into the structure of {\rm Aut}(\O_n) and leads to numerous new
examples. In particular, we completely classify all such automorphisms of
${\mathcal O}_2$ for the permutation unitaries in $\otimes^4 M_2$. We show that
the subgroup of {\rm Out}(\O_2) generated by these automorphisms contains a
copy of the infinite dihedral group ${\mathbb Z} \rtimes {\mathbb Z}_2$.Comment: 35 pages, slight changes, to appear on Trans. Amer. Math. So

### Asymptotic morphisms and superselection theory in the scaling limit II: analysis of some models

We introduced in a previous paper a general notion of asymptotic morphism of
a given local net of observables, which allows to describe the sectors of a
corresponding scaling limit net. Here, as an application, we illustrate the
general framework by analyzing the Schwinger model, which features confined
charges. In particular, we explicitly construct asymptotic morphisms for these
sectors in restriction to the subnet generated by the derivatives of the field
and momentum at time zero. As a consequence, the confined charges of the
Schwinger model are in principle accessible to observation. We also study the
obstructions, that can be traced back to the infrared singular nature of the
massless free field in d=2, to perform the same construction for the complete
Schwinger model net. Finally, we exhibit asymptotic morphisms for the net
generated by the massive free charged scalar field in four dimensions, where no
infrared problems appear in the scaling limit.Comment: 36 pages; no figure

### Scaling limit for subsystems and Doplicher-Roberts reconstruction

Given an inclusion $B \subset F$ of (graded) local nets, we analyse the
structure of the corresponding inclusion of scaling limit nets $B_0 \subset
F_0$, giving conditions, fulfilled in free field theory, under which the
unicity of the scaling limit of $F$ implies that of the scaling limit of $B$.
As a byproduct, we compute explicitly the (unique) scaling limit of the
fixpoint nets of scalar free field theories. In the particular case of an
inclusion $A \subset B$ of local nets with the same canonical field net $F$, we
find sufficient conditions which entail the equality of the canonical field
nets of $A_0$ and $B_0$.Comment: 31 page

### The Fourierâ€“Stieltjes algebra of a C*-dynamical system

In analogy with the Fourierâ€“Stieltjes algebra of a group, we associate to a unital discrete
twisted Câˆ—-dynamical system a Banach algebra whose elements are coefficients of
equivariant representations of the system. Building upon our previous work, we show
that this Fourierâ€“Stieltjes algebra embeds continuously in the Banach algebra of completely
bounded multipliers of the (reduced or full) Câˆ—-crossed product of the system.
We introduce a notion of positive definiteness and prove a Gelfandâ€“Raikov type theorem
allowing us to describe the Fourierâ€“Stieltjes algebra of a system in a more intrinsic way.
We also propose a definition of amenability for Câˆ—-dynamical systems and show that it
implies regularity. After a study of some natural commutative subalgebras, we end with
a characterization of the Fourierâ€“Stieltjes algebra involving Câˆ—-correspondences over the
(reduced or full) Câˆ—-crossed product

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