1,481 research outputs found
Endomorphisms of the Cuntz Algebras
This mainly expository article is devoted to recent advances in the study of
dynamical aspects of the Cuntz algebras O_n, with n finite, via their
automorphisms and, more generally, endomorphisms. A combinatorial description
of permutative automorphisms of O_n in terms of labeled, rooted trees is
presented. This in turn gives rise to an algebraic characterization of the
restricted Weyl group of O_n. It is shown how this group is related to certain
classical dynamical systems on the Cantor set. An identification of the image
in Out(O_n) of the restricted Weyl group with the group of automorphisms of the
full two-sided n-shift is given, for prime n, providing an answer to a question
raised by Cuntz in 1980. Furthermore, we discuss proper endomorphisms of O_n
which preserve either the canonical UHF-subalgebra or the diagonal MASA, and
present methods for constructing exotic examples of such endomorphisms.Comment: 2 figures, uses pictex, to appear in the Proceedings of the Workshop
on Noncommutative Harmonic Analysis, Bedlewo 201
The mathematical research of William Parry FRS
In this article we survey the mathematical research of the late William (Bill) Parry, FRS
Endomorphisms and automorphisms of locally covariant quantum field theories
In the framework of locally covariant quantum field theory, a theory is
described as a functor from a category of spacetimes to a category of
*-algebras. It is proposed that the global gauge group of such a theory can be
identified as the group of automorphisms of the defining functor. Consequently,
multiplets of fields may be identified at the functorial level. It is shown
that locally covariant theories that obey standard assumptions in Minkowski
space, including energy compactness, have no proper endomorphisms (i.e., all
endomorphisms are automorphisms) and have a compact automorphism group.
Further, it is shown how the endomorphisms and automorphisms of a locally
covariant theory may, in principle, be classified in any single spacetime. As
an example, the endomorphisms and automorphisms of a system of finitely many
free scalar fields are completely classified.Comment: v2 45pp, expanded to include additional results; presentation
improved and an error corrected. To appear in Rev Math Phy
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
Monic representations of the Cuntz algebra and Markov measures
We study representations of the Cuntz algebras \O_N. While, for fixed ,
the set of equivalence classes of representations of \O_N is known not to
have a Borel cross section, there are various subclasses of representations
which can be classified. We study monic representations of \O_N, that have a
cyclic vector for the canonical abelian subalgebra. We show that \O_N has a
certain universal representation which contains all positive monic
representations. A large class of examples of monic representations is based on
Markov measures. We classify them and as a consequence we obtain that different
parameters yield mutually singular Markov measure, extending the classical
result of Kakutani. The monic representations based on the Kakutani measures
are exactly the ones that have a one-dimensional cyclic -invariant
space
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