37 research outputs found
An introduction to Thompson knot theory and to Jones subgroups
We review a constructions of knots from elements of the Thompson groups due
to Vaughan Jones, which comes in two flavours: oriented and unoriented.Comment: Accepted for publication in the Special Issue of JKTR dedicated to
Vaughan Jone
A Spectral Triple for a Solenoid Based on the Sierpinski Gasket
The Sierpinski gasket admits a locally isometric ramified self-covering. A
semifinite spectral triple is constructed on the resulting solenoidal space,
and its main geometrical features are discussed
Diagonal automorphisms of the -adic ring -algebra
The -adic ring -algebra naturally contains a copy of
the Cuntz algebra and, a fortiori, also of its diagonal
subalgebra with Cantor spectrum. This paper is aimed at
studying the group of the
automorphisms of fixing pointwise. It turns out
that any such automorphism leaves globally invariant.
Furthermore, the subgroup is shown
to be maximal abelian in . Saying exactly what the
group is amounts to understanding when an automorphism of that
fixes pointwise extends to . A complete answer
is given for all localized automorphisms: these will extend if and only if they
are the composition of a localized inner automorphism with a gauge
automorphism.Comment: Improved exposition and corrected some typos and inaccuracie
A look at the inner structure of the 2-adic ring C*-algebra and its automorphism groups
We undertake a systematic study of the so-called 2-adic ring C\u87-algebra Q2. This is the
universal C\u87-algebra generated by a unitary U and an isometry S2 such that S2U = U2S2
and S2S\u87
2+US2S\u87
2U\u87 = 1. Notably, it contains a copy of the Cuntz algebra O2 = C\u87(S1;S2)
through the injective homomorphism mapping S1 to US2. Among the main results, the
relative commutant C\u87(S2)\u9c 9 Q2 is shown to be trivial. This in turn leads to a rigidity
property enjoyed by the inclusion O2 ` Q2, namely the endomorphisms of Q2 that restrict
to the identity on O2 are actually the identity on the whole Q2. Moreover, there is no
conditional expectation from Q2 onto O2. As for the inner structure of Q2, the diagonal
subalgebra D2 and C\u87(U) are both proved to be maximal abelian in Q2. The maximality
of the latter allows a thorough investigation of several classes of endomorphisms and
automorphisms of Q2. In particular, the semigroup of the endomorphisms xing U turns
out to be a maximal abelian subgroup of Aut(Q2) topologically isomorphic with C(T;T).
Finally, it is shown by an explicit construction that Out(Q2) is uncountable and non-
abelian
Spectral triples for noncommutative solenoidal spaces from self-coverings
Examples of noncommutative self-coverings are described, and spectral triples
on the base space are extended to spectral triples on the inductive family of
coverings, in such a way that the covering projections are locally isometric.
Such triples are shown to converge, in a suitable sense, to a semifinite
spectral triple on the direct limit of the tower of coverings, which we call
noncommutative solenoidal space. Some of the self-coverings described here are
given by the inclusion of the fixed point algebra in a C-algebra acted upon
by a finite abelian group. In all the examples treated here, the noncommutative
solenoidal spaces have the same metric dimension and volume as on the base
space, but are not quantum compact metric spaces, namely the pseudo-metric
induced by the spectral triple does not produce the weak topology on the
state space.Comment: v3: the paper will appear in the Journal of Mathematical Analysis and
Applications, 42 pages, no figure
The Homflypt polynomial and the oriented Thompson group
We show how to construct unitary representations of the oriented Thompson
group from oriented link invariants. In particular we show that the
suitably normalised HOMFLYPT polynomial defines a positive definite function of
.Comment: To appear in Quantum Topolog
Positive oriented Thompson links
We prove that the links associated with positive elements of the oriented
subgroup of the Thompson group are positive