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 22-adic ring C∗C^*-algebra

The $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$ naturally contains a copy of the Cuntz algebra $\mathcal{O}_2$ and, a fortiori, also of its diagonal subalgebra $\mathcal{D}_2$ with Cantor spectrum. This paper is aimed at studying the group ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ of the automorphisms of $\mathcal{Q}_2$ fixing $\mathcal{D}_2$ pointwise. It turns out that any such automorphism leaves $\mathcal{O}_2$ globally invariant. Furthermore, the subgroup ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ is shown to be maximal abelian in ${\rm Aut}(\mathcal{Q}_2)$. Saying exactly what the group is amounts to understanding when an automorphism of $\mathcal{O}_2$ that fixes $\mathcal{D}_2$ pointwise extends to $\mathcal{Q}_2$. 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 $\vec{F}$ from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of $\vec{F}$.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