arXiv subjects
Publications and source records attributed to Valeriano Aiello.
A Fejér-type theorem is proved within the framework of $C^*$-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.
A complete description is provided for the unitary normalizer of the diagonal Cartan subalgebra $\mathcal{D}_2$ in the $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$, which generalizes and unifies analogous results for Cuntz and Bunce-Deddens algebras. Furthermore, the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$ is proved not to be regular. Finally, countably many novel permutative endomorphisms of $\mathcal{Q}_2$ are exhibited with prescribed images of the generator $U$.
The Hilbert space $\mathcal H$ of backward renormalisation of an anyonic quantum spin chain affords a unitary representation of Thompson's group $F$ via local scale transformations. Given a vector in the canonical dense subspace of $\mathcal H$ we show how to calculate the corresponding spectral measure for any element of $F$ and illustrate with some examples. Introducing the "essential part" of an element we show that the spectral measure of any vector in $\mathcal H$ is, apart from possibly finitely many eigenvalues, absolutely continuous with respect to Lebesgue measure. The same considerations and results hold for the Brown-Thompson groups $F_n$ (for which $F=F_2$).
In [Jo14] and [Jo18] Vaughan Jones introduced a construction which yields oriented knots and links from elements of the oriented Thompson group $\vec{F}$. In this paper we prove, by analogy with Alexander's classical theorem establishing that every knot or link can be represented as a closed braid, that given an oriented knot/link $\vec{L}$, there exists an element $g$ in $\vec{F}$ whose closure $\vec{\mathcal{L}}(g)$ is $\vec L$.
Following a procedure due to V. Jones, using suitably normalized elements in a Temperley-Lieb-Jones (planar) algebra we introduce a 3-parametric family of unitary representations of the Thompson's group $F$ equipped with canonical (vacuum) vectors and study some of their properties. In particular, we discuss the behaviour at infinity of their matrix coefficients, thus showing that these representations do not contain any finite type component. We then focus on a particular representation known to be quasi-regular and irreducible and show that it is inequivalent to itself once composed with a classical automorphism of F. This allows us to distinguish three equivalence classes in our family. Finally, we investigate a family of stabilizer subgroups of $F$ indexed by subfactor Jones indices that are described in terms of the chromatic polynomial. In contrast to the first non-trivial index value for which the corresponding subgroup is isomorphic to the Brown-Thompson's group $F_3$, we show that when the index is large enough this subgroup is always trivial.
The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $\vec{F}$ and $\vec{T}$ of the Thompson groups $F$ and $T$, respectively, together with a procedure that associates an oriented link diagram to any element of these subgroups. Moreover, several specializations of some well-known polynomial link invariants can be seen as functions of positive type on the Thompson groups or the Jones-Thompson subgroups. One important example is provided by suitable evaluations of the Jones polynomial, which are thus associated with certain unitary representations of the groups $\vec{F}$ and $\vec{T}$. Within this framework, we discuss an alternative approach that relies on some partition function interpretation of the Jones polynomial, and also exhibit more examples associated with other link invariants, notably the two-variable Kauffman polynomial and the HOMFLY polynomial. In the unoriented case, extending our previous results, we also show by similar methods that certain evaluations of the Tutte polynomial and of the Kauffman bracket, suitably renormalized, yield functions of positive type on $T$.
In a recent paper Jones introduced a correspondence between elements of the Thompson group $F$ and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be reinterpreted as coefficients of certain unitary representations of $F$. We give a somewhat different and elementary proof of this fact for the Kauffman bracket evaluated at certain roots of unity by means of a statistical mechanics model interpretation. Moreover, by similar methods we show that, for some particular specializations of the variables, other familiar link invariants and graph polynomials, namely the number of $N$-colourings and the Tutte polynomial, can be viewed as positive definite functions on $F$.
We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,θ)$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in $2 \leq p<\infty$ generators to the $p$-adic ring $C^*$-algebra. For the case where $P$ is abelian and $C^*(G)$ is a maximal abelian subalgebra, we establish a picture for the automorphisms of the boundary quotient that fix $C^*(G)$ pointwise. This allows us to show that they form a maximal abelian subgroup of the entire automorphism group. The picture also leads to the surprising outcome that, for integral dynamics, every automorphism that fixes one of the natural Cuntz subalgebras pointwise is necessarily a gauge automorphism. Many of the automorphisms we consider are shown to be outer.
The notion of permutative representation is generalized to the $2$-adic ring $C^*$-algebra $\mathcal{Q}_{2}$. Permutative representations of $\mathcal{Q}_2$ are then investigated with a particular focus on the inclusion of the Cuntz algebra $\mathcal{O}_2\subset\mathcal{Q}_2$. Notably, every permutative representation of $\mathcal{O}_2$ is shown to extend automatically to a permutative representation of $\mathcal{Q}_2$ provided that an extension whatever exists. Moreover, all permutative extensions of a given representation of $\mathcal{O}_2$ are proved to be unitarily equivalent to one another. Irreducible permutative representations of $\mathcal{Q}_2$ are classified in terms of irreducible permutative representations of the Cuntz algebra. Apart from the canonical representation of $\mathcal{Q}_2$, every irreducible representation of $\mathcal{Q}_2$ is the unique extension of an irreducible permutative representation of $\mathcal{O}_2$. Furthermore, a permutative representation of $\mathcal{Q}_2$ will decompose into a direct sum of irreducible permutative subrepresentations if and only if it restricts to $\mathcal{O}_2$ as a regular representation in the sense of Bratteli-Jorgensen. As a result, a vast class of pure states of $\mathcal{O}_2$ is shown to enjoy the unique pure extension property with respect to the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$.
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.
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 $\vec{F}$.
We undertake a systematic study of the so-called $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$. This is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_2$ such that $S_2U=U^2S_2$ and $S_2S_2^*+US_2S_2^*U^*=1$. Notably, it contains a copy of the Cuntz algebra $\mathcal{O}_2=C^*(S_1, S_2)$ through the injective homomorphism mapping $S_1$ to $US_2$. Among the main results, the relative commutant $C^*(S_2)'\cap \mathcal{Q}_2$ is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$, namely the endomorphisms of $\mathcal{Q}_2$ that restrict to the identity on $\mathcal{O}_2$ are actually the identity on the whole $\mathcal{Q}_2$. Moreover, there is no conditional expectation from $\mathcal{Q}_2$ onto $\mathcal{O}_2$. As for the inner structure of $\mathcal{Q}_2$, the diagonal subalgebra $\mathcal{D}_2$ and $C^*(U)$ are both proved to be maximal abelian in $\mathcal{Q}_2$. The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of $\mathcal{Q}_2$. In particular, the semigroup of the endomorphisms fixing $U$ turns out to be a maximal abelian subgroup of ${\rm Aut}(\mathcal{Q}_2)$ topologically isomorphic with $C(\mathbb{T},\mathbb{T})$. Finally, it is shown by an explicit construction that ${\rm Out}(\mathcal{Q}_2)$ is uncountable and non-abelian.
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.