We relate permutative representations of discrete groups-those that permute an orthonormal basis-to quasi-regular representations associated with stabilizer subgroups, and use this correspondence to study representations of the Higman-Thompson groups. In particular, we study the connection between irreducible permutative representations of the Cuntz algebran and their restrictions to the Higman-Thompson groups. We focus on the family of representations pi x arising from the orbits of points x is an element of I under the dynamical system (I, f), where I = [0, 1) and f (x) = nx mod 1. We obtain a decomposition of pi 0|Fn into n irreducible representations, closely related to the open amenability problem for Fn, and prove that the group algebra C & lowast;pi x (Tn) is simple and traceless. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We consider a class of Markov interval maps f∈ℳ(I) with I an interval, whose associated transition matrix A_f is necessarily primitive. Then we search for subdynamics, i.e., a subset J⊂ I and g=f| _J∈ ℳ([J]) , with [J] the minimal closed interval containing J. The transition matrix A_g of g is obtained through successive state splittings of A_f , followed by the removal of appropriate row(s) and column(s). We prove the existence of such J ensuring g∈ℳ([J]) . We also consider the Cuntz–Krieger algebra 𝒪_A_f representation π _f,x on the Hilbert space associated to the f-orbit of each point x∈ J . We similarly obtain a representation π _g,x of 𝒪_A_g . We prove that π _g,x(𝒪_A_g) is a subalgebra of π _f,x(𝒪_A_f) . By exploring this further, we show that in fact 𝒪_A_g is a corner algebra of 𝒪_A_f by finding a projection p_J such that 𝒪_A_g=p_J 𝒪_A_fp_J . We explicitly enumerate such corner algebras for each splitting state. This method provides a systematic way to construct concrete examples of specific Cuntz–Krieger corner algebras.

We give an explicit injective representation of the universal C^∗-algebra that is generated by doubly non-commuting isometries. This injectivity allows us to prove that such universal algebras embed naturally into each other and also, when combined with Rieffel's theory of deformation, to show that they are nuclear and to compute their K-theory.

For every integer n >= 2 n 2 , we consider a family { pi w } w is an element of { 0 , 1 , .... , n - 1 } N of irreducible representations of the Cuntz algebra O n. All these representations (except one) are shown to be equivalent to those arising from the orbits of the interval map dynamical system ( I , f ) (I,f) with f ( x ) = n x ( mod 1 ) f(x)=nx 1) . We consider the embeddings V = V 2 subset of V n subset of O n V=V of the Thompson group in the Higman-Thompson group V n obtained by Birget and then from V n which was obtained independently by Birget and Nekrashevych. The restriction of pi w to V nis still irreducible; however, the restriction of pi w to is no longer irreducible, and we obtain the corresponding irreducible decomposition in terms of quasi-regular representations.

We show that the Banach *-algebra $\ell^1(G,A,\alpha)$, arising from a C*-dynamical system $(A,G,\alpha)$, is an hermitian Banach algebra if the discrete group $G$ is finite or abelian (or more generally, a finite extension of a nilpotent group). As a corollary, we obtain that $\ell^1(\mathbb{Z},C(X),\alpha)$ is hermitian, for every topological dynamical system $\Sigma = (X, \sigma)$, where $\sigma: X\to X$ is a homeomorphism of a compact Hausdorff space $X$ and the action is $\alpha_n(f)=f\circ \sigma^{-n}$ with $n\in\mathbb{Z}$.

We describe Markov interval maps via branching systems and develop the theory of relative branching systems, characterizing when the associated representations of relative graph C*-algebras are faithful. When the Markov interval maps f have escape sets, we use our results to characterize injectivity of the associated relative graph algebra representations, improving on previous work by the first, third, and fourth authors.

We establish the Schwarz inequality for a class of Banach *-algebras, and use it to derive some consequences, for example when a linear map between Banach *-algebras is a Jordan homomorphism. This applies to the class of Banach *-algebras $$\ell ^1(G,A;\alpha )$$ arising from C*-dynamical systems $$(A,G,\alpha )$$ with $$\alpha $$ an action of a discrete group G on a separable C*-algebra A.

Every representation of the Cuntz algebra On leads to a unitary representation of the Higman-Thompson group Vn. We consider the family {πx}x∈[0,1[ of permutative representations of On that arise from the interval map f(x)=nx (mod 1) acting on the Hilbert space that underlies each orbit, and then study the unitary equivalence and the irreducibility of the corresponding family {ρx}x∈[0,1[ of representations of Higman-Thompson group Vn, showing that these representations are indeed irreducible and moreover ρx and ρy are equivalent if and only if the orbits of x and y coincide.

Abstract We obtain an uncountable family of inequivalent and irreducible representations of the Higman–Thompson groups F n ⊂ T n ⊂ V n F_{n}\subset T_{n}\subset V_{n} . This is accomplished by considering a family of representations of the Higman–Thompson groups V n V_{n} that arise from representations of Cuntz algebras, each one acting on a Hilbert space built upon the orbit of a point x ∈ [ 0 , 1 ) x\in[0,1) under the dynamical system Φ ⁢ ( x ) = n ⁢ x ( mod 1 ) \Phi(x)=nx\pmod{1} . Every such representation is retrieved through the action of V n V_{n} on orb ⁡ ( x ) \operatorname{orb}(x) , and their restrictions to the subgroups F n F_{n} and T n T_{n} of V n V_{n} are studied using properties of the groups.

We consider, M(I), a certain class of Markov interval maps with domain contained in the interval I, whose associated transition matrix Af, necessarily primitive, plays a crucial role in the study of the underlying dynamics. We study the problem of deciding when a restriction g:=f|J of a map f∈M(I) to a subset J⊂I is in the class M([J]), where [J] is the minimal closed interval containing J. We establish natural conditions on J⊂I so that g=f|J is a Markov map. Then we tackle the central problem of deciding when the matrix Ag is primitive in this framework. We are able to enumerate the sets J, satisfying the referred conditions, through a systematic process of elimination of the rows/columns of the state splitting of Af associated to the so called removable states, which ensures the primitivity of the matrices Ag.

Suppose that n≥1 and that, for all i and j with 1≤i,j≤n and i≠j, zij∈T are given such that zji=z‾ij for all i≠j. If V1,…,Vn are isometries on a Hilbert space such that Vi⁎Vj=z‾ijVjVi⁎ for all i≠j, then (V1,…,Vn) is called an n-tuple of doubly non-commuting isometries. The generators of non-commutative tori are well-known examples. In this paper, we establish a simultaneous Wold decomposition for (V1,…,Vn). This decomposition enables us to classify such n-tuples up to unitary equivalence. We show that the joint listing of a unitary equivalence class of a representation of each of the 2n non-commutative tori that are naturally associated with the structure constants is a classifying invariant. A dilation theorem is also established, showing that an n-tuple of doubly non-commuting isometries can be extended to an n-tuple of doubly non-commuting unitary operators on an enveloping Hilbert space.

We produce and study a family of representations of relative graph algebras on Hilbert spaces that arise from the orbits of points of one dimensional dynamical systems, where the underlying Markov interval maps $f$ have escape sets. We identify when such representations are faithful in terms of the transitions to the escape subintervals.

We associate a representation of the Thompson group V (and thus of F and T by considering the restriction) to every representation of the Cuntz algebra O2. The well-developed theory of representations of the Cuntz algebras leads us to exhibit an uncountable family of unitary representations of V which are pairwise non equivalent, and two others for F and T.

We study the structure of the escape orbits for a certain class of interval maps. This structure is encoded in the escape transition matrix (A) over cap (f) of an interval map f, extending the traditional matrix A(f) which considers the transition among the Markov subintervals. We show that the escape transition matrix is a topological conjugacy invariant. We then characterize the 0-1 matrices that can be fabricated as escape transition matrices of Markov interval maps f with escape sets. This shows the richness of this class of interval maps.

We prove that the crossed product Banach algebra \(\ell ^1(G,A;\alpha )\) that is associated with a \({\mathrm C}^*\)-dynamical system \((A,G,\alpha )\) is amenable if G is a discrete amenable group and A is a commutative or finite dimensional \({\mathrm C}^*\)-algebra. Perspectives for further developments are indicated.

We prove that every representation of the Cuntz algebra ON on a separable Hilbert space H arises from a pure isometry V whose wandering space H⊖imV has dimension N. We identify the permutative representations in this construction.

We generate a representation of the Toeplitz C*-algebra T-Af on a Hilbert space H-x that encodes the orbit of an escape point x is an element of I of a Markov interval map f, with transition matrix A(f). This leads to a family of representations of T-Af labeled by points in all intervals I. The underlying dynamics of the interval map are used in the study of this family.

Given n >= 2, z(ij) epsilon T such that z(ij) = (Z) over bar (ji) for 1 <= i, j <= n and z(ii) = 1 for 1 <= i <= n, and integers p(1), . . . , P-n >= 1, we show that the universal C*-algebra generated by unitaries u(1), . . . , u(n) such that u(i)(Pi) u(j)(Pj) = Z(ij)u(j)(j)(P) u(i)(Pi) for 1 <= i, j <= n is not simple if at least one exponent p(i) is at least two. We indicate how the method of proof by "working with various quotients" can be used to establish nonsimplicity of universal C*-algebras in other cases.