arXiv subjects
Publications and source records attributed to Paulo R. Pinto.
We give an explicit injective representation of the universal $\mathrm{C}^\ast$-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.
We show that the Banach *-algebra $\ell^1(G,A,α)$, arising from a C*-dynamical system $(A,G,α)$, 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 $\ell^1(\mathbb{Z},C(X),α)$ is hermitian, for every topological dynamical system $Σ= (X, σ)$, where $σ: X\to X$ is a homeomorphism of a compact Hausdorff space $X$ and the action is $α_n(f)=f\circ σ^{-n}$ 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.
Every representation of the Cuntz algebra $\mathcal{O}_n$ leads to a unitary representation of the Higman-Thompson group $V_n$. We consider the family $\{π_x\}_{x\in [0,1[}$ of permutative representations of $\mathcal{O}_n$ 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 $\{ρ_x\}_{x\in [0,1[}$ of representations of Higman-Thompson group $V_n$, showing that that these representations are indeed irreducible and moreover $ρ_x$ and $ρ_y$ are equivalent if and only if the orbits of $x$ and $y$ coincide.
Suppose that $n\geq 1$ and that, for all $i$ and $j$ with $1\leq i,j\leq n$ and $i\neq j$, $z_{ij}\in{\mathbb T}$ are given such that $z_{ji}=\overline{z}_{ij}$ for all $i\neq j$. If $V_1,\dotsc, V_n$ are isometries on a Hilbert space such that $V_i^\ast V_j^{\phantom{\ast}}\!=\overline{z}_{ij} V_j^{\phantom{\ast}}\!V_i^\ast$ for all $i\neq j$, then $(V_1,\dotsc,V_n)$ 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 $(V_1,\dotsc,V_n)$. 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 $2^n$ 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 prove that the crossed product Banach algebra $\ell^1(G,A;α)$ that is associated with a ${\mathrm C}^\ast$-dynamical system $(A,G,α)$ is amenable if $G$ is a discrete amenable group and $A$ is a strongly amenable ${\mathrm C}^\ast$-algebra. This is a consequence of the combination of a more general result with Paterson's characterisation of strongly amenable unital $\mathrm{C}^\ast$-algebras in terms of invariant means for their unitary groups.
We prove that the crossed product Banach algebra $\ell^1(A,G,α)$ that is associated with a $\mathrm{C}^\ast$-dynamical system $(A,G,α)$ is amenable if $G$ is a discrete amenable group and $A$ is a commutative or finite dimensional $\mathrm{C}^\ast$-algebra. Perspectives for further developments are indicated.
Given $n\geq 2$, $z_{ij}\in\mathbb{T}$ such that $z_{ij}=\overline z_{ji}$ for $1\leq i,j\leq n$ and $z_{ii}=1$ for $1\leq i\leq n$, and integers $p_1,...,p_n\geq 1$, we show that the universal $\mathrm{C}^*$-algebra generated by unitaries $u_1,...,u_n$ such that $u_i^{p_i}u_j^{p_j}=z_{ij}u_j^{p_j}u_i^{p_i}$ for $1\leq i,j \leq n$ 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 $\mathrm{C}^*$-algebras in other cases.
We show that every point $x_0\in [0,1]$ carries a representation of a $C^*$-algebra that encodes the orbit structure of the linear mod 1 interval map $f_{β,α}(x)=βx +α$. Such $C^*$-algebra is generated by partial isometries arising from the subintervals of monotonicity of the underlying map $f_{β,α}$. Then we prove that such representation is irreducible. Moreover two such of representations are unitarily equivalent if and only if the points belong to the same generalized orbit, for every $α\in [0,1[$ and $β\geq 1$.
We define the profinite completion of a C*-algebra, which is a pro-C*-algebra, as well as the pro-C*-algebra of a profinite group. We show that the continuous representations of the pro-C*-algebra of a profinite group correspond to the unitary representations of the group which factor through a finite group. We define natural homomorphisms from the C*-algebra of a locally compact group and its profinite completion to the pro-C*-algebra of the profinite completion of the group. We give some conditions for injectivity or surjectivity of these homomorphisms, but an important question remains open.