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Justin R. Peters

Publications and source records attributed to Justin R. Peters.

18 recordsLinked to original sources
arXiv2023

Singly Generated Radical Operator Algebras

We examine two nonselfadjoint operator algebras: the weighted shift algebra, and the Volterra operator algebra. In both cases, the operator algebra is the norm closure of the polynomials in the operator norm. In the case of the weighted shift algebra, the existence of a gauge action allows us to apply Fourier analysis to study the ideals of the algebra. In the case of the Volterra operator algebra, there is no gauge action, and other methods are needed to study the norm structure and the ideals.

math.OA↗
arXiv2020

Representations of Dirichlet Operator Algebras

A Dirichlet operator algebra is a nonself-adjoint operator algebra $\mathcal{A}$ with the property that $\mathcal{A} + \mathcal{A}^*$ is norm-dense in the C$^*$-envelope of $\mathcal{A}.$ We show that, under certain restrictions, $\mathcal{A}$ has a family of completely contractive representations $\{π_i\}$ with the property that the invariant subspaces of $π_i(\mathcal{A})$ are totally ordered, and such that, for all $a \in \mathcal{A}, \ ||a|| = \sup_i ||π_i(a)||.$ The class of Dirichlet algebras includes strongly maximal triangular AF algebras, certain semicrossed product algebras, and gauge-invariant subalgebras of Cuntz C$^*$-algebras. The main tool is the duality theory for essentially principal etale groupoids.

math.OA↗
arXiv2019

Isomorphism of uniform algebras on the 2-torus

For $α$ a positive irrational, let $\mathcal{A}_α$ be the subalgebra of continuous functions on the two-torus whose Fourier transform vanishes at $(m, n)$ if $m + αn < 0.$ These algebras were studied by Wermer and others, who proved properties such as maximality and characterized the Gelfand space. One of the major themes of current work in operator algebras is classification, but none of the properties which were investigated earlier distinguished between $\mathcal{A}_α$ and $\mathcal{A}_β,$ if $β$ is another positive irrational. We address this question. We also determine the automorphism group of $\mathcal{A}_α.$

math.FA↗
arXiv2014

Ergodic extensions and Hilbert modules associated to endomorphisms of MASAS

We show that a class of ergodic transformations on a probability measure space $(X,μ)$ extends to a representation of $\mathcal{B}(L^2(X,μ))$ that is both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in this work. During the analysis of the existence and uniqueness of such a Cuntz family we give several results of individual interest. Most notably we prove a decomposition of $X$ for $N$-to-one local homeomorphisms that is connected to the orthonormal basis of Hilbert modules. We remark that the trivial Hilbert module of the Cuntz algebra $\mathcal{O}_N$ does not have a well-defined Hilbert module basis (moreover that it is unitarily equivalent to the module sum $\sum_{i=1}^n \mathcal{O}_N$ for infinitely many $n \in \mathbb{N}$).

math.OA↗
arXiv2014

Operator algebras and representations from commuting semigroup actions

Let $\sS$ be a countable, abelian semigroup of continuous surjections on a compact metric space $X$. Corresponding to this dynamical system we associate two operator algebras, the tensor algebra, and the semicrossed product. There is a unique smallest C$^*$-algebra into which an operator algebra is completely isometrically embedded, which is the C$^*$-envelope. We provide two distinct characterizations of the C$^*$-envelope of the tensor algebra; one developed in a general setting by Katsura, and the other using tools of projective and inductive limits, which gives the C$^*$-envelope as a crossed product C$^*$-algebra. We also study two natural classes of representations, the left regular representations and the orbit representations. The first is Shilov, and the second has a Shilov resolution.

math.OA↗
arXiv2014

Representations of C*-dynamical systems implemented by Cuntz families

Given a dynamical system $(A,\al)$ where $A$ is a unital $\ca$-algebra and $\al$ is a (possibly non-unital) *-endomorphism of $A$, we examine families $(π,\{T_i\})$ such that $π$ is a representation of $A$, $\{T_i\}$ is a Toeplitz-Cuntz family and a covariance relation holds. We compute a variety of non-selfadjoint operator algebras that depend on the choice of the covariance relation, along with the smallest $\ca$-algebra they generate, namely the $\ca$-envelope. We then relate each occurrence of the $\ca$-envelope to (a full corner of) an appropriate twisted crossed product. We provide a counterexample to show the extent of this variety. In the context of $\ca$-algebras, these results can be interpreted as analogues of Stacey's famous result, for non-automorphic systems and $n>1$. Our study involves also the one variable generalized crossed products of Stacey and Exel. In particular, we refine a result that appears in the pioneering paper of Exel on (what is now known as) Exel systems.

math.OA↗
arXiv2012

Diagonalizing hermitian matrices of continuous functions

The problem of diagonalizing hermitian matrices of continuous fiunctions was studied by Grove and Pederson in 1984. While diagonalization is not possible in general, in the presence of differentiability conditions we are able to obtain positive results in the case of $2\times 2$ matrices. It remains open whether our results can be extended to $n\times n$ matrices.

math.OA↗
arXiv2012

Semicrossed products of the disk algebra and the Jacobson radical

We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with zero hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.

math.OA↗
arXiv2010

Continuity of Translation Operators

For a Radon measure $μ$ on $\bbR,$ we show that $L^{\infty}(μ)$ is invariant under the group of translation operators $T_t(f)(x) = {$f(x-t)$}\ (t \in \bbR)$ if and only if $μ$ is equivalent to Lebesgue measure $m$. We also give necessary and sufficient conditions for $L^p(μ),\1 \leq p < \infty,$ to be invariant under the group $\{T_t\}$ in terms of the Radon-Nikodym derivative w.r.t. $m$.

math.CA↗
arXiv2008

The C$^*$-envelope of a semicrossed product and Nest Representations

Let $X$ be compact Hausdorff, and $ϕ: X \to X$ a continuous surjection. Let $\mathcal{A}$ be the semicrossed product algebra corresponding to the relation fU = Uf\circ ϕ$ or to the relation $Uf = f\circ ϕU.$ Then the C$^*$-envelope of $\mathcal{A}$ is the crossed product of a commutative C$^*$-algebra which contains $C(X)$ as a subalgebra, with respect to a homeomorphism which we construct. We also show there are"sufficiently many" nest representations.

math.OA↗
arXiv2006

The C$^*$-envelope of a semicrossed product and nest representations

Let $X$ be compact Hausdorff, and $ϕ: X \to X$ a continuous surjection. Let $\mathcal{A}$ be the semicrossed product algebra corresponding to the relation $fU = Uf\circ ϕ$. Then the C$^*$-envelope of $\mathcal{A}$ is the crossed product of a commutative C$^*$-algebra which contains $C(X)$ as a subalgebra, with respect to a homeomorphism which we construct. We also show there are``sufficiently many'' nest representations.

math.OA↗
arXiv2005

Compact Operators and Nest Representations of Limit Algebras

In this paper we study the nest representations of a strongly maximal TAF algebra, whose ranges contain non-zero compact operators. We introduce a particular class of such representations, the essential nest representations, and we show that their kernels coincide with the completely meet irreducible ideals. From this we deduce that there exist enough contractive nest representations, with non-zero compact operators in their range, to separate the points. Using nest representation theory, we also give a coordinate-free description of the fundamental groupoid for strongly maximal TAF algebras. For an arbitrary nest representation $ρ$ of a strongly maximal TAF algebra, we show that the presence of non-zero compact operators in the range implies that the nest is similar to a completely atomic one. If, in addition, the range of $ρ$ is norm closed, then every compact operator in the range of $ρ$ can be approximated by sums of rank one operators in the range of $ρ$. In the case of $\bbN$-ordered nest representations, we show that the range of $ρ$ contains finite rank operators iff $\ker ρ$ fails to be a prime ideal.

math.OA↗
arXiv2003

Subalgebras of the Cuntz C^*-Algebra

In this paper we exploit the fact that a Cuntz C$^*$-algebra is a groupoid C$^*$-algebra to facilitate the study of non-self-adjoint subalgebras of $O_n$. The Cuntz groupoid is not principal and the spectral theorem for bimodules does not apply in full generality. We characterize the bimodules (over a natural masa) which are determined by their spectra in the Cuntz groupoid; these are exactly the ones which are invariant under the guage automorphisms and exactly the ones which are generated by the Cuntz partial isometries which they contain. We investigate analytic subalgebras of $O_n, n$ finite, by studying cocycles on the Cuntz groupoid. In contrast to AF groupoids, there are no cocycles which are integer valued or bounded and vanish precisely on the natural diagonal. $O_n$ contains a canonical UHF subalgebra; each strongly maximal triangular subalgebra of the UHF subalgebra has an extension to a strongly maximal triangular subalgebra of $O_n$ and each trivially analytic subalgebra of the UHF subalgebra has a proper analytic extension. We also study the Volterra subalgebra of $O_n$. We identify the spectrum of the Volterra subalgebra and use this to prove a theorem of Power that the radical is equal to the closed commutator ideal. We also show that the Volterra subalgebra is maximal triangular but not strongly maximal triangular.

math.OA↗
arXiv2003

Semicrossed products generated by two commuting automorphisms

We study the semicrossed product of a finite dimensional C^*-algebra by two types of commuting automorphisms, and identify them with matrix algebras of analytic functions in two variables. We look at the connections with semicrossed products by Z_+ actions.

math.OA↗
arXiv2003

Partial dynamical systems and AF C*-algebras

We obtain a characterization in terms of dynamical systems of those r-discrete groupoids for which the groupoid C*-algebra is approximately finite-dimensional (AF). These ideas are then used to compute the K-theory for AF algebras by utilizing the actions of these partial homeomorphisms, and these K-theoretic calculations are applied to some specific examples of AF algebras. Finally, we show that, for a certain class of dimension groups, a groupoid can be obtained directly from the dimension group's structure whose associated C*-algebra has K_0 group isomorphic to the original dimension group.

math.OA↗
arXiv1999

Nest Representations of TAF Algebras

A nest representation of a strongly maximal TAF algebra $A$ is a representation $π$ for which $\operatorname{Lat} π(A) is totally ordered. We prove that if the spectrum of $A$ is totally ordered, or if $\operatorname{Lat} π(A)$ contains an atom, then $\operatorname{ker} π$ is a meet irreducible ideal.

math.OA↗