arXiv 0810.53642008-10-29math.OAmath.DS

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

Justin R. Peters

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

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.

2006-09-19math.DSmath.OA

$C^*$-envelopes of universal free products and semicrossed products for multivariable dynamics

We show that for a class of operator algebras satisfying a natural condition the $C^*$-envelope of the universal free product of operator algebras $A_i$ is given by the free product of the $C^*$-envelopes of the $A_i$. We apply this theorem to, in special cases, the $C^*$-envelope of the semicrossed products for multivariable dynamics in terms of the single variable semicrossed products of Peters.

2007-10-16math.FAmath.OA

Semicrossed products of C-algebras and their C-envelopes

Let $\mathcal{C}$ be a C*-algebra and $α:\mathcal{C} \rightarrow \mathcal{C}$ a unital *-endomorphism. There is a natural way to construct operator algebras which are called semicrossed products, using a convolution induced by the action of $α$ on $\mathcal{C}$. We show that the C*-envelope of a semicrossed product is (a full corner of) a crossed product. As a consequence, we get that, when $α$ is *-injective, the semicrossed products are completely isometrically isomorphic and share the same C*-envelope, the crossed product $\mathcal{C}_\infty \rtimes_{α_\infty} \mathbb{Z}$. We show that mini…

2014-10-03math.OA

C*-envelopes of semicrossed products by lattice ordered abelian semigroups

A semicrossed product is a non-selfadjoint operator algebra encoding the action of a semigroup on an operator or C*-algebra. We prove that, when the positive cone of a discrete lattice ordered abelian group acts on a C*-algebra, the C*-envelope of the associated semicrossed product is a full corner of a crossed product by the whole group. By constructing a C*-cover that itself is a full corner of a crossed product, and computing the Shilov ideal, we obtain an explicit description of the C*-envelope. This generalizes a result of Davidson, Fuller, and Kakariadis from $\mathbb{Z}_+^n$ to the clas…

2021-02-05math.OA

Semicrossed products of operator algebras and their C*-envelopes

Let $\A$ be a unital operator algebra and let $α$ be an automorphism of $\A$ that extends to a *-automorphism of its $\ca$-envelope $\cenv (\A)$. In this paper we introduce the isometric semicrossed product $\A \times_α^{\is} \bbZ^+ $ and we show that $\cenv(\A \times_α^{\is} \bbZ^+) \simeq \cenv (\A) \times_α \bbZ$. In contrast, the $\ca$-envelope of the familiar contractive semicrossed product $\A \times_α \bbZ^+ $ may not equal $\cenv (\A) \times_α \bbZ$. Our main tool for calculating $\ca$-envelopes for semicrossed products is the concept of a relative semicrossed product of an operator al…

2010-08-13math.FAmath.OA

Dilation theory, commutant lifting and semicrossed products

We take a new look at dilation theory for nonself-adjoint operator algebras. Among the extremal (co)extensions of a representation, there is a special property of being fully extremal. This allows a refinement of some of the classical notions which are important when one moves away from standard examples. We show that many algebras including graph algebras and tensor algebras of C*-correspondences have the semi-Dirichlet property which collapses these notions and explains why they have a better dilation theory. This leads to variations of the notions of commutant lifting and Ando's theorem. Th…

2011-09-01math.FAmath.OA

更多相关论文收起更多论文

Semicrossed products of the disc algebra

If $α$ is the endomorphism of the disk algebra, $\AD$, induced by composition with a finite Blaschke product $b$, then the semicrossed product $\AD\times_α \bZ^+$ imbeds canonically, completely isometrically into $\rC(\bT)\times_α \bZ^+$. Hence in the case of a non-constant Blaschke product $b$, the C*-envelope has the form $ \rC(§_{b})\times_{s} \bZ$, where $(§_{b}, s)$ is the solenoid system for $(\bT, b)$. In the case where $b$ is a constant, then the C*-envelope of $\AD\times_α \bZ^+$ is strongly Morita equivalent to a crossed product of the form $ \rC(§_{e})\times_{s} \bZ$, where $e \colo…

2011-04-07math.OA

Semicrossed Products of Operator Algebras by Semigroups

We examine the semicrossed products of a semigroup action by $*$-endomorphisms on a C*-algebra, or more generally of an action on an arbitrary operator algebra by completely contractive endomorphisms. The choice of allowable representations affects the corresponding universal algebra. We seek quite general conditions which will allow us to show that the C*-envelope of the semicrossed product is (a full corner of) a crossed product of an auxiliary C*-algebra by a group action. Our analysis concerns a case-by-case dilation theory on covariant pairs. In the process we determine the C*-envelope fo…

2016-01-11math.FAmath.OA

Operator algebras associated to modules over an integral domain

We use the Fock semicrossed product to define an operator algebra associated to a module over an integral domain. We consider the $C^*$-envelope of the semicrossed product, and then consider properties of these algebras as models for studying general semicrossed products.

2016-07-28math.OA

Semicrossed Products of Operator Algebras: A Survey

Semicrossed product algebras have been used to study dynamical systems since their introduction by Arveson in 1967. In this survey article, we discuss the history and some recent work, focussing on the conjugacy problem, dilation theory and C*-envelopes, and some connections back to the dynamics

2014-04-05math.FAmath.OA

Nonself-adjoint semicrossed products by abelian semigroups

Let $\mathcal{S}$ be the semigroup $\mathcal{S}=\sum^{\oplus k}_{i=1}\Sc{S}_i$, where for each $i\in I$, $\mathcal{S}_i$ is a countable subsemigroup of the additive semigroup $\B{R}_+$ containing 0. We consider representations of $\mathcal{S}$ as contractions $\{T_s\}_{s\in\mathcal{S}}$ on a Hilbert space with the Nica-covariance property: $T_s^*T_t=T_tT_s^*$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant represe…

2011-11-27math.OA

Dilating covariant representations of the non-commutative disc algebras

Let $ϕ$ be an isometric automorphism of the non-commutative disc algebra $\fA_n$ for $n \geq 2$. We show that every contractive covariant representation of $(\fA_n, ϕ)$ dilates to a unitary covariant representation of $(Ø_n, ϕ)$. Hence the C*-envelope of the semicrossed product $\fA_n \times_ϕ \bZ^+$ is $Ø_n \times_ϕ \bZ$.

2009-04-19math.FAmath.OA

Local maps and the representation theory of operator algebras

Using representation theory techniques we prove that various spaces of derivations or one-sided multipliers over certain operator algebras are reflexive. A sample result: any bounded local derivation (local left multiplier) on an automorphic semicrossed product is a derivation (resp. left multiplier). In the process we obtain various results of independent interest. In particular, the finite dimensional nest representations of the tensor algebra of a topological graph separate points.

2015-02-09math.OA

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 cros…

2014-10-06math.OAmath.RT

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 gaug…

2020-04-18math.OA

Meet Irreducible Ideals and Representations of Limit Algebras

In this paper we give a criterion for an ideal of a TAF algebra to be meet irreducible. We show that an ideal $J$ of $A$ is meet irreducile if and only if the C$^*$-envelope of the quotient $A/J$ is primitive. In that case, A/J admits a nest representation which extends to a *-representation of the C$^*$-envelope for $A/J.$ We also characterize the meet irreducible ideals as the kernels of nest representations; this settles the question of whether the n-primitive and meet irreducible ideals coincide.

2002-06-24math.OA

Maximal C-algebras of quotients and injective envelopes of C-algebras

A new C*-enlargement of a C*-algebra $A$ nested between the local multiplier algebra $M_{\text{loc}}(A)$ of $A$ and its injective envelope $I(A)$ is introduced. Various aspects of this maximal C*-algebra of quotients, $Q_{\text{max}}(A)$, are studied, notably in the setting of AW*-algebras. As a by-product we obtain a new example of a type I C*-algebra $A$ such that $M_{\text{loc}}(M_{\text{loc}}(A))\ne M_{\text{loc}}(A)$.

2007-04-27math.OAmath.RA