Crossed product of a C*-algebra by an endomorphism, coefficient algebras and transfer operators

The partial-isometric crossed products by semigroups of endomorphisms are Morita equivalent to crossed products by groups

Let $Γ^{+}$ be the positive cone of a totally ordered abelian discrete group $Γ$, and $α$ an action of $Γ^{+}$ by extendible endomorphisms of a $C^*$-algebra $A$. We prove that the partial-isometric crossed product $A\times_α^{\textrm{piso}}Γ^{+}$ is a full corner of a group crossed product $\mathcal{B}\times_βΓ$, where $\mathcal{B}$ is a subalgebra of $\ell^{\infty}(Γ,A)$ generated by a collection of faithful copies of $A$, and the action $β$ on $\mathcal{B}$ is induced by shift on $\ell^{\infty}(Γ,A)$. We then use this realization to show that $A\times_α^{\textrm{piso}}Γ^{+}$ has an essentia…

Partial-isometric crossed products by semigroups of endomorphisms

Let $Γ^+$ be the positive cone in a totally ordered abelian group $Γ$, and let $α$ be an action of $Γ^+$ by endomorphisms of a $C^*$-algebra $A$. We consider a new kind of crossed-product $C^*$-algebra $A\times_αΓ^+$, which is generated by a faithful copy of $A$ and a representation of $Γ^+$ as partial isometries. We claim that these crossed products provide a rich and tractable family of Toeplitz algebras for product systems of Hilbert bimodules, as recently studied by Fowler, and we illustrate this by proving detailed structure theorems for actions by forward and backward shifts.

The Partial-Isometric Crossed Products by Semigroups of Endomorphisms as Full Corners

Suppose $Γ^{+}$ is the positive cone of a totally ordered abelian group $Γ$, and $(A,Γ^{+},α)$ is a system consisting of a $C^*$-algebra $A$, an action $α$ of $Γ^{+}$ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A\times_α^{\piso}Γ^{+}$ is a full corner in the subalgebra of $Ł(\ell^{2}(Γ^{+},A))$, and that if $α$ is an action by automorphisms of $A$, then it is the isometric-crossed product $(B_{Γ^{+}}\otimes A)\times^{\iso}Γ^{+}$, which is therefore a full corner in the usual crossed product of system by a group of automorphisms. We use these realiza…

Crossed products by endomorphisms and reduction of relations in relative Cuntz-Pimsner algebras

Starting from an arbitrary endomorphism αof a unital C*-algebra A we construct a crossed product. It is shown that the natural construction depends not only on the C*-dynamical system (A,α) but also on the choice of an ideal orthogonal to kernel of α. The article gives an explicit description of the internal structure of this crossed product and, in particular, discusses the interrelation between relative Cuntz-Pimsner algebras and partial isometric crossed products. We present a canonical procedure that reduces any given C*-correspondence to the 'smallest' C*-correspondence yielding the same …

The composition series of ideals of the partial-isometric crossed product by semigroup of endomorphisms

Let $Γ^{+}$ be the positive cone in a totally ordered abelian group $Γ$, and $α$ an action of $Γ^{+}$ by extendible endomorphisms of a $C^{\ast}$-algebra $A$. Suppose $I$ is an extendible $α$-invariant ideal of $A$. We prove that the partial-isometric crossed product $\mathcal{I}:=I\times_α^{\textrm{piso}}Γ^{+}$ embeds naturally as an ideal of $A\times_α^{\textrm{piso}}Γ^{+}$, such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal $\mathcal{I}$ together with the kernel of a natural homomorphism $ϕ: A\times_α^{\textrm{piso}}Γ^{+}\righta…

Dilations of semigroup crossed products as crossed products of dilations

Laca constructed a minimal automorphic dilation for every semigroup dynamical system arising from an action of an Ore semigroup by injective endomorphisms of a unital $C^*$-algebra. Here we show that the semigroup crossed product with its action by inner endomorphisms given by the implementing isometries has as minimal automorphic dilation the group crossed product of the original dilation. Applications include recent examples studied by Cuntz and the second named author.

Crossed product by an arbitrary endomorphism

Starting from an arbitrary endomorphism δof a unital C*-algebra A we construct a crossed product. It is shown that the natural construction depends not only on the C*-dynamical system (A,δ) but also on the choice of an ideal J orthogonal to Ker δ.

Stacey crossed products associated to Exel systems

There are many different crossed products by an endomorphism of a C*-algebra, and constructions by Exel and Stacey have proved particularly useful. Here we show that every Exel crossed product is isomorphic to a Stacey crossed product, though by a different endomorphism of a different C*-algebra. We apply this result to a variety of Exel systems, including those associated to shifts on the path spaces of directed graphs.

Two families of Exel-Larsen crossed products

Larsen has recently extended Exel's construction of crossed products from single endomorphisms to abelian semigroups of endomorphisms, and here we study two families of her crossed products. First, we look at the natural action of the multiplicative semigroup $\mathbb{N}^\times$ on a compact abelian group $Γ$, and the induced action on $C(Γ)$. We prove a uniqueness theorem for the crossed product, and we find a class of connected compact abelian groups $Γ$ for which the crossed product is purely infinite simple. Second, we consider some natural actions of the additive semigroup $\mathbb{N}^2$ …

C*-algebras of endomorphisms of groups with finite cokernel and partial actions

In this paper we extend the constructions of Boava and Exel to present the C*-algebra associated with an injective endomorphism of a group with finite cokernel as a partial group algebra and consequently as a partial crossed product. With this representation we present another way to study such C*-algebras, only using tools from partial crossed products.

Ideal structure of crossed products by endomorphisms via reversible extensions of $C^*$-dynamical systems

We consider an extendible endomorphism $α$ of a $C^*$-algebra $A$. We associate to it a canonical $C^*$-dynamical system $(B,β)$ that extends $(A,α)$ and is `reversible' in the sense that the endomorphism $β$ admits a unique regular transfer operator $β_*$. The theory for $(B,β)$ is analogous to the theory of classic crossed products by automorphisms, and the key idea is to describe the counterparts of classic notions for $(B,β)$ in terms of the initial system $(A,α)$. We apply this idea to study the ideal structure of a non-unital version of the crossed product $C^*(A,α,J)$ introduced recentl…

Exel and Stacey crossed products, and Cuntz-Pimsner algebras

There are many different crossed products by an endomorphism of a C*-algebra, and constructions by Exel and Stacey have proved particularly useful. Here we show that every Exel crossed product is isomorphic to a Stacey crossed product (though by a different endomorphism of a different C*-algebra), that every Stacey crossed product is an Exel crossed product, and answer some questions raised by Ionescu and Muhly. While this manuscript is not yet in its final form(s) several people have already used our results, so we are posting it now.

Crossed product of a C*-algebra by a semigroup of bounded positive linear maps. Interactions

The paper presents a construction of the crossed product of a C*-algebra by a commutative semigroup of bounded positive linear maps generated by partial isometries. In particular, it generalizes Antonevich, Bakhtin, Lebedev's crossed product by an endomorphism, and is related to Exel's interactions. One of the main goals is the Isomorphism Theorem established in the case of actions by endomorphisms.

From endomorphisms to automorphisms and back: dilations and full corners

When S is a discrete subsemigroup of a discrete group G such that G = S^{-1} S, it is possible to extend circle-valued multipliers from S to G; to dilate (projective) isometric representations of S to (projective) unitary representations of G; and to dilate/extend actions of S by injective endomorphisms of a C*-algebra to actions of G by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilate…

The Rokhlin property for endomorphisms and strongly self-absorbing C*-algebras

In this paper we define a Rokhlin property for automorphisms of non-unital C*-algebras and for endomorphisms. We show that the crossed product of a C*-algebra by a Rokhlin automorphism preserves absorption of a strongly self-absorbing C*-algebra, and use this result to deduce that the same result holds for crossed products by endomorphisms in the sense of Stacey. This generalizes earlier results of the second named author and W. Winter.

Crossed products by abelian semigroups via transfer operators

We propose a generalisation of Exel's crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler's theory of Toeplitz and Cuntz-Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz-Pimsner algebras of product systems.

A new look at the Crossed-Product of a C*-Algebra by a Semigroup of Endomorphisms

Let G be a group and let P be a subsemigroup of G. In order to describe the crossed product of a C*-algebra A by an action of P by unital endomorphisms we find that we must extend the action to the whole group G. This extension fits into a broader notion of "interaction groups" which consists of an assignment of a positive operator V_g on A for each g in G, obeying a partial group law, and such that (V_g,V_{g^{-1}}) is an interaction for every g, as defined in a previous paper by the author. We then develop a theory of crossed products by interaction groups and compare it to other endomorphism…

Crossed products by endomorphisms of $C_0(X)$-algebras

In the first part of the paper, we develop a theory of crossed products of a $C^*$-algebra $A$ by an arbitrary (not necessarily extendible) endomorphism $α:A\to A$. We consider relative crossed products $C^*(A,α;J)$ where $J$ is an ideal in $A$, and describe up to Morita-Rieffel equivalence all gauge invariant ideals in $C^*(A,α;J)$ and give six term exact sequences determining their $K$-theory. We also obtain certain criteria implying that all ideals in $C^*(A,α;J)$ are gauge invariant, and that $C^*(A,α;J)$ is purely infinite. In the second part, we consider a situation where $A$ is a $C_0(X…

The Crossed Product by a Partial Endomorphism

Given a closed ideal I in a C*-algebra A, an ideal J (not necessarily closed) in I, a *-homomorphism \al:A --> M(I) and a map L:J --> A with some properties, based on [3] and [9] we define a C*-algebra O(A,\al,L) which we call the "Crossed Product by a Partial Endomorphism." In the second section we introduce the Crossed Product by a Partial Endomorphism O(X,\al,L) induced by a local homeomorphism σ:U --> X where X is a compact Hausdorff space and U is an open subset of X.The main result of this section is that every nonzero gauge invariant ideal of O(X,\al,L) has nonzero intersection with C(X…

The Crossed Product by a Partial Endomorphism and the Covariance Algebra

Given a local homeomorphism σ:U -> X where U is a clopen subset of an compact and Hausdorff topological space X, we obtain the possible transfer operators L_ρwhich may occur for \al:C(X) -> C(U) given by \al(f)=fσ. We obtain examples of partial dynamical systems (X_A,σ_A) such that the construction of the covariance algebra C^*(X_A,σ_A) and the crossed product by partial endomorphism O(X_A,\al,L) associated to this system are not equivalent, in the sense that there does not exists invertible function ρin C(U) such that O(X_A,\al,L_ρ)=C^*(X_A,σ).

A New Look at The Crossed-Product of a C*-algebra by an Endomorphism

We give a new definition for the crossed-product of a C*-algebra A by a *-endomorphism α, which depends not only on the pair (A,α) but also on the choice of a transfer operator (defined in the paper). With this we generalize some of the earlier constructions in the situations in which they behave best (e.g. for monomorphisms with hereditary range), but we get a different and perhaps more natural outcome in other situations. For example, we show that the Cuntz-Krieger algebra O_A arises as the result of our construction when applied to the corresponding Markov subshift and a very natural transf…

Crossed Products by Endomorphisms, Vector Bundles and Group Duality

We construct the crossed product of a C(X)-algebra by an endomorphism, in such a way that the endomorphism itself becomes induced by the bimodule of continuous sections of a vector bundle. Some motivating examples for such a construction are given. Furthermore, we study the C*-algebra of G-invariant elements of the Cuntz-Pimsner algebra associated with a G-vector bundle, where G is a (noncompact, in general) group. In particular, the C*-algebra of invariant elements w.r.t. the action of the group of special unitaries of the given vector bundle is a crossed product in the above sense. We also s…

Actions of symbolic dynamical systems on $C^*$-algebras II. Simplicity of $C^*$-symbolic crossed products and some examples

We have introduced a notion of $C^*$-symbolic dynamical system in [K. Matsumoto: Actions of symbolic dynamical systems on $C^*$-algebras, to appear in J. Reine Angew. Math.], that is a finite family of endomorphisms of a $C^*$-algebra with some conditions. The endomorphisms are indexed by symbols and yield both a subshift and a $C^*$-algebra of a Hilbert $C^*$-bimodule. The associated $C^*$-algebra with the $C^*$-symbolic dynamical system is regarded as a crossed product by the subshift. We will study a simplicity condition of the $C^*$-algebras of the $C^*$-symbolic dynamical systems. Some ex…

Gauge-equivariant Hilbert bimodules and crossed products by endomorphisms

C*-endomorphisms arising from superselection structures with non-trivial centre define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz-Pimsner algebra of a vector bundle having the above-mentioned rank and first Chern class, and can be used to construct a duality for abstract (nonsymmetric) tensor categories vs. group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e., the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with …

Representations of Hecke algebras and dilations of semigroup crossed products

We consider a family of Hecke C*-algebras which can be realised as crossed products by semigroups of endomorphisms. We show by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group.

Universal C*-algebras of endomorphisms of groups and partial actions

We present a different way to study the C*-algebra associated with an injective endomorphism of a group G of infinite cokernel. We follow the work of Boava and Exel to construct a partial crossed product representation of that C*-algebra and show properties only using such tools.

Crossed-Products by Finite Index Endomorphisms and KMS states

Given a unital C*-algebra A, an injective endomorphism α:A --> A preserving the unit, and a conditional expectation E from A to the range of αwe consider the crossed-product of A by αrelative to the transfer operator L=α^{-1}E. When E is of index-finite type we show that there exists a conditional expectation G from the crossed-product to A which is unique under certain hypothesis. We define a "gauge action" on the crossed-product algebra in terms of a central positive element h and study its KMS states. The main result is: if h>1 and E(ab)=E(ba) for all a,b in A (e.g. when A is commutative) t…

Exel's crossed product for non-unital C*-algebras

We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for α. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.

Exel's Crossed Product and Relative Cuntz-Pimsner Algebras

We consider Exel's new construction of a crossed product of a C*-algebra A by an endomorphism α. We prove that this crossed product is universal for an appropriate family of covariant representations, and we show that it can be realised as a relative Cuntz-Pimsner algbera. We describe a necessary and sufficient condition for the canonical map from A into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product.

Exel's crossed product and crossed products by completely positive maps

We introduce crossed products of a $C^*$-algebra $A$ by a completely positive map $\varrho:A\to A$ relative to an ideal in $A$. They generalize various crossed products by endomorphisms when $\varrho$ is multiplicative. When $A$ is commutative they include $C^*$-algebras associated to Markov operators by Ionescu, Muhly, Vega, and to topological relations by Brenken, but in general they are not modeled by topological quivers popularized by Muhly and Tomforde. We show that Exel's crossed product $A\rtimes_{α,L} \mathbb{N}$, generalized to the case where $A$ is not necessarily unital, is the cros…

Realising the C*-algebra of a higher-rank graph as an Exel crossed product

We use the boundary-path space of a finitely-aligned k-graph Λto construct a compactly-aligned product system X, and we show that the graph algebra C^*(Λ) is isomorphic to the Cuntz-Nica-Pimsner algebra NO(X). In this setting, we introduce the notion of a crossed product by a semigroup of partial endomorphisms and partially-defined transfer operators by defining it to be NO(X). We then compare this crossed product with other definitions in the literature.

Toeplitz algebras associated to Endomorphisms of Ore semigroups

In this paper, we consider the Toeplitz algebra associated to actions of Ore semigroups on $C^{*}$-algebras. In particular, we consider injective and surjective actions of such semigroups. We use the theory of groupoid dynamical systems to represent the Toeplitz algebra as a groupoid crossed product. We also discuss the K-theory of the Toeplitz algebra in some examples. For instance, we show that for the semigroup of positive matrices, the K-theory of the associated Toeplitz algebra vanishes.

Partial-isometric crossed products of dynamical systems by left LCM semigroups

Let P be a left LCM semigroup, and $α$ an action of $P$ by endomorphisms of a $C^{*}$-algebra $A$. We study a semigroup crossed product $C^{*}$-algebra in which the action $α$ is implemented by partial isometries. This crossed product gives a model for the Nica-Teoplitz algebras of product systems of Hilbert bimodules (associated with semigroup dynamical systems) studied first by Fowler, for which we provide a structure theorem as it behaves well under short exact sequences and tensor products.

Extension of automorphisms to C*-crossed products with non-trivial centre

Given a quasi-special endomorphism $ρ$ of a C*-algebra A with nontrivial center, we study an extension problem for automorphisms of A to a minimal cross-product B of A by $ρ$. Exploiting some aspects of the underlying generalized Doplicher-Roberts duality theory based on Pimsner algebras, an obstruction to the existence of such extensions is found and described in terms of sections of a suitable group bundle.

Phase transition on Exel crossed products assocaited to dilation matrices

An integer matrix $A\in M_d(\Z)$ induces a covering $σ_A$ of $\T^d$ and an endomorphism $α_A:f\mapsto f\circ σ_A$ of $C(\T^d)$ for which there is a natural transfer operator $L$. In this paper, we compute the KMS states on the Exel crossed product $C(\T^d)\rtimes_{α_A,L}\N$ and its Toeplitz extension. We find that $C(\T^d)\rtimes_{α_A,L}\N$ has a unique KMS state, which has inverse temperature $β=\log|\det A|$. Its Toeplitz extension, on the other hand, exhibits a phase transition at $β=\log|\det A|$, and for larger $β$ the simplex of KMS$_β$ states is isomorphic to the simplex of probability …

半群 $\mathbb{N}^{2}$ 的偏等距交叉积理想的合成列

The composition series of ideals of the partial-isometric crossed product by the semigroup $\mathbb{N}^{2}$

本文利用半群 $\mathbb{N}^{2}$ 偏等距交叉积到群 $\mathbb{Z}^{2}$ 交叉积的完整角嵌入，构造其本质理想合成列，并确定各子商对应的已知代数。

Crossed products for interactions and graph algebras

We consider Exel's interaction $(V,H)$ over a unital $C^*$-algebra $A$, such that $V(A)$ and $H(A)$ are hereditary subalgebras of $A$. For the associated crossed product, we obtain a uniqueness theorem, ideal lattice description, simplicity criterion and a version of Pimsner-Voiculescu exact sequence. These results cover the case of crossed products by endomorphisms with hereditary ranges and complementary kernels. As model examples of interactions not coming from endomorphisms we introduce and study in detail interactions arising from finite graphs. The interaction $(V,H)$ associated to a gra…

A crossed product of the CAR algebra in the Cuntz algebra

In this paper we show that the Cuntz algebra can be represented as a C*-crossed product by endomorphism of the canonical anticommutation relations (CAR) algebra, generated by the standard recursive fermion system.

Skew-products of higher-rank graphs and crossed products by semigroups

We consider a free action of an Ore semigroup on a higher-rank graph, and the induced action by endomorphisms of the $C^*$-algebra of the graph. We show that the crossed product by this action is stably isomorphic to the $C^*$-algebra of a quotient graph. Our main tool is Laca's dilation theory for endomorphic actions of Ore semigroups on $C^*$-algebras, which embeds such an action in an automorphic action of the enveloping group on a larger $C^*$-algebra.

Nica-Toeplitz algebras associated with product systems over right LCM semigroups

We prove uniqueness of representations of Nica-Toeplitz algebras associated to product systems of $C^*$-correspondences over right LCM semigroups by applying our previous abstract uniqueness results developed for $C^*$-precategories. Our results provide an interpretation of conditions identified in work of Fowler and Fowler-Raeburn, and apply also to their crossed product twisted by a product system, in the new context of right LCM semigroups, as well as to a new, Doplicher-Roberts type $C^*$-algebra associated to the Nica-Toeplitz algebra. As a derived construction we develop Nica-Toeplitz cr…

Spectral triples on irreversible $C^*$-dynamical systems

Given a spectral triple on a $C^*$-algebra $\mathcal A$ together with a unital injective endomorphism $α$, the problem of defining a suitable crossed product $C^*$-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and of Hawkins, Skalski, White and Zacharias, and on our previous papers. The embedding of $α(\mathcal A)$ in $\mathcal A$ can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricit…

Discrete product systems and twisted crossed products by semigroups

A product system E over a semigroup P is a family of Hilbert spaces {E_s:s\in P} together with multiplications E_s \times E_t\to E_{st}. We view E as a unitary- valued cocycle on P, and consider twisted crossed products A \times_{β,E} P involving E and an action βof P by endomorphisms of a C*-algebra A. When P is quasi-lattice ordered in the sense of Nica, we isolate a class of covariant representations of E, and consider a twisted crossed product B_P \times_{τ,E} P which is universal for covariant representations of E when E has finite-dimensional fibres, and in general is slightly larger. In…

On transfer operators for C*-dynamical systems

The theme of the paper is the question of existence and basic structure of transfer operators for endomorphisms of a unital C*-algebra. We establish a complete description of non-degenerate transfer operators, characterize complete transfer operators and clarify their role in crossed product constructions. Also, we give necessary and sufficient conditions for existence of transfer operators for commutative systems, and discuss their form for endomorphisms of B(H) which is relevant to Kadison-Singer problem.

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…

关于半群动力系统和乘积系统的简化的$C^*$-代数的一些注记

Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems

研究半群动力系统的简化的交叉积和乘积系统的简化的$C^*$-代数的精确性，证明在合理假设下，简化的交叉积精确当且仅当原代数精确。

Extensions of C*-dynamical systems to systems with complete transfer operators

Starting from an arbitrary endomorphism $α$ of a unital C*-algebra $A$ we construct a bigger C*-algebra $B$ and extend $α$ onto $B$ in such a way that the extended endomorphism $α$ has a unital kernel and a hereditary range, i.e. there exists a unique non-degenerate transfer operator for $(B,α)$, called the complete transfer operator. The pair $(B,α)$ is universal with respect to a suitable notion of a covariant representation and depends on a choice of an ideal in $A$. The construction enables a natural definition of the crossed product for arbitrary $α$.

Infinite-dimensional transfer operators, endomorphisms, and measurable partitions

We develop a new duality between endomorphisms of measure spaces, on the one hand, and a certain family of positive operators, called transfer operators, acting in spaces of measurable functions on, on the other. A framework of standard Borel spaces is adopted; and this generality is wide enough to cover a host of applications. While the mathematical structures of positive operators, endomorphisms, transfer operators, measurable partitions, and Markov processes arise in a host of settings, both pure and applied, we propose here a unified study. This is the general setting of dynamics in Borel …

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…

Lower bounds in $L^p$-transference for crossed-products

Let $Γ\curvearrowright Ω$ be a measure-preserving action and $\mathcal{L} Γ\hookrightarrow L^\infty(Ω) \rtimes Γ$ the natural inclusion of the group von Neumann algebra into the crossed product. When $μ(Ω) = \infty$, we have that this natural embedding is not trace-preserving and therefore does not extends boundedly to the associated noncommutative $L^p$-spaces. Nevertheless, we show that when $Ω$ has an invariant mean there is an isometric embedding of $L^p(\mathcal{L} Γ)$ into an ultrapower of $L^p(Ω\rtimes Γ)$ that intertwines Fourier multipliers and is $\mathcal{L} Γ$-bimodular. As a conse…

Purely infinite simple $C^*$-algebras arising from free product constructions

Examples of simple, separable, unital, purely infinite $C^*$--algebras are constructed, including: (1) some that are not approximately divisible; (2) those that arise as crossed products of any of a certain class of $C^*$--algebras by any of a certain class of non--unital endomorphisms; (3) those that arise as reduced free products of pairs of $C^*$--algebras with respect to any from a certain class of states.

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…

QWEP性质在扭曲交叉积下的稳定性

QWEP stability under twisted crossed products

本文研究扭曲交叉积的QWEP性质，证明其不被超线性群交叉积保持，给出非QWEP例子，并建立标量扭曲下QWEP的等价条件及正则圈积的限制定理，为Fourier乘子转移提供输入并识别范数障碍。

Crossed Product Equivalence of Quantum Automorphism Groups

We compare the algebras of the quantum automorphism group of finite-dimensional C$^\ast$-algebra $B$, which includes the quantum permutation group $S_N^+$, where $N = \dim B$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $Γ$ lead to isomorphic $\ast$-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $1$-boundedness between the various algebras associated with these quantum groups.

具有弱迹罗克林性质的紧群作用

Crossed products by compact group actions with the weak tracial Rokhlin property

本文研究了具有弱迹罗克林性质的紧群作用，证明了相关稳定性性质可从前代数转移到交叉积，并给出了非平凡例子。