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) then the KMS_βstates are precisely those of the form ψ= ϕG, where ϕis a trace on A satisfying the identity ϕ(a) = ϕ(L(h^{-β}ind(E)a)), where ind(E) is the Jones-Kosaki-Watatani index of E.
We provide a general description of the KMS states for flows whose fixed point algebra satisfies a certain regularity condition. This is the applied to crossed products by discrete groups, and in particular to certain flows on crossed products by discrete abelian groups where the methods can be combined with spectral analysis for abelian automrphism groups.
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 …
We extend ultragraph shift spaces and the realization of ultragraph C*-algebras as partial crossed products to include ultragraphs with sinks (under a mild condition, called (RFUM2), which allow us to dismiss the use of filters) and we describe the associated transformation groupoid. Using these characterizations we study continuous orbit equivalence of ultragraph shift spaces (via groupoids) and KMS and ground states (via partial crossed products).
It is shown that the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism is purely infinite when it is simple, provided the endomorphism does not leave a trace state invariant and maps the unit to a full projection.
We construct the crossed product of a C(X)-algebra by an endomorphism, in such a way that it becomes induced by a Hilbert C(X)-bimodule. Furthermore we introduce the notion of C(X)-category, and discuss relationships with crossed products and duality for compact groups.
Let $φ:X\to X$ be a homeomorphism of a compact metric space $X$. For any continuous function $F:X\to \mathbb{R}$ there is a one-parameter group $α^{F}$ of automorphisms on the crossed product $C^*$-algebra $C(X)\rtimes_φ\mathbb{Z}$ defined such that $α^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and $U$ is the canonical unitary in the construction of the crossed product. In this paper we study the KMS states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS-states can be very rich and complicated. Our resul…
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…
We study the crossed product $C^*$-algebra associated to injective endomorphisms, which turns out to be equivalent to study the crossed product by the dilated autormorphism. We prove that the dilation of the Bernoulli $p$-shift endomorphism is topologically free. As a consequence, we have a way to twist any endomorphism of a $\D$-absorbing $C^*$-algebra into one whose dilated automorphism is essentially free and have the same $K$-theory map than the original one. This allows us to construct purely infinite crossed products $C^*$-algebras with diverse ideal structures.
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.
We describe simplicity of the Stacey crossed product A\times_β\N in terms of conditions of the endomorphism β. Then, we use a characterization of the graph C*-algebras C*(E) as the Stacey crossed product C*(E)^γ\times_{β_E}\N to study its ideal properties, in terms of the (non-classical) C*-dynamical system (C*(E)^γ, β_E). Finally, we give sufficient conditions for the Stacey crossed product A\times_β\N being a purely infinite simple C*-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).
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…
We study C*-algebra endomorphims which are special in a weaker sense w.r.t. the notion introduced by Doplicher and Roberts. We assign to such endomorphisms a geometrical invariant, representing a cohomological obstruction for them to be special in the usual sense. Moreover, we construct the crossed product of a C*-algebra by the action of the dual of a (nonabelian, noncompact) group of vector bundle automorphisms. These crossed products supply a class of examples for such generalized special endomorphisms.
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…
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.
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…
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 …
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 …
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…
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…
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.
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.
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 $\mathbb{N}^2$ …
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.
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…
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.
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.
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…
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.
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.
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.
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…
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…
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…
We describe KMS and ground states arising from a generalized gauge action on ultragraph C*-algebras. We focus on ultragraphs that satisfy Condition~(RFUM), so that we can use the partial crossed product description of ultragraph C*-algebras recently described by the second author and Danilo Royer. In particular, for ultragraphs with no sinks, we generalize a recent result by Toke Carlsen and Nadia Larsen: Given a time evolution on the C*-algebra of an ultragraph, induced by a function on the edge set, we characterize the KMS states in five different ways and ground states in four different way…
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.
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.
We study the Toeplitz algebra $\TT(\N\rtimes\N^\times)$ and three quotients of this algebra: the $C^*$-algebra $\qn$ recntly introduced by Cuntz, and two new ones, which we call the additive and multiplicative boundary quotients. These quotients are universal for Nica-covariant representations of $\N\rtimes\N^\times$ satisfying extra relations, and can be realised as partial crossed products. We use the structure theory for partial crossed products to prove a uniqueness theorem for the additive boundary quotient, and use the recent analysis of KMS states on $\TT(\nxnx)$ to describe the KMS sta…
A classical construction associates to a transient random walk on a discrete group $Γ$ a compact $Γ$-space $\partial_M Γ$ known as the Martin boundary. The resulting crossed product $C^*$-algebra $C(\partial_M Γ) \rtimes_r Γ$ comes equipped with a one-parameter group of automorphisms given by the Martin kernels that define the Martin boundary. In this paper we study the KMS states for this flow and obtain a complete description when the Poisson boundary of the random walk is trivial and when $Γ$ is a torsion free non-elementary hyperbolic group. We also construct examples to show that the stru…
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…
Based on an embedding formula of the CAR algebra into the Cuntz algebra ${\mathcal O}_{2^p}$, properties of the CAR algebra are studied in detail by restricting those of the Cuntz algebra. Various $\ast$-endomorphisms of the Cuntz algebra are explicitly constructed, and transcribed into those of the CAR algebra. In particular, a set of $\ast$-endomorphisms of the CAR algebra into its even subalgebra are constructed. According to branching formulae, which are obtained by composing representations and $\ast$-endomorphisms, it is shown that a KMS state of the CAR algebra is obtained through the a…
Given a topologically free action of a countable group $G$ on a compact metric space $X$, there is a canonical correspondence between continuous 1-cocycles for this group action and diagonal 1-parameter groups of automorphisms of the reduced crossed product C*-algebra. The KMS spectrum is defined as the set of inverse temperatures for which there exists a KMS state. We prove that the possible KMS spectra depend heavily on the nature of the acting group $G$. For groups of subexponential growth, we prove that the only possible KMS spectra are $\{0\}$, $[0,+\infty)$, $(-\infty,0]$ and $\mathbb{R}…
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.
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.
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…
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.
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.
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.
The relative graph $C^*$-algebras introduced by Muhly and Tomforde are generalizations of both graph algebras and their Toeplitz extensions. For an arbitrary graph $E$ and a subset $R$ of the set of regular vertices of $E$ we show that the relative graph $C^*$-algebra $C^*(E, R)$ is isomorphic to a partial crossed product for an action of the free group generated by the edge set on the relative boundary path space. Given a time evolution on $C^*(E, R)$ induced by a function on the edge set, we characterize the KMS$_β$ states and ground states using an abstract result of Exel and Laca. Guided b…
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.
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.
We examine Nica-Pimsner algebras associated with semigroup actions of $\mathbb{Z}_+^n$ on a C*-algebra $A$ by $*$-endomorphisms. We give necessary and sufficient conditions on the dynamics for exactness and nuclearity of the Nica-Pimsner algebras. Furthermore we parameterize the KMS states at finite temperature by the tracial states on $A$. A parametrization is also shown for KMS states at zero temperature (resp. ground states) by the tracial states on $A$ (resp. states on $A$). Finally we give a formula for obtaining tracial states on the Nica-Pimsner algebras.
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 $α$.
