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We obtain criteria for when a ring with enough idempotents is left/right artinian or noetherian in terms of local criteria defined by the associated complete set of idempotents for the ring. We apply these criteria to object unital category graded rings in general and, in particular, to the class of skew category algebras. Thereby, we generalize results by Nastasescu-van Oystaeyen, Bell, Park, and Zelmanov from the group graded case to groupoid, and in some cases category, gradings. [ABSTRACT FROM AUTHOR]

Bernal et al. provided a necessary and sufficient condition for a linear code to be realized as an ideal in a finite group algebra and De La Cruz and Willems proved a similar result for ideals in twisted group algebras. In this paper, we extend this characterization to crossed products. Furthermore, we determine conditions for some crossed product codes to be self-dual. [ABSTRACT FROM AUTHOR]

For finite groups X, G and the right G-action on X by group automorphisms, the non-balanced quantum double D(X; G) is defined as the crossed product (ℂXop)* ⋊ ℂG. We firstly prove that D(X; G) is a finite-dimensional Hopf C*-algebra. For any subgroup H of G, D(X; H) can be defined as a Hopf C*-subalgebra of D(X; G) in the natural way. Then there is a conditonal expectation from D(X; G) onto D(X; H) and the index is [G; H]. Moreover, we prove that an associated natural inclusion of non-balanced quantum doubles is the crossed product by the group algebra. [ABSTRACT FROM AUTHOR]

We examine the structure of two possible candidates of isometry groups for the spectral triples on AF-algebras introduced by Christensen and Ivan. In particular, we completely determine the isometry group introduced by Park and observe that these groups coincide in the case of the Cantor set. We also show that the construction of spectral triples on crossed products given by Hawkins, Skalski, White, and Zacharias is suitable for the purpose of lifting isometries. [ABSTRACT FROM AUTHOR]

Let R be a left-symmetric conformal algebra and Q be a C [ ∂ ] -module. We introduce the notion of a unified product for left-symmetric conformal algebras and apply it to construct an object H R 2 (Q , R) to describe and classify all left-symmetric conformal algebra structures on the direct sum E = R ⊕ Q as a C [ ∂ ] -module such that R is a subalgebra of E up to isomorphism whose restriction on R is the identity map. Moreover, we study H R 2 (Q , R) in detail when Q, R are free as C [ ∂ ] -modules and rank Q = 1 . Some special products such as crossed product and bicrossed product are also investigated. [ABSTRACT FROM AUTHOR]

The tracial cyclic Rokhlin property for automorphisms of simple not necessarily unital C*-algebras is investigated. We show that the tracial cyclic Rokhlin property is preserved by going to certain restrictions to subalgebras and taking direct limit or tensor products of actions. We also show that under certain conditions properties such as real rank zero, the tracial rank zero, stable rank one, (tracial) Z-stability, Property (SP), strict comparison on projections are passed to crossed products under automorphisms with the tracial cyclic Rokhlin property. [ABSTRACT FROM AUTHOR]

We examine inclusions of C*-algebras of the form AH⊆A⋊r​G, where G and H are groups acting on a unital simple C*-algebra A by outer automorphisms and H is finite. It follows from a theorem of Izumi that AH⊆A is C*-irreducible, in the sense that all intermediate C*-algebras are simple. We show that AH⊆A⋊r​G is C*-irreducible for all G and H as above if and only if G and H have trivial intersection in the outer automorphisms of A, and we give a~Galois type classification of all intermediate C*-algebras in the case when H is abelian and the two actions of G and H on A commute. We illustrate these results with examples of outer group actions on the irrational rotation C*-algebras. We exhibit, among other examples, C*-irreducible inclusions of AF-algebras that have intermediate C*-algebras that are not AF-algebras; in fact, the irrational rotation C*-algebra appears as an intermediate C*-algebra. [ABSTRACT FROM AUTHOR]

Let N and H be groups, and let G be an extension of H by N. In this article, we describe the structure of the complex group ring of G in terms of data associated with N and H. In particular, we present conditions on the building blocks N and H guaranteeing that G satisfies the zero-divisor and idempotent conjectures. Moreover, for central extensions involving amenable groups we present conditions on the building blocks guaranteeing that the Kadison–Kaplansky conjecture holds for the group C ∗ -algebra of G. [ABSTRACT FROM AUTHOR]

We give a survey of Rokhlin-type properties of group actions on operator algebras. Though we briefly review the classification of group actions on von Neumann algebras, we mainly overview group actions on C ∗ -algebras, focusing on classification of group actions and preservation of properties under taking C ∗ -crossed products for actions with Rokhlin-type properties. [ABSTRACT FROM AUTHOR]

We give efficient conditions under which a C*-subalgebra A ⊆ B separates ideals in a C*-algebra B, and B is purely infinite if every positive element in A is properly infinite in B. We specialise to the case when B is a crossed product for an inverse semigroup action by Hilbert bimodules or a section C*-algebra of a Fell bundle over an étale, possibly non-Hausdorff, groupoid. Then our theory works provided B is the recently introduced essential crossed product and the action is essentially exact and residually aperiodic or residually topologically free. These last notions are developed in the article. [ABSTRACT FROM AUTHOR]

In his seminal paper [On the XY-model on two-sided infinite chain, Publ. RIMS Kyoto Univ. 20 (1984) 277–296], Araki introduced an elegant extension of the Jordan–Wigner transformation which establishes a precise connection between quantum spin systems and Fermi lattice gases in one dimension in the so-called infinite system idealization of quantum statistical mechanics. His extension allows in particular for the rigorous study of numerous aspects of the prominent XY chain over the two-sided infinite discrete line without having to resort to a thermodynamic limit procedure at an intermediate or at the final stage. We rigorously review and elaborate this extension from scratch which makes the paper rather self-contained. In the course of the construction, we also present a simple and concrete realization of Araki's crossed product extension. [ABSTRACT FROM AUTHOR]

Let G be a group and F be a field of characteristic p ≥ 0 . In this paper we provide some necessary and sufficient conditions for a twisted group algebra F τ G to satisfies a non-matrix identity. This allows us to show that a crossed product F σ τ G is Lie nilpotent if and only if σ is trivial, G is nilpotent and p-abelian, G has a unique normal Sylow p-subgroup P and G = P × Q for some central p ′ -subgroup Q and F τ G is stably untwisted. Also, Z σ τ G is Lie nilpotent if and only if Z σ τ G is commutative. Some generalized group identities on the unit group of (HTML translation failed) are also investigated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Given a spectral triple on a unital C * -algebra A and an equicontinuous action of a discrete group G on A , a spectral triple on the reduced crossed product C * -algebra A ⋊ r G was constructed by Hawkins, Skalski, White and Zacharias in [On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], extending the construction by Belissard, Marcolli and Reihani in [Dynamical systems on spectral metric spaces, preprint (2010), arXiv:1008.4617], by using the Kasparov product to make an ansatz for the Dirac operator. Supposing that the triple on A is equivariant for an action of G , we show that the triple on A ⋊ r G is equivariant for the dual coaction of G. If moreover an equivariant real structure J is given for the triple on A , we give constructions for two inequivalent real structures on the triple A ⋊ r G. We compute the KO-dimension with respect to each real structure in terms of the KO-dimension of J and show that the first- and the second-order conditions are preserved. Lastly, we characterize an equivariant orientation cycle on the triple on A ⋊ r G coming from an equivariant orientation cycle on the triple on A. We show, along the paper, that our constructions generalize the respective constructions of the equivariant spectral triple on the noncommutative 2 -torus. [ABSTRACT FROM AUTHOR]

For p ∈ [ 1 , ∞) , we show that every unital L p -operator algebra contains a unique maximal C ⁎ -subalgebra, which is always abelian if p ≠ 2. Using this, we canonically associate to every unital L p -operator algebra A an étale groupoid G A , which in many cases of interest is a complete invariant for A. By identifying this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case p = 2 ; the most striking one being that of crossed products by topologically free actions. Our rigidity results give answers to questions concerning the existence of isomorphisms between different algebras. Among others, we show that for the L p -analog O 2 p of the Cuntz algebra, there is no isometric isomorphism between O 2 p and O 2 p ⊗ p O 2 p , when p ≠ 2. In particular, we deduce that there is no L p -version of Kirchberg's absorption theorem, and that there is no K -theoretic classification of purely infinite simple amenable L p -operator algebras for p ≠ 2. Our methods also allow us to recover a folklore fact in the case of C*-algebras (p = 2), namely that no isomorphism O 2 ≅ O 2 ⊗ O 2 preserves the canonical Cartan subalgebras. [ABSTRACT FROM AUTHOR]

A Novikov–Poisson algebra (A , ∘ , ·) is a vector space with a Novikov algebra structure (A , ∘) and a commutative associative algebra structure (A , ·) satisfying some compatibility conditions. Give a Novikov–Poisson algebra (A , ∘ , ·) and a vector space V. A natural problem is how to construct and classify all Novikov–Poisson algebra structures on the vector space E = A ⊕ V such that (A , ∘ , ·) is a subalgebra of E up to isomorphism whose restriction on A is the identity map. This problem is called extending structures problem. In this paper, we introduce the definition of a unified product for Novikov–Poisson algebras, and then construct an object GH 2 (V , A) to answer the extending structures problem. Note that unified product includes many interesting products such as bicrossed product, crossed product and so on. Moreover, the special case when dim (V) = 1 is investigated in detail. [ABSTRACT FROM AUTHOR]

Suppose that (G , G +) is a lattice-ordered abelian group with positive cone G + . Adji and Raeburn have shown a result about the structure of the primitive ideal space of the C ∗ -algebra B G + × α G + for a totally ordered abelian group (Adji and Raeburn in Integral Equ Oper Theory 48:281–293, 2004, Theorem 3.1). In this paper, we show if ∑ (G) is the chain of subgroups H of G, where H : = H + - H + and H + is any hereditary subsemigroup of G + . Then, there exists a well-defined map F from the disjoint union ⨆ { H ^ : H ∈ ∑ G } to the primitive ideals of the C ∗ -algebra B G + × α G + . Our result is interesting and has more challenges, since we are working under the assumption that the group G is lattice-ordered and extends their result to more general cases. [ABSTRACT FROM AUTHOR]

We study the Fredholm solvability for a new class of nonlocal boundary value problems associated with group actions on smooth manifolds. Namely, we consider the case in which the group action is defined on an ambient manifold without boundary and does not preserve the manifold with boundary on which the problem is stated. In particular, the group action does not map the boundary into itself. The orbits of the boundary under the group action split the manifold into subdomains, and this decomposition, being combined with the -algebra techniques, plays an important role in our approach to the analysis of the problem. [ABSTRACT FROM AUTHOR]

We prove that if G is a discrete group and (A , G , α) is a C*-dynamical system such that the reduced crossed product A ⋊ r , α G possesses property (SOAP) then every completely compact Herz–Schur (A , G , α) -multiplier can be approximated in the completely bounded norm by Herz–Schur (A , G , α) -multipliers of finite rank. As a consequence, if G has the approximation property (AP) then the completely compact Herz–Schur multipliers of A(G) coincide with the closure of A(G) in the completely bounded multiplier norm. We study the class of invariant completely compact Herz–Schur multipliers of A ⋊ r , α G and provide a description of this class in the case of the irrational rotation algebra. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the paper, Matsumoto ['Subshifts, $\lambda $ -graph bisystems and $C^*$ -algebras', J. Math. Anal. Appl. 485 (2020), 123843]. A $\lambda $ -graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift $\Lambda $ , there exists a $\lambda $ -graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ with shift automorphism $\rho _{\mathcal {L}}$ from a $\lambda $ -graph bisystem $({\mathcal {L}}^-,{\mathcal {L}}^+)$ , and define a $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ by the crossed product. It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If $\lambda $ -graph bisystems come from two-sided subshifts, these $C^*$ -algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ and the K-theory formulas of the $C^*$ -algebras ${\mathcal {F}}_{\mathcal {L}}$ and ${\mathcal R}_{\mathcal {L}}$. The K-group for the AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ is regarded as a two-sided extension of the dimension group of subshifts. [ABSTRACT FROM AUTHOR]

Given a non-necessarily commutative unital ring R and a unital partial representation Θ of a group G into the Picard semigroup PicS (R) of the isomorphism classes of partially invertible R -bimodules, we construct an abelian group C (Θ / R) formed by the isomorphism classes of partial generalized crossed products related to Θ and identify an appropriate second partial cohomology group of G with a naturally defined subgroup C 0 (Θ / R) of C (Θ / R). Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of non-necessarily commutative rings R ⊆ S with the same unity and a unital partial representation G → S R (S) of an arbitrary group G into the monoid S R (S) of the R -subbimodules of S. This generalizes the works by Kanzaki and Miyashita. [ABSTRACT FROM AUTHOR]

We consider the tracial crossed product algebra M = A ⋊ Λ arising from a trace preserving action σ : Λ ↷ A of a discrete group Λ on a tracial von Neumann algebra A. For a unitary subgroup G ⊂ U (M) , we study when this G can be conjugated into U (A) ⋅ Λ in M. We provide a general sufficient condition for this to happen. Our result generalizes [14, Theorem 3.1] which treats the case when M is the group von Neumann algebra L (Λ). [ABSTRACT FROM AUTHOR]

We prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products C(X) \rtimes _\lambda G, where G is a countable discrete group, and X is a compact Hausdorff space which G acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of C(X) \rtimes _\lambda G and to Kawabe's generalized space of amenable subgroups \operatorname {Sub}_a(X,G). This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if C(Y) \subseteq C(X) is an inclusion of unital commutative G-C*-algebras with X minimal and C(Y) \rtimes _\lambda G simple, then any intermediate C*-algebra A satisfying C(Y) \rtimes _\lambda G \subseteq A \subseteq C(X) \rtimes _\lambda G is simple. [ABSTRACT FROM AUTHOR]

Recent work by Baum et al. ['Expanders, exact crossed products, and the Baum–Connes conjecture', Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. ['Exotic crossed products and the Baum–Connes conjecture', J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action $(C,\unicode[STIX]{x1D6FE})$ , then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact, and if $(C,\unicode[STIX]{x1D6FE})$ is the action by translation on $\ell ^{\infty }(G)$ , we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the $E$ -ization functor we defined earlier, where $E$ is a large ideal of $B(G)$. [ABSTRACT FROM AUTHOR]

Given a spectral triple on a C ∗ -algebra 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 [A. Hawkins, A. Skalski, S. White and J. Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], and on our previous papers [V. Aiello, D. Guido and T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448(2) (2017) 1378–1412; V. Aiello, D. Guido and T. Isola, A spectral triple for a solenoid based on the Sierpinski gasket, SIGMA Symmetry Integrability Geom. Methods Appl. 17(20) (2021) 21]. The embedding of α () in 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 isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on and α (). [ABSTRACT FROM AUTHOR]

In this note we correct two propositions from the paper "Exact sequences in the enchilada category". Moreover, we present our further investigation on monomor- phisms and epimorphisms in the enchilada category. [ABSTRACT FROM AUTHOR]

If E is a directed graph and K is a field, the Leavitt path algebra L K (E) of E over K is naturally graded by the group of integers ℤ. We formulate properties of the graph E which are equivalent with L K (E) being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of L K (E) are also characterized in terms of the pre-ordered group properties of the Grothendieck ℤ -group of L K (E). If E has finitely many vertices, we characterize when L K (E) is strongly graded in terms of the properties of K 0 Γ (L K (E)). Our proof also provides an alternative to the known proof of the equivalence L K (E) is strongly graded if and only if E has no sinks for a finite graph E. We also show that, if unital, the algebra L K (E) is strongly graded and graded unit-regular if and only if L K (E) is a crossed product. In the process of showing the main result, we obtain conditions on a group Γ and a Γ -graded division ring K equivalent with the requirements that a Γ -graded matrix ring R over K is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group Γ on the Grothendieck Γ -group K 0 Γ (R). [ABSTRACT FROM AUTHOR]

Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let α : G → Aut(A) be an action of G on A which has the weak tracial Rokhlin property. Let Aα be the fixed point algebra. Then the radius of comparison satisfies rc(Aα) ≤ rc(A) and rc(C* (G, A, α)) ≤ (1/card(G)) ⋅ rc(A). The inclusion of Aα in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu(Aα) to the fixed points of the purely positive part of Cu(A), and the purely positive part of Cu(C*(G, A, α)) is isomorphic to this semigroup. We construct an example in which G = Z/ 2Z, A is a simple unital AH algebra, α has the Rokhlin property, rc(A) > 0 , rc(Aα) = rc(A), and rc(C* (G, A, α)) = (1/2) rc(A). [ABSTRACT FROM AUTHOR]

We define operator-valued Schur and Herz–Schur multipliers in terms of module actions, and show that the standard properties of these multipliers follow from well-known facts about these module actions and duality theory for group actions. These results are applied to study the Herz–Schur multipliers of an abelian group acting on its Pontryagin dual: it is shown that a natural subset of these Herz–Schur multipliers can be identified with the classical Herz–Schur multipliers of the direct product of the group with its dual group. [ABSTRACT FROM AUTHOR]

Using the recently developed theory of Herz–Schur multipliers of a C ∗ -dynamical system, we prove equality of the Haagerup constants for a C ∗ -algebra and its crossed product by an amenable action. [ABSTRACT FROM AUTHOR]

We develop a group graded Morita theory over a G-graded G-acted algebra, where G is a finite group.

We study the K-theory ranks for crossed products of C∗-algebras by the group of integers. As an application, we obtain certain estimates for the K-theory ranks of the group C∗-algebras of torsion free, finitely generated, nilpotent or solvable discrete groups, written as successive semi-direct products.

In this paper, we introduce all Green's relations on the ternary semihypergroup and study some properties of these equivalence relations in view of those obtained in binary and ternary semigroups. We also define crossed hyperproduct of two hypergroups. [ABSTRACT FROM AUTHOR]

A Novikov–Poisson algebra (A,∘,·) is a vector space with a Novikov algebra structure (A,∘) and a commutative associative algebra structure (A,·) satisfying some compatibility conditions. Give a Novikov–Poisson algebra (A,∘,·) and a vector space V. A natural problem is how to construct and classify all Novikov–Poisson algebra structures on the vector space E=A⊕V such that (A,∘,·) is a subalgebra of E up to isomorphism whose restriction on A is the identity map. This problem is called extending structures problem. In this paper, we introduce the definition of a unified product for Novikov–Poisson algebras, and then construct an object GH2(V,A) to answer the extending structures problem. Note that unified product includes many interesting products such as bicrossed product, crossed product and so on. Moreover, the special case when dim(V)=1 is investigated in detail.

The aim of this article is to provide an answer to the C [ ∂ ] -split extending structures problem for Leibniz conformal algebras, which asks that how to describe all Leibniz conformal algebra structures on E = R ⊕ Q up to an isomorphism such that R is a Leibniz conformal subalgebra. For this purpose, a unified product of Leibniz conformal algebras is introduced. Using this tool, two cohomological type objects are constructed to classify all such extending structures up to an isomorphism. Then this general theory is applied to the special case when R is a free C [ ∂ ] -module and Q is a free C [ ∂ ] -module of rank one. Finally, the twisted product, crossed product, and bicrossed product between two Leibniz conformal algebras are introduced as special cases of the unified product, and some examples are given. [ABSTRACT FROM AUTHOR]

We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizing previous results from the unital case to a non-unital situation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain a presentation for crossed product of some infinite groups and then find its complete rewriting system. Hence, we present normal form structure of elements of crossed product of infinite groups which yield solvability of the word problem. [ABSTRACT FROM AUTHOR]

In this paper we consider the algebraic crossed product ${\mathcal{A}}:=C_{K}(X)\rtimes _{T}\mathbb{Z}$ induced by a homeomorphism $T$ on the Cantor set $X$ , where $K$ is an arbitrary field with involution and $C_{K}(X)$ denotes the $K$ -algebra of locally constant $K$ -valued functions on $X$. We investigate the possible Sylvester matrix rank functions that one can construct on ${\mathcal{A}}$ by means of full ergodic $T$ -invariant probability measures $\unicode[STIX]{x1D707}$ on $X$. To do so, we present a general construction of an approximating sequence of $\ast$ -subalgebras ${\mathcal{A}}_{n}$ which are embeddable into a (possibly infinite) product of matrix algebras over $K$. This enables us to obtain a specific embedding of the whole $\ast$ -algebra ${\mathcal{A}}$ into ${\mathcal{M}}_{K}$ , the well-known von Neumann continuous factor over $K$ , thus obtaining a Sylvester matrix rank function on ${\mathcal{A}}$ by restricting the unique one defined on ${\mathcal{M}}_{K}$. This process gives a way to obtain a Sylvester matrix rank function on ${\mathcal{A}}$ , unique with respect to a certain compatibility property concerning the measure $\unicode[STIX]{x1D707}$ , namely that the rank of a characteristic function of a clopen subset $U\subseteq X$ must equal the measure of $U$. [ABSTRACT FROM AUTHOR]

Let M be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes _\sigma \Gamma $. When Γ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes _\sigma \Gamma $ is full if and only if M is full and the quotient map $\overline {\sigma } : \Gamma \rightarrow {\rm out}(M)$ has finite kernel and discrete image. This answers the question of Jones from [11]. When M is full and Γ is arbitrary, we give a sufficient condition for $M \rtimes _\sigma \Gamma $ to be full which generalizes both Jones' criterion and Choda's criterion. In particular, we show that if M is any full factor (possibly of type III) and Γ is a non-inner amenable group, then the crossed product $M \rtimes _\sigma \Gamma $ is full. [ABSTRACT FROM AUTHOR]

The article considers construction of generalized crossed product of an arbitrary quaternions skew field and Klein four-group relative to factor system. It is well known that such crossed product is semisimple ring. Under specific conditions it is easy to show that crossed product of simple algebra and its inner automorphism group is central simple algebra. Finding out the conditions under which the crossed product is division algebra we face to difficulties of general case linked to analysis of system of linear equations defined over non-commutative rings. In the terms of anisotropic quadratic forms, there are sufficient conditions under those the crossed product of skew field of quaternions and four-group relative to factor system is division algebra. In addition, it is proved that the crossed product is the tensor product of two quaternions skew fields.

In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ( C ∗ -algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions — via the construction of a sectional algebra — are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; semidirect products of bundles correspond to "naïve" crossed products of algebras; skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever 𝜃 is a ∧-preaction of a discrete inverse semigroupoid S on an ample (possibly non-Hausdorff) groupoid 𝒢 , the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of 𝒢 by S. This is a far-reaching generalization of analogous results which had been proven in particular cases. [ABSTRACT FROM AUTHOR]

We prove a group graded Morita equivalences version of the \butter y theorem" on character triples. This gives a method to construct an equivalence between block extensions from another related equivalence. [ABSTRACT FROM AUTHOR]

Let D be a division ring of fractions of a crossed product F[G, η, α], where F is a skew field and G is a group with Conradian left-order ≤. For D we introduce the notion of freeness with respect to ≤ and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space F((G)) of all formal power series in G over F with respect to ≤. From this we obtain that all division rings of fractions of F[G, η, α] which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, F[G, η, α] possesses a division ring of fraction which is free in this sense if and only if the rational closure of F[G, η, α] in the endomorphism ring of the corresponding right F-vector space F((G)) is a skew field. [ABSTRACT FROM AUTHOR]

Given an inverse semigroup S endowed with a partial action on a topological space X , we construct a groupoid of germs S ⋉ X in a manner similar to Exel's groupoid of germs, and similarly a partial action of S on an algebra A induces a crossed product A ⋊ S. We then prove, in the setting of partial actions, that if X is locally compact Hausdorff and zero-dimensional, then the Steinberg algebra of the groupoid of germs S ⋉ X is isomorphic to the crossed product A R (X) ⋊ S , where A R (X) is the Steinberg algebra of X. We also prove that the converse holds, that is, that under natural hypotheses, crossed products of the form A R (X) ⋊ S are Steinberg algebras of appropriate groupoids of germs of the form S ⋉ X. We introduce a new notion of topologically principal partial actions, which correspond to topologically principal groupoids of germs, and study orbit equivalence for these actions in terms of isomorphisms of the corresponding groupoids of germs. This generalizes previous work of the second-named author as well as from others, which dealt mostly with global actions of semigroups or partial actions of groups. We finish the article by comparing our notion of orbit equivalence of actions and orbit equivalence of graphs. [ABSTRACT FROM AUTHOR]

We define exact sequences in the enchilada category of C-algebras and correspondences, and prove that the reduced-crossed-product functor is not exact for the enchilada categories. Our motivation was to determine whether we can have a better understanding of the Baum-Connes conjecture by using enchilada categories. Along the way we prove numerous results showing that the enchilada category is rather strange. [ABSTRACT FROM AUTHOR]

A Morita equivalence similar to that found by Green for crossed products by groups will be established for crossed products by inverse semigroups. More precisely, let S be an inverse semigroup, T a finite sub-inverse semigroup of S and A an S-algebra or a T-algebra. Then the crossed product A ⋊ T is Morita equivalent to a certain crossed product B ⋊ S . [ABSTRACT FROM AUTHOR]