Narrow Results

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.

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.

Let G be a second countable locally compact group and A a separable continuous trace C*-algebra. To each pointwise unitary coaction δ of G on A one can associate a proper G-bundle , π × μ → π. We show that two pointwise unitary coactions δ and ∊ of G on A are exterior equivalent if and only if the proper G-bundles and are isomorphic. Thus, if A is stable, there exists a bijection between the isomorphism classes of proper G-bundles over and the exterior equivalence classes of pointwise unitary coactions of G on A. Moreover, when G is abelian we recover a theorem of Olesen and Raeburn.

We show that if p:F→E is a covering of directed graphs, then the Cuntz–Krieger algebra C*(F) of F can be viewed as a crossed product of C*(E) by a coaction of a homogeneous space for the fundamental group π₁(E). Combining this result with information about Cuntz–Krieger algebras gives some interesting corollaries which suggest conjectures about crossed products by coactions of homogeneous spaces of discrete groups. We then prove these conjectures.

A coaction δ of a locally compact group G on a C*-algebra A is maximal if a certain natural map from onto is an isomorphism. All dual coactions on full crossed products by group actions are maximal; a discrete coaction is maximal if and only if A is the full cross-sectional algebra of the corresponding Fell bundle. For every nondegenerate coaction of G on A, there is a maximal coaction of G on an extension of A such that the quotient map induces an isomorphism of the crossed products.

We construct the crossed product of a C(X)-algebra by an endomorphism ρ, in such a way that ρ becomes induced by the bimodule of continuous sections of a vector bundle ℰ → X. 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 , where G is a (noncompact, in general) group acting on ℰ. In particular, the C^*-algebra of invariant elements with respect to the action of the group of special unitaries of ℰ is a crossed product in the above sense. We also study the analogous construction on certain Hilbert bimodules, called "noncommutative pullbacks".

In this paper, we describe the commutant of an arbitrary subalgebra A of the algebra of functions on a set X in a crossed product of A with the integers, where the latter act on A by a composition automorphism defined via a bijection of X. The resulting conditions which are necessary and sufficient for A to be maximal abelian in the crossed product are subsequently applied to situations where these conditions can be shown to be equivalent to a condition in topological dynamics. As a further step, using the Gelfand transform, we obtain for a commutative completely regular semi-simple Banach algebra a topological dynamical condition on its character space which is equivalent to the algebra being maximal abelian in a crossed product with the integers.

In the paper, we provide some examples of Blackadar and Kirchberg's MF algebras by considering minimal and maximal tensor products of MF algebras and crossed products of MF algebras by finite groups or an integer group. We also present some examples of C*-algebras, which BDF extension semigroups are not groups. These examples include, for example, , and with 2 ≤ |H₁| < ∞ and 3 ≤ |H₂| < ∞ where |H₁| and |H₂| are the orders of the groups H₁ and H₂ respectively, and several others similarly.

Based on the fact that, for a subfactor N of a II₁ factor M, the first nontrivial Jones index is two and then M is decomposed as the crossed product of N by an outer action of ℤ₂, we study pairs {N, uNu*} from the view-point of entropy for two subalgebras of M in connection with the entropy for automorphisms, where the inclusion of II₁ factors N ⊂ M is given with M being the crossed product of N by a finite group of outer automorphisms and u is a unitary in M.

In order to give numerical characterizations of the notion of "mutual orthogonality", we introduce two kinds of family of positive definite matrices for a unitary u in a finite von Neumann algebra M. They are arising from u naturally depending on the decompositions of M. One corresponds to the tensor product decomposition and the other does to the crossed product decomposition. By using the von Neumann entropy for these positive definite matrices, we characterize the notion of mutual orthogonality between subalgebras.

We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways, based on certain pairs of quantum group representations and based on covariant Hilbert space representations, respectively. We establish basic properties of the twisted tensor product and study some examples.

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 recently by the author and A. V. Lebedev. This crossed product depends on the choice of an ideal J in (ker α)^⊥, and if J = (ker α)^⊥ it is a modification of Stacey's crossed product that works well with non-injective α's. We provide descriptions of the lattices of ideals in C*(A, α, J) consisting of gauge-invariant ideals and ideals generated by their intersection with A. We investigate conditions under which these lattices coincide with the set of all ideals in C*(A, α, J). In particular, we obtain simplicity criteria that besides minimality of the action require either outerness of powers of α or pointwise quasinilpotence of α.

Let $G$ be a discrete group acting on a von Neumann algebra $M$ by properly outer ∗-automorphisms. In this paper, we study the containment $M\subseteq M{⋊}_{\alpha }G$ of $M$ inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the $M$-bimodules that are closed in the Bures topology and which coincide with the $w^{\ast }$-closed ones under a mild hypothesis on $G$. We use these results to obtain a general version of Mercer’s theorem concerning the extension of certain isometric $w^{\ast }$-continuous maps on $M$-bimodules to ∗-automorphisms of the containing von Neumann algebras.

Landstad–Vaes theory deals with the structure of the crossed product of a ${\mathrm{\text{C}}}^{\ast }$-algebra by an action of locally compact (quantum) group. In particular, it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of $G$-dynamical system introducing the concept of weak action of quantum groups on ${\mathrm{\text{C}}}^{\ast }$-algebras. It is still possible to define crossed product (by weak action) and characterize the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.

Given a spectral triple on a $C^{\ast }$-algebra ${𝒜}$ together with a unital injective endomorphism $\alpha$, the problem of defining a suitable crossed product $C^{\ast }$-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 $\alpha ({𝒜})$ in ${𝒜}$ 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 isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on ${𝒜}$ and $\alpha ({𝒜})$.

In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^{\ast }$-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.

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 $\mathbb{Z}.$ 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 $\mathbb{Z}$-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 $K_0^{\mathrm{\Gamma }}(L_K(E)).$ 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 $\mathrm{\Gamma }$ and a $\mathrm{\Gamma }$-graded division ring $K$ equivalent with the requirements that a $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$ on the Grothendieck $\mathrm{\Gamma }$-group $K_0^{\mathrm{\Gamma }}(R).$

Starting with a complex Hilbert space, using inductive limits, we build Lie algebras, and find families of representations. They include those often studied in mathematical physics in order to model quantum statistical mechanics or quantum fields. We explore natural actions on infinite tensor algebras T(H) built with a functorial construction, starting with a fixed Hilbert space H.

While our construction applies also when H is infinite-dimensional, the case with N ≔ dim H finite is of special interest as the symmetry group we consider is then a copy of the non-compact Lie group U(N, 1). We give the tensor algebra T(H) the structure of a Hilbert space, i.e. the unrestricted infinite tensor product Fock space . The tensor algebra T(H) is naturally represented as acting by bounded operators on , and U(N, 1) as acting as a unitary representation. From this we built a covariant system, and we explore how the fermion, the boson, and the q on Hilbert spaces are reduced by the representations. In particular we display the decomposition into irreducible representations of the naturally defined U(N, 1) representation.

In this paper, we provide necessary and sufficient conditions for a cleft right H-comodule algebra (A, ϱ_A) over a Hopf quasigroup H to be isomorphic as an algebra to the crossed product A_H♯_σAHH, where A_H is the coinvariants subalgebra of A and σ_AH is a morphism between H ⊗ H and A_H. As a consequence, we obtain the corresponding version in the nonassociative setting of the result given by Blattner, Cohen and Montgomery for projections of Hopf algebras with coalgebra splitting. Concrete examples satisfying the obtained conditions are provided.

Let H be a bialgebra. Let σ : H ⊗ H → A be a linear map, where A is a left H-comodule coalgebra, and an algebra with a left H-weak action. Let B be a right H-module algebra and also a comodule coalgebra. In this paper, we provide necessary and sufficient conditions for the one-sided crossed product algebra A#_σ H # B and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which we call the crossed double biproduct. Majid's double biproduct is recovered from this. Moreover, necessary and sufficient conditions are given for Brzeziński's crossed product equipped with the smash coproduct coalgebra structure to be a bialgebra. The celebrated Radford's biproduct in [The structure of Hopf algebra with a projection, J. Algebra92 (1985) 322–347], the unified product defined by Agore and Militaru in [Extending structures II: The quantum version, J. Algebra336 (2011) 321–341] and the Wang–Jiao–Zhao's crossed product in [Hopf algebra structures on crossed products Comm. Algebra26 (1998) 1293–1303] are all derived as special cases.