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In this work, we present methods for constructing representations of polynomial covariance type commutation relations A B = B F (A) by linear integral operators in Banach spaces L p . We derive necessary and sufficient conditions on the kernel functions for the integral operators to satisfy the covariance type commutation relation for general polynomials F, as well as for important cases, when F is arbitrary affine or quadratic polynomial, or arbitrary monomial of any degree. Using the obtained general conditions on the kernels, we construct concrete examples of representations of the covariance type commutation relations by integral operators on L p . Also, we derive useful general reordering formulas for the integral operators representing the covariance type commutation relations, in terms of the kernel functions. [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]

The aim of this paper is to study a problem raised by Phillips concerning the existence of Takai duality for L p operator crossed products F p (G , A , α) , where G is a locally compact Abelian group, A is an L p operator algebra and α is an isometric action of G on A. Inspired by Williams' proof for the Takai duality theorem for crossed products of C ∗ -algebras, we construct a homomorphism Φ from F p (G ^ , F p (G , A , α) , α ^) to K (l p (G)) ⊗ p A which is a natural L p -analog of Williams' map. For countable discrete Abelian groups G and separable unital L p operator algebras A which have unique L p operator matrix norms, we show that Φ is an isomorphism if and only if either G is finite or p = 2 ; in particular, Φ is an isometric isomorphism in the case that p = 2 . Moreover, it is proved that Φ is equivariant for the double dual action α ^ ^ of G on F p (G ^ , F p (G , A , α) , α ^) and the action Ad ρ ⊗ α of G on K (l p (G)) ⊗ p A . [ABSTRACT FROM AUTHOR]

We discuss various aspects of noncommutative geometry of smooth subalgebras of Bunce-Deddens-Toeplitz algebras. [ABSTRACT FROM AUTHOR]

In the article, we find new dilatation results on non-commutative L p spaces. We prove that any self-adjoint, unital, positive measurable Schur multiplier on some B (L 2 (Σ)) admits, for all 1 ⩽ p < ∞ , an invertible isometric dilation on some non-commutative L p -space. We obtain a similar result for self-adjoint, unital, completely positive Fourier multiplier on VN(G), when G is a unimodular locally compact group. Furthermore, we establish multivariable versions of these results. [ABSTRACT FROM AUTHOR]

We establish several new characterizations of amenable $W^*$ - and $C^*$ -dynamical systems over arbitrary locally compact groups. In the $W^*$ -setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz–Schur multipliers of $(M,G,\alpha)$ converging point weak* to the identity of $G\bar {\ltimes }M$. In the $C^*$ -setting, we prove that amenability of $(A,G,\alpha)$ is equivalent to an analogous Herz–Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$ , as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted $C^*$ -dynamical systems, Hilbert $C^*$ -modules and regularity. Münster J. Math.5 (2012), 183–208] (generalized to the locally compact setting). When $Z(A^{**})=Z(A)^{**}$ , it follows that amenability is equivalent to the 1-positive approximation property of Exel and Ng [Approximation property of $C^*$ -algebraic bundles. Math. Proc. Cambridge Philos. Soc.132(3) (2002), 509–522]. In particular, when $A=C_0(X)$ is commutative, amenability of $(C_0(X),G,\alpha)$ coincides with topological amenability of the G-space $(G,X)$. [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]

We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of $\ast$ -automorphisms of a compact quantum metric space yields a field of crossed product algebras which varies continuously in Rieffel's quantum Gromov–Hausdorff distance. Finally, we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney $C^{1}$ -topology. [ABSTRACT FROM AUTHOR]

In this paper, we construct, for a certain class of semigroup dynamical systems, two operator algebras that are universal with respect to their corresponding covariance conditions: one being selfadjoint, and another being nonselfadjoint. We prove that the C*-envelope of the nonselfadjoint operator algebra is precisely the selfadjoint one. This result leads to a number of new examples of operator algebras and their C*-envelopes, with many from number fields and commutative rings. We further establish the functoriality of these operator algebras along with their applications. [ABSTRACT FROM AUTHOR]

Let M be an arbitrary collection of linear-fractional non-automorphism self-maps of the unit disk D . We consider the unital C ∗ -algebras C ∗ (T z , { C φ : φ ∈ M }) and C ∗ ({ C φ : φ ∈ M } , K) generated by the composition operators induced by the maps in M and either the unilateral shift T z or the ideal of compact operators K on the Hardy space H 2 (D) . We describe the structures of these C ∗ -algebras, modulo the ideal of compact operators, for all finite collections M as well as all collections M that have finite boundary behavior. This work completes a line of research investigating the structures, modulo the ideal of compact operators, of Toeplitz-composition and composition C ∗ -algebras induced by linear-fractional non-automorphism self-maps of D that has unfolded over the last decade and half. While all results in this paper are stated for H 2 (D) , the descriptions of the structures of these C ∗ -algebras, modulo the ideal of compact operators, also apply to the weighted Bergman spaces A α 2 (D) for α > - 1 . [ABSTRACT FROM AUTHOR]

We define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case, our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of clopen sets. We show that for a large class of ultragraphs the basis elements of the topology are compact. We examine shift morphisms between these shift spaces, and, for the locally compact case, show that if two (possibly infinite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras are isomorphic. To prove this last result we realize the relevant ultragraph C*-algebras as partial crossed products. [ABSTRACT FROM AUTHOR]

Let ( G , Σ ) be a (partially) ordered abelian group with Haar measure μ , let ( A , G , α ) be a dynamical system and let A ⋊ α Σ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism A ⋊ α Σ ∼ s ( A ⊗ K ( G , Σ , μ ) ) ⋊ α ⊗ Ad ρ G , where K ( G , Σ , μ ) denotes the compact operators in the CSL algebra Alg L ( G , Σ , μ ) and ρ denotes the right regular representation of G . We also show that there exists a complete lattice isomorphism between the α ˆ -invariant ideals of A ⋊ α Σ and the ( α ⊗ Ad ρ ) -invariant ideals of A ⊗ K ( G , Σ , μ ) . Using Takai duality we also continue our study of the Radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors. Among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra. We also show that the permanence of semisimplicity fails for crossed products by R . A final section of the paper is devoted to the study of radically tight dynamical systems, i.e., dynamical systems ( A , G , α ) for which the identity Rad ( A ⋊ α G ) = ( Rad A ) ⋊ α G persists. A broad class of such dynamical systems is identified. [ABSTRACT FROM AUTHOR]

Let be a single vertex k-graph and let be the von Neumann algebra induced from the Gelfand–Naimark–Segal (GNS) representation of a distinguished state ω of its k-graph C*-algebra . In this paper we prove the factoriality of , and furthermore determine its type when either has the little pullback property, or the intrinsic group of has rank 0. The key step to achieving this is to show that the fixed-point algebra of the modular action corresponding to ω has a unique tracial state. [ABSTRACT FROM PUBLISHER]

For families of magnetic self-adjoint operators on $$\mathbb Z^d$$ whose symbols and magnetic fields depend continuously on a parameter $$\epsilon $$ , it is shown that the spectrum of these operators also varies continuously with respect to $$\epsilon $$ . The proof is based on an algebraic setting involving twisted crossed product $$C^*$$ -algebras. [ABSTRACT FROM AUTHOR]

We prove a Beurling-Blecher-Labuschagne theorem for $${H^\infty}$$ -invariant spaces of $${L^p(\mathcal{M},\tau)}$$ when $${0 < p \leq\infty}$$ , using Arveson's non-commutative Hardy space $${H^\infty}$$ in relation to a von Neumann algebra $${\mathcal{M}}$$ with a semifinite, faithful, normal tracial weight $${\tau}$$ . Using the main result, we are able to completely characterize all $${H^\infty}$$ -invariant subspaces of $${L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}$$ , where $${\mathcal{M} \rtimes_\alpha \mathbb{Z} }$$ is a crossed product of a semifinite von Neumann algebra $${\mathcal{M}}$$ by the integer group $${\mathbb{Z}}$$ , and $${H^\infty}$$ is a non-selfadjoint crossed product of $${\mathcal{M}}$$ by $${\mathbb{Z}^+}$$ . As an example, we characterize all $${H^\infty}$$ -invariant subspaces of the Schatten p-class $${S^p(\mathcal{H})}$$ , where $${H^\infty}$$ is the lower triangular subalgebra of $${B(\mathcal{H})}$$ , for each $${0 < p \leq\infty}$$ . .. [ABSTRACT FROM AUTHOR]