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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 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]

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]

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]

This paper is devoted to the study of operator-valued Triebel-Lizorkin spaces. We develop some Fourier multiplier theorems for square functions as our main tool, and then study the operator-valued Triebel-Lizorkin spaces on Rd. As in the classical case, we connect these spaces with operator-valued local Hardy spaces via Bessel potentials. We show the lifting theorem, and get interpolation results for these spaces. We obtain Littlewood-Paley type, as well as the Lusin type square function characterizations in the general way. Finally, we establish smooth atomic decompositions for the operator-valued Triebel-Lizorkin spaces. These atomic decompositions play a key role in our recent study of mapping properties of pseudo-differential operators with operator-valued symbols. [ABSTRACT FROM AUTHOR]