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Register a new userCommutativity and ideals in algebraic crossed products
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We investigate properties of commutative subrings and ideals in non-commutative algebraic crossed products for actions by arbitrary groups. A description of the commutant of the base coefficient subring in the crossed product ring is given. Conditions for commutativity and maximal commutativity of the commutant of the base subring are provided in terms of the action as well as in terms of the intersection of ideals in the crossed product ring with the base subring, specially taking into account both the case of base rings without non-trivial zero-divisors and the case of base rings with non-trivial zero-divisors.
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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 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 $mu$ on $X$. To do so, we present a general construction of an approximating sequence of $*$-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 $*$-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 $mu$, namely that the rank of a characteristic function of a clopen subset $U subseteq X$ must equal the measure of $U$.
We consider a twisted action of a discrete group G on a unital C*-algebra A and give conditions ensuring that there is a bijective correspondence between the maximal invariant ideals of A and the maximal ideals in the associated reduced C*-crossed product.
Let $X$ be an infinite compact metric space with finite covering dimension and let $alpha, beta : Xto X$ be two minimal homeomorphisms. We prove that the crossed product $C^*$-algebras $C(X)rtimes_alphaZ$ and $C(X)rtimes_beltaZ$ are isomorphic if and only if they have isomorphic Elliott invariant. In a more general setting, we show that if $X$ is an infinite compact metric space and if $alpha: Xto X$ is a minimal homeomorphism such that $(X, alpha)$ has mean dimension zero, then the tensor product of the crossed product with a UHF-algebra of infinite type has generalized tracial rank at most one. This implies that the crossed product is in a classifiable class of amenable simple $C^*$-algebras.
In this paper we present exact solutions of the Dirac equation on the non-commutative plane in the presence of crossed electric and magnetic fields. In the standard commutative plane such a system is known to exhibit contraction of Landau levels when the electric field approaches a critical value. In the present case we find exact solutions in terms of the non-commutative parameters $eta$ (momentum non-commutativity) and $theta$ (coordinate non-commutativity) and provide an explicit expression for the Landau levels. We show that non-commutativity preserves the collapse of the spectrum. We provide a dual description of the system: (i) one in which at a given electric field the magnetic field is varied and the other (ii) in which at a given magnetic field the electric field is varied. In the former case we find that momentum non-commutativity ($eta$) splits the critical magnetic field into two critical fields while coordinates non-commutativity ($theta$) gives rise to two additional critical points not at all present in the commutative scenario.
We study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. We develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. We complement our generic results with the detailed study of many important special cases. In particular we study crossed products of tensor algebras, triangular AF algebras and various associated C*-algebras. We make contributions to the study of C*-envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. We also answer questions from the pertinent literature.
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