Searcharxiv⌕ Search

arXiv subjects

Siegfried Echterhoff

Publications and source records attributed to Siegfried Echterhoff.

49 records · Page 3Linked to original sources
arXiv2006

The structure of crossed products of irrational rotation algebras by finite subgroups of SL_2 (Z)

Let F be a finite subgroup of SL_2 (Z) (necessarily isomorphic to one of Z/2Z, Z/3Z, Z/4Z, or Z/6Z), and let F act on the irrational rotational algebra A_θ via the restriction of the canonical action of SL_2 (Z). Then the crossed product of A_θ by F, and the fixed point algebra for the action of F on A_θ, are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of Z/2Z on any simple d-dimensional noncommutative torus A_Θ. Along the way, we prove a number of general results which should have useful applications in other situations.

math.OA↗
arXiv2006

Inducing Primitive Ideals

We study conditions on a $C^*$-dynamical system $(A,G,α)$ under which induction of primitive ideals (resp. irreducible representations) from stabilizers for the action of $G$ on the primitive ideal space $\Prim(A)$ give primitive ideals (resp. irreducible representations) of the crossed product $A\rtimes_αG$. The results build on earlier results of Sauvageot and others, and will correct a (possibly overly optimistic) statement of the first author.

math.OA↗
arXiv2005

A Categorical Approach to Imprimitivity Theorems for C*-Dynamical Systems

Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossed-product C*-algebras. In this work, we show that the Imprimitivity Theorem for induced algebras, Green's Imprimitivity Theorem for actions of groups, and Mansfield's Imprimitivity Theorem for coactions of groups can all be viewed as natural equivalences between various crossed-product functors among certain equivariant categories. The categories involved have C*-algebras with actions or coactions (or both) of a fixed locally compact group G as their objects, and equivariant equivalence classes of right-Hilbert bimodules as their morphisms. Composition is given by the balanced tensor product of bimodules. The functors involved arise from taking crossed products; restricting, inflating, and decomposing actions and coactions; inducing actions; and various combinations of these. Several applications of this categorical approach are also presented, including some intriguing relationships between the Green and Mansfield bimodules, and between restriction and induction of representations.

math.OA↗
arXiv2002

Central twisted transformation groups and group $C^*$-algebras of central group extensions

We examine the structure of central twisted transformation group \cs-algebras $C_{0}(X)\rtimes_{\id,u}G$, and apply our results to the group \cs-algebras of central group extensions. Our methods require that we study Moore's cohomology group $H^{2}\bigl(G,C(X,\T)\bigr)$, and, in particular, we prove an inflation result for pointwise trivial cocyles which may be of use elsewhere.

math.OA↗
arXiv2001

Full duality for coactions of discrete groups

Using the strong relation between coactions of a discrete group G on C*-algebras and Fell bundles over G, we prove a new version of Mansfield's imprimitivity theorem for coactions of discrete groups. Our imprimitivity theorem works for the universally defined full crossed products and arbitrary subgroups of G, as opposed to the usual theory which uses the spatially defined reduced crossed products and normal subgroups of G. Moreover, our theorem factors through the usual one by passing to appropriate quotients. As applications we show that a Fell bundle over a discrete group is amenable in the sense of Exel if and only if the double dual action is amenable in the sense that the maximal and reduced crossed products coincide. We also give a new characterization of induced coactions in terms of their dual actions.

math.OA↗
arXiv2001

Maximal Coactions

A coaction d of a locally compact group G on a C*-algebra A is maximal if a certain natural map from A times_d G times_{d hat} G onto A otimes K(L^2(G)) 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.

math.OA↗
arXiv2000

Naturality and Induced Representations

We show that induction of covariant representations for C*-dynamical systems is natural in the sense that it gives a natural transformation between certain crossed-product functors. This involves setting up suitable categories of C*-algebras and dynamical systems, and extending the usual constructions of crossed products to define the appropriate functors. From this point of view, Green's Imprimitivity Theorem identifies the functors for which induction is a natural equivalence. Various spcecial cases of these results have previously been obtained on an ad hoc basis.

math.OA↗
arXiv1998

Induced coactions of discrete groups on C*-algebras

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction of a quotient group G/N of a discrete group G to a C*-coaction of G itself on an induced C*-algebra. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products of the original and the induced coactions are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

math.OA↗
arXiv1997

Locally Inner Actions on $C_0(X)$-Algebras

We make a detailed study of locally inner actions on C*-algebras whose primitive ideal spaces have locally compact Hausdorff complete regularizations. We suppose that $G$ has a representation group and compactly generated abelianization $G_{ab}$. Then if the complete regularization of $\Prim(A)$ is $X$, we show that the collection of exterior equivalence classes of locally inner actions of $G$ on $A$ is parameterized by the group $\E_G(X)$ of exterior equivalence classes of $C_0(X)-actions of $G$ on $C_0(X,\K)$. Furthermore, we exhibit a group isomorphism of $\E_G(X)$ with the direct sum $H^1(X,\sheaf \hat{G_{ab}}) \oplus C(X,H^2(G,\T))$. As a consequence, we can compute the equivariant Brauer group $\Br_G(X)$ for $G$ acting trivially on $X$.

funct-an↗
arXiv1997

Crossed products by $C_0(X)$-actions

Suppose that $G$ has a representation group $H$, that $G_{ab}:= G/\bar{[G,G]}$ is compactly generated, and that $A$ is a \cs-algebra for which the complete regularization of $\Prim(A)$ is a locally compact Hausdorff space $X$. In a previous article, we showed that there is a bijection $α\mapsto (Z_α,f_α)$ between the collection of exterior equivalence classes of locally inner actions $α:G\to\Aut(A)$, and the collection of principal $\hgab$-bundles $Z_α$ together with continuous functions $f_α:X\to H^2(G,\T)$. In this paper, we compute the crossed products $A\rtimes_αG$ in terms of the data $Z_α$, $f_α$, and~$\cs(H)$.

funct-an↗
arXiv1996

Crossed Products by Dual Coactions of Groups and Homogeneous Spaces

Mansfield showed how to induce representations of crossed products of C*-algebras by coactions from crossed products by quotient groups and proved an imprimitivity theorem characterising these induced representations. We give an alternative construction of his bimodule in the case of dual coactions, based on the symmetric imprimitivity theorem of the third author; this provides a more workable way of inducing representations of crossed products of C*-algebras by dual coactions. The construction works for homogeneous spaces as well as quotient groups, and we prove an imprimitivity theorem for these induced representations.

funct-an↗
arXiv1994

Crossed products whose primitive ideal spaces are generalized trivial $\widehat G$-bundles

We characterize when the primitive ideal space of a crossed product $\acg$ of a \cs-algebra $A$ by a locally compact abelian group $G$ is a $σ$-trivial $\ghat G$-space for the dual $\ghat G$-action. Specifically, we show that $\Prim(\acg)$ is $σ$-trivial if and only if the quasi-orbit space is Hausdorff, the map which assigns to each quasi-orbit $\w$ a certain subgroup $\ttg(α^\w)$ of the Connes spectrum of the system $(A_\w,G,α^\w)$ is continuous, and there is a generalized Green twisting map for $(A,G,α)$. Our proof requires a substantial generalization of a theorem of Olesen and Pedersen in which we show that there is a generalized Green twisting map for $(A,G,α)$ if and only if $\acg$ is isomorphic to a generalized induced algebra.

funct-an↗