Questions tagged [crossed-products]

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A completely positive equivariant map $\varphi: A \to B$ induces a map on the full crossed products

Let $G$ be a discrete group. Let $(A,\alpha)$ and $(B,\beta)$ be $G$-$C^*$-algebras and $\varphi: A \to B$ be $G$-equivariant and completely positive. All crossed products in this post are full (= ...
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A completely positive equivariant map $\varphi: A \to B$ induces a map $A \rtimes_r \Gamma \to B \rtimes_r \Gamma.$

Recall the construction of the reduced crossed product: Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra with an action $\alpha: \Gamma\to \operatorname{Aut}(A)$. Consider the $*$-algebra $...
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2 votes
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Subalgebra of a crossed product central division algebra, generated by powers of group elements

Let $k=\mathbb C$, let $K$ be a finite extension of the field $k(X_1,X_2,X_3)$ of rational functions in $3$ variables, let $L/K$ be a finite Galois extension of commutative fields and let the Galois ...
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10 votes
0 answers
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Twisted crossed product von Neumann Algebras

I asked a question over on Math.stackexchange a few days ago, but it didn't get much activity. Hopefully this question isn't considered too elementary by the standards of Mathoverflow. Here is what I ...
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3 votes
1 answer
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Covariant representations and crossed products of von Neumann algebras

Let $(M,G,\alpha)$ be a $W^\ast$-dynamical system with $G$ locally compact abelian (I am mostly interested in the case $G=\mathbb{R})$. A covariant representation of $(M,G,\alpha)$ is a pair $(\pi,u)$ ...
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5 votes
0 answers
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Non-existence of projections in crossed product

If $X$ is a smooth manifold on which a Lie group $G$ acts properly and cocompactly (meaning $X/G$ is compact), then one can find a compactly support cut-off function $c:X\rightarrow\mathbb{R}$ such ...
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2 votes
1 answer
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Crossed products and unitaries implementing $\mathbb{Z}_n$-actions

I'm working through Li's and Barlak's Cartan Subalgebras and the UCT Problem but I'm stuck at one of the simpler proofs of the paper. On page 9 they deal with masas (maximal abelian subalgebras) of a ...
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6 votes
1 answer
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Classification of quasitriangular Hopf algebras

The classification of hopf algebras is a big and open problem, containing various subproblems (such as: the classification of groups, of Lie algebras, the study of special classes such as (co)...
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2 votes
0 answers
96 views

Amenability for Actions twited with 2-cocycles

Let $A \subset B(H)$ be a unital $C^\ast$-algebra and $\theta: G \rightarrow \mathrm{Aut}(A)$ an action and let $\omega: G \times G \rightarrow U(\mathcal{Z}(A))$ be a $2$-cocyle with respect to $\...
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5 votes
1 answer
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commutative diagram with $K_{i+1}(A)\to K_i(A\rtimes_{\rho} \mathbb{R})$ (for $C^*$-algebras)

I have a question about a proof in Rosenberg and Schochet's paper "the Künneth theorem and the Universal Coefficient Theorem for Kasparov's generalized K-functor", proposition 2.6. First of all, the ...
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4 votes
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When is a crossed-product algebra a division algebra?

Let $L/K$ be a finite Galois extension with Galois group $G$. For every 2-cocycle $\gamma$ of $G$ with values in $L^\times$ there is the crossed-product $K$-algebra $$S(L,G,\gamma) = \bigoplus_{g\in ...
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1 answer
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types of crossed product von Neumann algebras

Let $M$ be a type $II_1$ factor von Neumann algebra, and let $G$ be a discrete group acting on $M$ which is free and ergodic. Is the crossed product von Neumann algebra $M \rtimes G$ type $II_1$ ...
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1 vote
1 answer
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An unconventional definition of the $ C^{*} $-algebraic reduced crossed product

Let $ (A,G,\alpha) $ be a $ C^{*} $-dynamical system, i.e., $ A $ is a $ C^{*} $-algebra, $ G $ is a locally compact Hausdorff group and $ \alpha $ is a strongly continuous action of $ G $ on $ A $ by ...
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4 votes
1 answer
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Crossed Products by Permutation Groups

What can be said about the following crossed product $C^*$-algebra? Let $A$ be a Kirchberg algebra with $K_0(A) = \mathbb{Q}$ and $K_1(A) = 0$. Consider the direct sum of $n$ copies of $A$, i.e. $B = ...
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8 votes
2 answers
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Extending a $ * $-Representation of $ ({C_{c}}(G,\mathscr{A}),\star,^{*}) $ to a $ * $-Representation of $ \mathscr{A} \rtimes_{\alpha} G $

Let $ (\mathscr{A},G,\alpha) $ be a $ C^{*} $-dynamical system, and consider the twisted convolution $ * $-algebra $ ({L^{1}}(G,\mathscr{A}),\star,^{*}) $ defined by \begin{align*} \forall \phi,\psi \...
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3 votes
1 answer
259 views

‘Non-Induced’ Left Regular Representations of $ C^{*} $-Dynamical Systems

In what follows, a ‘$ * $-representation’ always means a non-degenerate $ * $-representation. Let $ (\mathscr{A},G,\alpha) $ be a $ C^{*} $-dynamical system, and let $ \pi: \mathscr{A} \to B(\mathcal{...
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6 votes
1 answer
284 views

States/functionals on crossed product C*-algebras

Let $A$ be a C*-algebra, $\alpha$ a strongly continuous automorphic action by a locally compact group $G$ on $A$, and consider the crossed product $A\rtimes_\alpha G$. I am looking for references ...
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6 votes
2 answers
168 views

Reference for a Maschke lemma for crossed products

Can someone provide a reference for the following Maschke lemma: If $C$ is a semisimple algebra over a field of characteristic zero and $G$ is a finite group acting on $C$ by automorphisms, then $C \...
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0 votes
0 answers
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For a separable v-N algebra M, how to see $M \rtimes \mathbb{R}$ as a subalgebra of $M \otimes B(L^2)$?

For a v-N algebra $M$ acting as bounded operators on a separable Hilbert space $H$, how to see $M \rtimes \mathbb{R}$ as a subalgebra of $M \otimes B(L^2(\mathbb{R})$? Why I am confused is because $...
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5 votes
2 answers
793 views

Universal characterization and explicit description of elements of the group von Neumann algebra and the crossed product

Group von Neumann algebras and crossed products for a locally compact group G can be constructed in many different ways. For example, one can take the von Neumann algebra generated by certain ...
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