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## Crossed product

What do you want to do in September for MO's tenth anniversary? Related 2. In Section 9. The brief Section 9. Our main interest is in structural results for crossed products. We want simplicity, but we really want much more than that. In many cases, one settles for related weaker structural results, such as stable rank one, real rank zero, order on traces determined by projections, strict comparison of positive elements, or Z -stability. Some results with conclusions of this sort are stated in these notes, but mostly without proof.

We give two simplicity proofs, using very different methods. One works for free minimal actions, and the method gives further information, as well as some information when the action is not minimal; see Theorems The second proof is a special case of a more general simplicity theorem; the case we prove allows some simplification of the argument.

Our theorem is Theorem The full theorem is stated as Theorem Much of the text of this part is taken directly from [], which is a survey of applications of large subalgebras based on lectures given at the University of Wyoming in the summer of That survey assumes much more background than these notes it starts with the material here , there are some differences in the organization, and it contains some open problems and other discussion omitted here because they are too far off the topic of these notes.

The notion of amenability in its most basic combinatorial sense captures the idea of internal finite approximation from a measure-theoretic perspective. It plays a pivotal role not only in combinatorial and geometric group theory but also in the theory of operator algebras through its various linear manifestations like hyperfiniteness, semidiscreteness, injectivity, and nuclearity.

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In this section we will review the theory of amenability for discrete groups see [34, 76] for general references , and then move in Section As discussed in Section Let F be a nonempty finite subset of a discrete group G. The topological analogue of amenability for discrete groups is local finiteness. The group G is said to be locally finite if every finite subset of G generates a finite subgroup.

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Equivalently, G is the increasing union of finite subgroups. Obviously every finite group is locally finite.

## Crossed Product of a C*-Algebra by a Semigroup of Interactions : Demonstratio Mathematica

An example of a countably infinite locally finite group is the group of all permutations of N which fix all but finitely many elements. There are uncountably many pairwise nonisomorphic countable locally finite groups, and there is a countable locally finite group U which has the universal property that it contains a copy of every finite group and any two monomorphisms of a finite group into U are conjugate by an inner automorphism [47].

Note that every locally finite group is torsion. The converse is the general Burnside problem and is false, as was shown by Golod.

In fact a torsion group need not even be amenable, which is the measure-theoretic analogue of local finiteness to be discussed below. Complementing the internal finite modelling property of local finiteness is the external property of local embeddability into finite groups, which is the topological i.

## Crossed Products of \(C^{*}\)-Algebras

This notion was introduced by Gordon and Vershik in [33]. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question.

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