Applications of model theory to C*-dynamics

Abstract

We initiate the study of compact group actions on C*-algebras from the perspective of model theory, and present several applications to C*-dynamics. Firstly, we prove that the continuous part of the central sequence algebra of a strongly self-absorbing action is indistinguishable from the continuous part of the sequence algebra, and in fact equivariantly isomorphic under the Continuum Hypothesis. As another application, we present a unified approach to several dimensional inequalities in C*-algebras, which is done through the notion of order zero dimension for an (equivariant) *-homomorphism. Finiteness of the order zero dimension implies that the dimension of the target algebra can be bounded by the dimension of the domain. The dimension can be, among others, decomposition rank, nuclear dimension, or Rokhlin dimension. As a consequence, we obtain new inequalities for these quantities.As a third application we obtain the following result: if a C*-algebra A absorbs a strongly self-absorbing C*algebra D, and α is an action of a compact group G on A with finite Rokhlin dimension with commuting towers, then α absorbs any strongly self-absorbing action of G on D. This has a number of interesting consequences, already in the case of the trivial action on D. For example, we deduce that D-stability passes from A to the crossed product. Additionally, in many cases of interest, our result restricts the possible values of the Rokhlin dimension to 0, 1 and ∞, showing a striking parallel to the behavior of the nuclear dimension for simple C*-algebras. We also show that an action of a finite group with finite Rokhlin dimension with commuting towers automatically has the Rokhlin property if the algebra is UHF-absorbing.2000 Mathematics Subject Classification. Primary 03C98, 46L55; Secondary 28D05, 46L40, 46M07. 1 2 EUSEBIO GARDELLA AND MARTINO LUPINIproperty. In the context of C*-algebras, relative commutants were used in connection with the Rokhlin property for group actions in the work of Herman-Jones [37], Kishimoto [51], Izumi [42], , and the first-named author [28]. The study of Rokhlin dimension has also made extensive use of these tools, for example in [28] and [38], as well as the more recent work on strongly self-absorbing actions [64]. As is clear from these works, the use of sequence algebras in the equivariant setting becomes even more delicate when the acting group is not discrete, since a continuous action on an operator algebra may induce a discontinuous action on its relative commutant. As such, equivariant (central) sequence algebras are interesting objects whose systematic study is justified by their wide application in the literature.The present work takes up this task. For a given compact second countable group G, we consider actions of G on C*-algebras (G-C*-algebras) as structures in the framework of continuous model theory. When the group G is finite, one can regard a G-action as a usual metric structure by adding a function symbol for every element of the group. This does not work for a general compact g...

Strongly outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras

“…The following lemma is well known in the non-equivariant setting. In our context, it follows from the fact that equivariant ultrapowers are (countably) saturated; see Subsection 2.2.4 in [7]. Lemma 5.2.…”

“…If f (j) g ∈ I G,k , for g ∈ G and j = 0, 1, 2, are positive contractions as in the conclusion of Proposition 4.9, the positive contractions θ(f (j) g ) ∈ A ω ∩ A ′ satisfy the conditions of Definition 4.3 up to ǫ. Since ε > 0 is arbitrary, the result follows using a reindexation argument (or from countable saturation of the equivariant ultrapower A ω ; see Subsection 2.2.4 in [7]).…”

Partial C*-dynamics and Rokhlin dimension

“…The properties listed in (1) through (5) are well known to satisfy (E) and (M); see respectively [24,19,21,4]. Finally, they also satisfy (C) by Corollary 4.25 in [14] and Theorem 3.20 in [13].…”

An Invitation to Model Theory and C*-Algebras

“…It is shown in [48] that G-C*-algebras form an axiomatizable class in such a language. This perspective, and the corresponding notion of positive existential embedding, has been used implicitly in [9] and explicitly in [49] to give a model-theoretic characterization of the Rokhlin property for G-C*-algebras. In turn, this characterization has been used to provide a unified approach to several preservation results for fixed point algebras and crossed products with respect to Rokhlin actions.…”

“…In turn, this characterization has been used to provide a unified approach to several preservation results for fixed point algebras and crossed products with respect to Rokhlin actions. More generally, a modeltheoretic characterization of Rokhlin dimension is considered in [49]. This is applied to obtain preservation results of finite nuclear dimension and finite composition rank for fixed point algebras and crossed products with respect to actions with finite Rokhlin dimension.…”