1 Introduction
Connes’ classification of automorphisms on the hyperfinite II
$_1$
factor
$\mathcal {R}$
[Reference Connes7, Reference Connes8] paved the way toward a classification of symmetries of simple operator algebras. Over the next decade, this was followed by V. F. R. Jones’ [Reference Jones29] classification of finite group actions on
$\mathcal {R}$
and Ocneanu’s [Reference Ocneanu36] classification of actions of countable amenable groups on
$\mathcal {R}$
. To achieve these classification results, an important role is played by adaptations of Connes’ noncommutative Rokhlin lemma, which yields that outer group actions on
$\mathcal {R}$
satisfy a condition often called the Rokhlin property that is analogous to properties of ergodic measure preserving actions of amenable groups on probability spaces [Reference Ornstein and Weiss38, Reference Rokhlin42]. In the C
$^*$
-setting, the analogous property is not automatic. However, there has been substantial progress in the classification of those group actions on C
$^*$
-algebras that satisfy the Rokhlin property [Reference Evans and Kishimoto12, Reference Gardella and Santiago16, Reference Herman and Jones18–Reference Herman and Ocneanu20, Reference Izumi22, Reference Izumi23, Reference Nawata35]. Very recently, groundbreaking results toward a classification of group actions without the need for the Rokhlin property have appeared [Reference Gabe and Szabó15, Reference Izumi and Matui25, Reference Izumi and Matui26].
Connes, V. F. R. Jones, and Ocneanu also classify group homomorphisms
$G\rightarrow \mathrm {Out}(\mathcal {R})$
up to outer conjugacy [Reference Connes8, Reference Jones29, Reference Ocneanu36]. Such a homomorphism is called a G-kernel on
$\mathcal {R}$
. The classification of G-kernels on injective factors was completed by Katayama and Takesaki [Reference Katayama and Takesaki31]. These can be understood as the first classification results for quantum symmetries of
$\mathcal {R}$
which do not arise as group actions. Quantum symmetry is a broad term that encapsulates generalized notions of symmetry that appear in topological and conformal field theories. These symmetries are often encoded through the action of a higher category equipped with a product operation such that the category weakly resembles a group. In the case of G-kernels, these can be understood as actions of
$2$
-groups or tensor categories [Reference Evington and Girón Pacheco13, Reference Jones27]. The study of quantum symmetries of
$\mathcal {R}$
was developed through the subfactor theory of Jones [Reference Jones30] culminating in Popa’s classification of subfactors
$\mathcal {N}\subset \mathcal {R}$
with amenable standard invariant [Reference Popa40].
In comparison to the success in understanding the existence and classification of G-kernels on von Neumann algebras, the study of G-kernels on C
$^*$
-algebras has up to recently been underdeveloped. In [Reference Jones27], C. Jones studies the closely related notion of
$\omega $
-anomalous action.Footnote
1
In his paper, C. Jones provides a C
$^*$
-adaptation of V. F. R. Jones’ work [Reference Jones28], laying out a systematic way to construct anomalous actions on C
$^*$
-crossed products. C. Jones also establishes existence and no-go theorems for anomalous actions on abelian C
$^*$
-algebras. In [Reference Evington and Girón Pacheco13], Evington and the author lay out an algebraic K-theory obstruction to the existence of anomalous actions on tracial C
$^*$
-algebras. Recently, Izumi [Reference Izumi24] has developed a cohomological invariant for G-kernels. This invariant introduces new obstructions to the existence of G-kernels which also apply in the non-tracial setting. Further, Izumi uses this invariant to classify G-kernels of some poly-
$\mathbb {Z}$
groups on strongly self-absorbing Kirchberg algebras satisfying the universal coefficient theorem (abbreviated as UCT).
This paper provides a classification of anomalous actions with the Rokhlin property on C
$^*$
-algebras where K-theoretic obstructions vanish. The Rokhlin property for finite group actions was first systematically studied by Izumi [Reference Izumi22, Reference Izumi23]. In his work, Izumi uses the Rokhlin property to boost existing classification results of Kirchberg algebras in the UCT class [Reference Kirchberg32, Reference Phillips39] and unital, simple, separable, nuclear, tracially approximate finite-dimensional (TAF) algebras in the UCT class [Reference Lin34] by their K-theory, to a classification of finite group actions with the Rokhlin property on these classes of C
$^*$
-algebras by the induced module structure on K-theory [Reference Izumi23, Theorems 4.2 and 4.3].Footnote
2
The strategy of this paper is to bootstrap Izumi’s classification of G actions with the Rokhlin property, for finite groups G, to achieve analogous classification results for anomalous actions. To do this, we will assume that our C
$^*$
-algebra A satisfies a UHF absorbing condition. To be precise, that the A is stable under tensoring with the UHF algebra
$M_{|G|^\infty }\cong \bigotimes _{i\in \mathbb {N}} M_{|G|}$
. This property is considered, for example, in [Reference Barlak and Szabó2, Reference Gardella and Santiago16] and in some cases follows immediately from the existence of Rokhlin G actions on A ([Reference Izumi23, Theorems 3.4 and 3.5], [16, Theorem 5.2]). Further assuming the Rokhlin property, we will establish a model action absorption result (Proposition 4.5). Second, we will use the model action absorption combined with a trick, that builds on ideas of Connes in the cyclic group case [Reference Connes8, Section 6]. This trick lets us use the existence of anomalous action on the UHF-algebra
$M_{|G|^\infty }$
to reduce the classification of anomalous actions to the classification of cocycle actions. We may not apply this method by replacing
$M_{|G|^\infty }$
by
$\mathcal {Z}$
or
$\mathcal {O}_\infty $
due to the obstruction results of [Reference Evington and Girón Pacheco13, Theorem A] and [Reference Izumi24, Theorem 3.6]. This argument allows us to prove the following.
Theorem A (cf. Theorems 5.2 and 5.3) Let G be a finite group, and let
$A\cong A\otimes M_{|G|^\infty }$
be either a Kirchberg algebra in the UCT class or a unital, simple, separable, nuclear TAF algebra in the UCT class. If
$(\alpha ,u),(\beta ,v)$
are anomalous G actions on A with the Rokhlin property, then
$(\alpha ,u)$
is cocycle conjugate to
$(\beta ,v)$
through an automorphism that is trivial on K-theory if and only if
$K_i(\alpha _g)=K_i(\beta _g)$
for all
$g\in G$
and the anomalies of
$(\alpha ,u)$
and
$(\beta ,v)$
coincide.
Similarly, we can boost Nawata’s classification of Rokhlin G actions on
$\mathcal {W}$
(see [Reference Nawata35]) to a classification of anomalous actions on
$\mathcal {W}$
.
Theorem B (cf. Theorem 5.4) Let G be a finite group, and let
$(\alpha ,u), (\beta ,v)$
be anomalous G actions on
$\mathcal {W}$
with the Rokhlin property, then
$(\alpha ,u)$
is cocycle conjugate to
$(\beta ,v)$
if and only if the anomalies of
$(\alpha ,u)$
and
$(\beta ,v)$
coincide.
As a consequence of the results of [Reference Gardella and Santiago16], we may also apply this strategy to classify anomalous actions with the Rokhlin property on C
$^*$
-algebras that arise as inductive limits of one-dimensional non commutative CW complexes (see Theorem 5.6).
The procedure utilized for the proof of Theorem A can be expected to work in more generality. The reason we restrict to unital, simple, nuclear TAF algebras in the tracial setting is due to the need to apply classification results for (cocycle) group actions. With more novel stably finite classification results in hand [Reference Carrión, Gabe, Schafhauser, Tikuisis and White5], and using similar techniques to [Reference Izumi22, Reference Izumi23], a classification of finite group actions with the Rokhlin property on simple, separable, nuclear,
$\mathcal {Z}$
-stable C
$^*$
-algebra satisfying the UCT through the induced module structure on the Elliott invariant is plausible. A strategy to approach this classification problem has been proposed by Szabó in private communications. With such a result in hand, one could apply the abstract Lemma 5.1 to yield the equivalent to Theorem A in the generality of simple, separable, nuclear,
$M_{|G|^\infty }$
-stable C
$^*$
-algebras satisfying the UCT.
Recent advances in the classification of more general symmetries on C
$^*$
-algebras pave the way toward a classification of quantum symmetries. Significant results in this direction are the classification of AF-actions of fusion categories on AF-algebras [Reference Chen, Palomares and Jones6], as well as Yuki Arano’s announcement of an adaptation of Izumi’s techniques in [Reference Izumi22] to actions of fusion categories with the Rokhlin property. In the final section of this paper, we connect our results to the work in [Reference Chen, Palomares and Jones6]. We demonstrate the existence of an AF
$\omega $
-anomalous G-action with the Rokhlin property on
$M_{|G|^\infty }$
which we denote by
$\theta _G^{\omega }$
. This has structural implications for anomalous actions with the Rokhlin property on any AF-algebra A. Indeed, combined with Theorem A, the existence of
$\theta _G^{\omega }$
implies that every anomalous action on A with the Rokhlin property, that consists of automorphisms that act trivially on K-theory, is automatically AF (see Corollary 6.3). Under some assumptions on the anomaly, an application of the classification results of [Reference Chen, Palomares and Jones6] establishes the converse (see Corollary 6.3). This partial converse exhibits a difference in behavior between anomalous actions and group actions (see the discussion following Corollary 6.3).
The paper is organized as follows. In Section 2, we recall some necessary background on anomalous actions. Section 3 recalls the construction of model anomalous actions on UHF algebras. In Section 4, we prove a model action absorbing result for finite group anomalous actions. In Section 5, we set out an abstract lemma for the classification of anomalous actions (Lemma 5.1) which we use to prove our main results. Finally, in Section 6, we discuss an application of the classification result to AF-actions.
2 Preliminaries
Throughout, A and B will be used to denote C
$^*$
-algebras and
$G,\Gamma ,K$
will be used to denote countable discrete groups. We let
$\mathbb {T}\subset \mathbb {C}$
be the circle group. We denote the multiplier algebra of A by
$M(A)$
. Any automorphism
$\alpha \in \mathrm {Aut}(A)$
extends uniquely to an automorphism of
$M(A)$
, we denote this extension also by
$\alpha $
. For a unitary
$u\in M(A)$
, we write
$\mathrm {Ad}(u)$
for the automorphism
$a\mapsto uau^*$
of A and the group of inner automorphisms on A by
$\mathrm {Inn}(A)$
. Recall that a G-kernel of A is a group homomorphism
$G\rightarrow \mathrm {Aut}(A)/\mathrm {Inn}(A)=\mathrm {Out}(A)$
. We now recall the definition of an anomalous action from [Reference Jones27, Definition 1.1]. In the case that A has trivial center, this notion coincides with a lift of a G-kernel into
$\mathrm {Aut}(A)$
.
Definition 2.1 An anomalous action of a countable discrete group G on a C
$^*$
-algebra A consists of a pair
$(\alpha ,u)$
where
$$ \begin{align*}\alpha: G\rightarrow \mathrm{Aut}(A),\end{align*} $$
$$ \begin{align*}u:G\times G\rightarrow U(M(A))\end{align*} $$
are a pair of maps such that
$$ \begin{align} \alpha_g\alpha_h&=\mathrm{Ad}(u_{g,h})\alpha_{gh},\ \text{for all}\ g,h\in G, \end{align} $$
$$ \begin{align} \alpha_g(u_{h,k})&u_{g,hk}u_{gh,k}^*u_{g,h}^* \in \mathbb{T}\cdot 1_{M(A)},\ \text{for all}\ g,h,k\in G. \end{align} $$
First, note that in (2.1) and (2.2), we have used the subscript notation
$\alpha _g$
and
$u_{g,h}$
instead of
$\alpha (g)$
and
$u(g,h)$
for
$g,h\in G$
. We will use this throughout when notationally convenient.
As shown in [Reference Eilenberg and MacLane10, Lemma 7.1], the formula in (2.2) defines a circle valued
$3$
-cocycle, i.e., an element of
$Z^3(G,\mathbb {T})$
. We will call this the anomaly of the action and denote it by
$o(\alpha ,u)$
. For
$\omega \in Z^3(G,\mathbb {T})$
, we say
$(\alpha ,u)$
is a
$(G,\omega )$
action on A to mean that
$(\alpha ,u)$
is an anomalous action of G on A with anomaly
$\omega $
.Footnote
3
If
$\omega =1$
, then we call
$(\alpha ,u)$
a cocycle action. Note that any anomalous action
$(\alpha ,u)$
induces a G-kernel when passing to the quotient group
$\mathrm {Out}(A)$
, we denote its associated G-kernel by
$\overline {\alpha }$
. For any G-kernel
$\overline {\alpha }$
on A, we denote by
$\operatorname {ob}(\overline {\alpha })\in H^3(G,Z(U(M(A))))$
its
$3$
-cohomology invariant (see, e.g., [Reference Evington and Girón Pacheco13, Section 2.1]).
The reader should be warned that there is a slight variation in Definition 2.1 to the definitions of anomalous actions in [Reference Evington and Girón Pacheco13, Reference Jones27]. Given our conventions in Definition 2.1, a
$(G,\omega )$
action induces an
$\overline {\omega }$
anomalous action as in [Reference Jones27, Definition 1.1], this is seen by taking
$m_{g,h}=u_{g,h}^*$
.
Throughout this paper, we will denote the algebra of bounded sequences of A quotiented by those sequences going to zero in norm by
$A_\infty $
. For a
$^*$
-closed subset S of
$A_{\infty }$
, we may consider the commutant C
$^*$
-algebra
$A_\infty \cap S'=\{x\in A_{\infty }:[x,S]=0\}$
and the annihilator
$A_{\infty }\cap S^{\perp }=\{x\in A_{\infty }: xS=Sx=0\}.$
We may then denote Kirchberg’s sequence algebra by
$$ \begin{align*}F(S,A_{\infty})=(A_{\infty}\cap S')/(A_{\infty}\cap S^{\perp}).\end{align*} $$
In the case that S is the C
$^*$
-algebra of constant sequences in
$A_{\infty }$
, we denote this simply by
$F(A)=F(A,A_{\infty })$
and
$F(A)$
the central sequence algebra of A. Note that
$F(A)$
is a unital C
$^*$
-algebra whenever A is
$\sigma $
-unital. Indeed, the unit is given by
$h=(h_n)$
for any sequential approximate unit
$h_n$
for A.
Any automorphism
$\theta \in \mathrm {Aut}(A)$
induces an automorphism
$\theta $
of
$A_\infty $
through
$(a_n)\mapsto (\theta (a_n))$
for any
$(a_n)\in A_\infty $
.Footnote
4
If a subset S of
$A_\infty $
is invariant under both
$\theta $
and
$\theta ^{-1}$
, then so are
$A_\infty \cap S'$
and
$A_\infty \cap S^{\perp }$
and
$\theta $
induces an automorphism of
$F(S,A_\infty )$
.
Definition 2.2 For an anomalous action
$(\alpha ,u)$
of a group G on a C
$^*$
-algebra A and a
$^*$
-closed subset
$S\subset A_{\infty }$
, we say S is
$(\alpha ,u)$
-invariant if
$\alpha _g(S)\subset S$
for all
$g\in G$
and
$u_{g,h}S+Su_{g,h}\subset S$
for all
$g,h\in G$
.
Note that whenever S is
$(\alpha ,u)$
-invariant, then the automorphisms
$\mathrm {Ad}(u_{g,h})$
also preserve S for all
$g,h\in G$
and so
$\alpha _g^{-1}=\mathrm {Ad}(u_{g,g^{-1}})\alpha _g$
preserve S for all
$g\in G$
.
Remark 2.1 When A is equipped with a
$(G,\omega )$
action
$(\alpha ,u)$
, it induces a
$(G,\omega )$
action on
$A_\infty $
. In fact,
$\alpha $
induces a group action on
$F(A)$
as
$\mathrm {Ad}(u)(x)-x\in A_\infty \cap A^{\perp }$
for any
$x\in A_\infty \cap A'$
and
$u\in U(M(A))$
. Similarly, if
$S=S^*$
is an
$(\alpha ,u)$
invariant subset of
$A_{\infty }$
, then
$\alpha $
induces a group action on
$F(S,A_{\infty })$
(see [Reference Szabó45, Remarks 1.8 and 1.10]).
We will be interested in anomalous actions with the Rokhlin property. This notion was introduced in [Reference Izumi22, Definition 3.10] for actions of finite groups on unital C
$^*$
-algebras and later generalized by Nawata and Santiago for non-unital C
$^*$
-algebras (see [Reference Nawata35, Reference Santiago43]). Its definition in the setting of anomalous actions is ad verbatim, we will only require it for
$\sigma $
-unital C
$^*$
-algebras.
Definition 2.3 An anomalous action
$(\alpha ,u)$
of a finite group G on a
$\sigma $
-unital C
$^*$
-algebra A is said to have the Rokhlin property, if there exist projections
$p_g\in F(A)$
for
$g\in G$
such that:
-
(1)
$\sum _{g\in G} p_g=1,$
-
(2)
$\alpha _g(p_h)=p_{gh}.$
Remark 2.2 The Rokhlin property also makes sense for G-kernels. In this case, a G-kernel
$\overline {\alpha }$
of a finite group G on a
$\sigma $
-unital C
$^*$
-algebra A satisfies the Rokhlin property if for any/some lift
$(\alpha ,u)$
of
$\overline {\alpha }$
there exists a partition of unity of projections
$p_g\in F(A)$
for
$g\in G$
such that
$\alpha _g(p_h)=p_{gh}$
for all
$g,h\in G$
.
Our main goal is to classify anomalous actions with the Rokhlin property. To make sense of this question, we first need to introduce equivalence relations for anomalous actions. Before we do so, we start by introducing some notation that will allow us to streamline future definitions.
Definition 2.4 Let
$(\alpha ,u)$
be an anomalous action of a group G on a C
$^*$
-algebra A. If 
for
$g\in G$
, then the pair 
with
is an anomalous action. We say that 
is a unitary perturbation of
$(\alpha ,u)$
.
It is a straightforward that 
for any map 
.
Definition 2.5 Let
$A,B$
be C
$^*$
-algebras, let
$(\alpha ,u)$
be an anomalous G action on A, and let
$(\beta ,v)$
be an anomalous action on B. Then we say that:
-
(i)
$(\alpha ,u)$
is conjugate to
$(\beta ,v)$
if there exists an isomorphism
$\theta :A\rightarrow B$
such that
$\alpha _g=\theta \beta _g\theta ^{-1}$
and
$v_{g,h}=\theta (u_{g,h})$
for all
$g,h\in G$
. -
(ii)
$(\alpha ,u)$
is cocycle conjugate to
$(\beta ,v)$
if there exist unitaries
$s_g\in U(M(A))$
for
$g\in G$
such that
$(\alpha ^{s},u^{s})$
is conjugate to
$(\beta ,v)$
. We denote this by
$(\alpha ,u)\simeq (\beta ,v)$
. -
(iii) If A and B are equal and
$(\alpha ,u)\simeq (\beta ,v)$
with the conjugacy holding through an automorphism
$\theta $
such that
$K_i(\theta )=\operatorname {id}_{K_i(A)}$
for
$i=1,2$
, we say
$(\alpha ,u)$
and
$(\beta ,v)$
are K-trivially cocycle conjugate. We denote this by
$(\alpha ,u)\simeq _{K}(\beta ,v)$
.
Finally, recall the definition of a unitary one cocycle.
Definition 2.6 Let
$\alpha $
be a
$(G,\omega )$
action on a C
$^*$
-algebra A. We call a map
$v: G\rightarrow U(M(A))$
such that
$v_g\alpha _g(v_h)=v_{gh}$
an
$\alpha $
-cocyle.
3 Model actions
Given a finite group G and
$\omega \in Z^3(G,\mathbb {T})$
a normalized
$3$
-cocycle, [Reference Evington and Girón Pacheco13, Theorem C] constructs a
$(G,\omega )$
action on
$M_{|G|^\infty }$
. This result is based on a construction of C. Jones in [Reference Jones27] which in turn is based on a construction of V. F. R. Jones in the setting of von Neumann algebras [Reference Jones28].
In this section, we recall this construction as we will need its specific form to deduce properties of the action. First, recall that a
$3$
-cocycle
$\omega :G^{\times 3}\rightarrow \mathbb {T}$
is called normalized if
$\omega (g,h,k)=1$
whenever either
$g,h$
or k are the identity. In [Reference Jones27], C. Jones shows that if
$\omega $
is a normalized
$3$
-cocycle and one has the following data:
-
• a group
$\Gamma $
and a surjection
$\rho :\Gamma \twoheadrightarrow G$
such that
$\rho ^*(\omega )$
is a coboundary, -
• a normalized
$2$
-cochain
$c:\Gamma \times \Gamma \rightarrow \mathbb {T}$
such that
$\rho ^*(\omega )=dc$
, -
• a C
$^*$
-algebra B and an action
$\pi :\Gamma \rightarrow \mathrm {Aut}(B)$
,
one can induce a
$(G,\omega )$
action on the twisted reduced crossed product
$B\rtimes _{\pi ,\overline {c}}^r K$
, with
$K=\ker (\rho )$
(see [Reference Busby and Smith4] for a reference on twisted crossed products).Footnote
5
The automorphic data of this
$(G,\omega )$
action are given by
$$ \begin{align} \theta_g\left(\sum_{k\in K}a_kv_k\right)=\sum_{k\in K} c_{\hat{g}k\hat{g}^{-1},\hat{g}^{-1}}\overline{c_{\hat{g},k}}\pi_{\hat{g}}(a_k)v_{\hat{g}k\hat{g}^{-1}}, \end{align} $$
for
$a_k\in B$
,
$v_k$
the canonical unitaries in
$M(B\rtimes _{\pi ,\overline {c}}^r K)$
,
$g\in G$
and
$g\mapsto \hat {g}$
a choice of set-theoretic section to
$\rho : \Gamma \rightarrow G$
.Footnote
6
In fact, given an arbitrary finite group G, C. Jones constructs a finite group
$\Gamma $
, a surjection
$\rho $
, and a
$2$
cochain c with the conditions needed above and additionally
$c|_{\ker (\rho )}=1$
. Additionally, to
$\Gamma $
and c, the extra data considered in [Reference Evington and Girón Pacheco13, Theorem C] are:
-
•
$B=\bigotimes _{i\in \mathbb {N}} \mathcal {B}(l^2(\Gamma ))$
, -
•
$\pi =\mathrm {Ad}(\lambda _\Gamma )^{\otimes \infty }$
,
with
$\lambda _\Gamma $
the left regular representation and
$\mathrm {Ad}(\lambda _\Gamma )_\gamma (T)=\lambda _\Gamma (\gamma )T\lambda _\Gamma (\gamma )^*$
for all
$T\in \mathcal {B}(l^2(\Gamma ))$
and
$\gamma \in \Gamma $
. In this case, the crossed product
$B\rtimes _\pi ^r K$
is shown to be isomorphic to the UHF algebra
$M_{|G|^\infty }$
. C. Jones’ construction then yields a
$(G,\omega )$
action on
$M_{|G|^\infty }$
through (3.1) for any
$\omega \in Z^3(G,\mathbb {T})$
, we denote it by
$(s_G^\omega ,u_G^{\omega })$
.
Proposition 3.1 Let G be a finite group and
$\omega \in Z^3(G,\mathbb {T})$
, then
$(s_G^\omega ,u_G^{\omega })$
has the Rokhlin property.
Proof We use the notation set up in the previous paragraphs. Furthermore, denote by
$r_i:\mathcal {B}(l^2(\Gamma ))\rightarrow B$
the unital embedding into the ith tensor factor. As
$A=B\rtimes _{\pi } K$
is unital,
$F(A)$
coincides with
$A_{\infty }\cap A'$
, so it suffices to find a partition of unity
$p_g\in A_{\infty }\cap A'$
for
$g\in G$
such that
$\alpha _g(p_h)=p_{gh}$
for all
$g,h\in G$
.
Let
$e_K$
in
$\mathcal {B}(l^2(\Gamma ))$
be the projection onto
$l^2(K)$
, that is,
$$ \begin{align*}e_K\left(\sum_{\gamma\in \Gamma}\mu_\gamma \gamma\right)=\sum_{\gamma\in K}\mu_\gamma \gamma\end{align*} $$
for any complex scalars
$\mu _\gamma $
. Let
$p_n=r_n(e_K)$
for
$n\in \mathbb {N}$
. Note that the projection
$p=(p_n) \in B_\infty $
commutes with any constant sequence of elements in B. Moreover, p commutes with the subalgebra
$C^*(K)\subset (B\rtimes K)_\infty $
. Indeed,
$e_K$
is invariant under
$\mathrm {Ad}(\lambda _\Gamma )_{k}$
for any
$k\in K$
and therefore for any
$n\in \mathbb {N}$
and
$k\in K$
,
$$ \begin{align*} v_kp_nv_k^*&=\mathrm{Ad}(\lambda_\Gamma)_{k}^{\otimes \infty}(r_n(e_K))\\ &=r_n(\mathrm{Ad}(\lambda_\Gamma)_{k}e_K))\\ &=r_n(e_K)\\ &=p_n. \end{align*} $$
Therefore,
$p\in A_{\infty }\cap A'$
.
We claim that the projections
$p_g:=s_G^{\omega }(g)(p)=(s_G^{\omega }(g)(p_n))_{n\in \mathbb {N}}$
form a set of Rokhlin projections. We start by showing that the sum
$\sum _{g\in G} s_G^\omega (g)(p)=1$
. Let
$n\in \mathbb {N}$
and
$g\in G$
, then as the cochain c is normalized, it follows from (3.1) that
$$ \begin{align} s_G^{\omega}(g)(p_n)&=\pi_{\hat{g}}(p_n)\\ &=\mathrm{Ad}(\lambda_{\Gamma})^{\otimes\infty}_{\hat{g}}(p_n)\notag \\ &=\mathrm{Ad}(\lambda_{\Gamma})^{\otimes\infty}_{\hat{g}}(r_n(e_K))\notag \\ &=r_n(\mathrm{Ad}(\lambda_{\Gamma})_{\hat{g}}(e_K))\notag. \end{align} $$
The maps
$r_n$
are unital, so it suffices to show that
$\sum _{g\in G}\mathrm {Ad}(\lambda _{\Gamma })_{\hat {g}}(e_K)=1_{\mathcal {B}(l^2(\Gamma ))}$
. To see this, let
$\gamma \in \Gamma $
,
$g\in G$
, and
$\delta _\gamma \in l^2(\Gamma )$
the point mass at
$\gamma $
, then
$$ \begin{align} \mathrm{Ad}(\lambda_{\Gamma})_{\hat{g}}(e_K)(\delta_\gamma)&=\lambda_\Gamma(\hat{g})e_K\lambda_\Gamma(\hat{g}^{-1})(\delta_\gamma)\\ &=\lambda_\Gamma(\hat{g})e_K(\delta_{\hat{g}^{-1}\gamma})\notag\\ &=\begin{cases}\notag \delta_{\gamma},\ \text{if}\ \gamma\in \hat{g}K,\\ 0,\ \text{otherwise}. \end{cases} \end{align} $$
The left K cosets are pairwise disjoint and cover the whole group
$\Gamma $
. Therefore, it follows that
$\sum _{g\in G}\mathrm {Ad}(\lambda _{\Gamma })_{\hat {g}}(e_K)(\delta _\gamma )=\delta _\gamma $
for every
$\gamma \in \Gamma $
. As the operators
$\sum _{g\in G}\mathrm {Ad}(\lambda _{\Gamma })_{\hat {g}}(e_K)$
and
$\operatorname {id}_{\mathcal {B}(l^2(\Gamma ))}$
coincide on a spanning set of
$l^2(\Gamma )$
, these operators are equal.
It remains to show that for
$g,h\in G$
the projections
$s_G^{\omega }(g)p_h=p_{gh}$
. This follows as
$s_G^{\omega }(g)p_h=s_G^{\omega }(g)s_G^{\omega }(h)p=\mathrm {Ad}(u^{\omega }_G(g,h))s_G^{\omega }(gh)p=\mathrm {Ad}(u_G^{\omega }(g,h))p_{gh}=p_{gh}$
where the last equality in the chain holds as
$p_{gh}$
commutes with A.
4 Absorption of model actions
In this section, we show that any Rokhlin anomalous action of a finite group G, on an
$M_{|G|^\infty }$
-stable C
$^*$
-algebra, absorbs the action
$$ \begin{align*}s_G=\bigotimes_{i=0}^{\infty}\mathrm{Ad}(\lambda_G)\end{align*} $$
up to cocycle conjugacy.Footnote
7
This result is similar in nature to
$(i)\Rightarrow (iii)$
of [Reference Gardella and Santiago16, Theorem 5.2]. The methods utilized in this chapter are an adaptation of V. F. R. Jones’ work [Reference Jones29] to the C
$^*$
-setting.
In his work [Reference Szabó44–Reference Szabó46], Szabó establishes the theory of strongly self-absorbing C
$^*$
-dynamical systems as an equivariant version of strongly self-absorbing C
$^*$
-algebras that were introduced in [Reference Toms and Winter47]. We recall the main definition below.
Definition 4.1 Let G be a locally compact group. A group action
$\gamma $
on a unital, separable C
$^*$
-algebra
$\mathcal {D}$
is called strongly self-absorbing if there exists an equivariant isomorphism
$\varphi :(\mathcal {D},\gamma )\rightarrow (\mathcal {D}\otimes \mathcal {D},\gamma \otimes \gamma )$
such that there exist unitaries
$u_n\in U(\mathcal {D}\otimes \mathcal {D})$
fixed by
$\gamma \otimes \gamma $
with
$$ \begin{align*}\lim\limits_{n\rightarrow \infty}\|\varphi(a)-u_n(a\otimes 1_{\mathcal{D}})u_n^*\|=0\end{align*} $$
for all
$a\in \mathcal {D}$
. That is, the maps
$\varphi $
and
$\operatorname {id}_{\mathcal {D}}\otimes 1_{\mathcal {D}}$
are approximately unitarily equivalent or in short
$\varphi \approx _{a.u}\operatorname {id}_{\mathcal {D}}\otimes 1_{\mathcal {D}}$
.
The relevant example of a strongly self-absorbing action for this paper is
$s_G$
. That
$s_G$
is strongly self-absorbing follows as a consequence of [Reference Szabó45, Example 5.1].
In [Reference Szabó45, Theorem 3.7], Szabó shows equivalent conditions for a cocycle action to tensorially absorb a strongly self-absorbing action. Although Szabó’s theory only treats the case of cocycle actions absorbing a given strongly self-absorbing group action, many of the arguments follow in exactly the same way when replacing cocycle actions by anomalous actions that may have nontrivial anomaly. The proofs of [Reference Szabó45, Lemma 2.1 and Theorem 2.6] and [Reference Szabó45, Theorem 3.7 and Corollary 3.8], for example, make no use of the anomaly associated with
$(\alpha ,u)$
and
$(\beta ,w)$
being trivial. Under this observation, we can state a specific case of [Reference Szabó45, Corollary 3.8].
Theorem 4.1 (cf. [Reference Szabó45, Theorem 2.8])
Let A and
$\mathcal {D}$
be separable C
$^*$
-algebras, and let G be a finite group. Assume that
$(\alpha ,u):G\curvearrowright A$
is an anomalous action. Let
$\gamma :G\curvearrowright \mathcal {D}$
be a group action such that
$(\mathcal {D},\gamma )$
is strongly self-absorbing. If there exists an equivariant and unital
$^*$
-homomorphism
$$ \begin{align*}(\mathcal{D},\gamma)\rightarrow (F(A),\alpha),\end{align*} $$
then
$(A,\alpha ,u)$
is cocycle conjugate to
$(A\otimes \mathcal {D},\alpha \otimes \gamma ,u\otimes 1_{\mathcal {D}})$
through a map
$\varphi :A\rightarrow A\otimes \mathcal {D}$
that is approximately unitarily equivalent to
$\operatorname {id}_A\otimes 1_{\mathcal {D}}$
.
We still require a few more results before we can achieve the model action absorption. These are based on known results in the setting of finite group actions on unital C
$^*$
-algebras. These generalize line by line to anomalous actions of finite groups on unital C
$^*$
-algebras, we adapt the arguments also for non-unital C
$^*$
-algebras.
Lemma 4.2 (cf. [Reference Hirshberg and Winter21, Theorem 3.3])
Let A be a C
$^*$
-algebra, let G be a finite group, and let
$(\alpha ,u)$
be an anomalous action of G on A with the Rokhlin property. If
$B=B^*$
is a separable
$(\alpha ,u)$
-invariant subset of
$A_\infty $
and there exists a unital
$^*$
-homomorphism
$M\rightarrow F(B,A_{\infty })$
for some separable, unital C
$^*$
-algebra M, then there exists a unital
$^*$
-homomorphism
$M\rightarrow F(B,A_\infty )^{\alpha }$
.
Proof Fix a unital homomorphism
$\psi : M\rightarrow F(B,A_{\infty })$
and choose a linear lift
$\psi _0:M\rightarrow A_{\infty }\cap B'$
. Then one has that:
-
(i)
$(\psi _0(m)\psi _0(m')-\psi _0(mm'))b=0,\quad \forall m,m'\in M, b\in B,$
-
(ii)
$(\psi _0(m^*)-\psi _0(m)^*)b=0,\quad \forall m\in M, b\in B,$
-
(iii)
$\psi _0(1)b-b=0,\quad \forall b\in B.$
Let
$S=B\cup _{g\in G}\alpha _g(\psi _0(M))\cup _{g\in G}\alpha _g(\psi _0(M))^*$
, so
$S=S^*$
. By the Rokhlin property followed by a standard reindexing argument, there exist positive contractions
$f_g\in A_{\infty }\cap S'$
such that:
-
(iv)
$(\alpha _g(f_h)-f_{gh})a=0,\quad \forall g,h\in G, a\in S,$
-
(iiv)
$(\sum _{g\in G} f_g)a-a=0\quad \forall a\in S,$
-
(iiiv)
$f_gf_ha-\delta _{g,h}a=0\quad \forall g,h\in G, a\in S.$
Now consider the linear mapping
$\varphi :M\rightarrow A_{\infty }\cap B'$
given by
$$ \begin{align*}\varphi(m)=\sum_{g\in G}\alpha_g(\psi_0(m))f_g.\end{align*} $$
First, for
$m,m'\in M$
and
$b\in B$
, it follows from (i) and (iiiv) that
$$ \begin{align*} \varphi(m)\varphi(m')b&=\sum_{g,h\in G}\alpha_g(\psi_0(m))f_g\alpha_h(\psi_0(m'))f_hb\\ &=\sum_{g,h\in G} \alpha_g(\psi_0(m))\alpha_h(\psi_0(m'))f_gf_hb\\ &=\sum_{g\in G}\alpha_g(\psi_0(m))\alpha_g(\psi_0(m'))bf_g\\ &=\sum_{g\in G}\alpha_g(\psi_0(mm'))bf_g\\ &=\sum_{g\in G}\alpha_g(\psi_0(mm'))f_gb\\ &=\varphi(mm')b. \end{align*} $$
Also for
$k\in G, m\in M$
and
$b\in B$
it follows using (iv) that
$$ \begin{align*} \alpha_k(\varphi(m))b&=\sum_{g\in G}\alpha_k\left(\alpha_g(\psi_0(m))\right)\alpha_k(f_g)b\\ &=\sum_{g\in G}\mathrm{Ad}(u_{k,g})(\alpha_{kg}(\psi_0(m)))f_{kg}b\\ &=\sum_{g\in G}\mathrm{Ad}(u_{k,g})(\alpha_{kg}(\psi_0(m)))bf_{kg}\\ &=\varphi(m)b. \end{align*} $$
Where in the last line we have used that B is u invariant and so the observation in Remark 2.1 applies. Therefore, the map
$$ \begin{align*}m\mapsto \varphi(m)+A_\infty\cap B^{\perp}\end{align*} $$
defines a homomorphism from M into
$(F(B,A_\infty ))^\alpha $
. This homomorphism is unital through combining (iii) and (iiv) and
$^*$
-preserving by (ii).
In the next lemma, recall that if
$\alpha $
is an action of a group G on a C
$^*$
-algebra A, an
$\alpha $
-cocycle is a family of unitaries
$v_g\in U(M(A))$
for
$g\in G$
such that
$v_g\alpha _g(v_h)=v_{gh}$
.
Lemma 4.3 (cf. [Reference Herman and Jones19, Lemma III.1])
Let A be a separable C
$^*$
-algebra, and let G be a finite group. Let
$(\alpha ,u)$
be an anomalous action of G on A with the Rokhlin property. Let
$B=B^*$
be a separable
$(\alpha ,u)$
-invariant subset of
$A_\infty $
. For any
$\alpha $
-cocycle
$v_g$
for the action induced by
$\alpha $
on
$F(B,A_\infty )$
, there exists a unitary
$u\in F(B,A_\infty )$
with
$u^*\alpha _g(u)=v_g$
.
Proof Let
$v_g\in U(F(B,A_\infty ))$
be an
$\alpha $
-cocycle. Choosing lifts
$v_g'\in A_{\infty }\cap B'$
for
$v_g$
, one has:
-
(i)
$v_g'(v_g')^*b-b=0,\quad \forall g\in G, b\in B,$
-
(ii)
$(v_g')^*v_g'b-b=0,\quad \forall g\in G, b\in B,$
-
(iii)
$v_g'\alpha _g(v_h')b-v_{gh}'b=0,\quad \forall g,h\in G, b\in B.$
Let
$S=B\cup \{\alpha _h(v_g'), \alpha _h(v_g')^* :g,h\in G\}$
. As in the previous lemma, one may apply the Rokhlin property combined with a reindexing argument to get a family of positive elements
$f_g\in A_{\infty }\cap S'$
such that:
-
(iv)
$(\alpha _g(f_h)-f_{gh})a=0,\quad \forall g,h\in G, a\in S,$
-
(iiv)
$\sum _{g\in G}f_ga-a=0, \quad \forall a\in S,$
-
(iiiv)
$f_gf_ha-\delta _{g,h}a=0\quad \forall g,h\in G, a\in S.$
Let
$u=\sum _{g\in G}v_g'f_g \in A_{\infty }\cap B'$
. Then, for any
$b\in B$
by (ii), (iiv), and (iiiv), it follows that
$$ \begin{align*} u^*ub&=\sum_{g,h\in G}f_g(v_g')^*v_h'f_hb\\ &=\sum_{g,h\in G}(v_g')^*v_h'bf_gf_h\\ &=\sum_{g\in G}(v_g')^*v_g'bf_g\\ &=b. \end{align*} $$
Similarly,
$uu^*b=b$
for any
$b\in B$
. Moreover, (iii), (i), (iv) and (iiv) imply that for
$b\in B$
and
$g\in G$
,
$$ \begin{align*} u\alpha_g(u^*)b&=\sum_{h,k}v_h'f_h\alpha_g(f_k)\alpha_g(v_k')^*b\\ &=\sum_{h,k}v_h'\alpha_g(v_k')^*bf_h\alpha_g(f_k)\\ &=\sum_k v_{gk}'\alpha_g(v_k')^*bf_{gk}\\ &=\sum_k v_{g}'bf_{gk}\\ &=v_g'b. \end{align*} $$
Therefore, by passing to the quotient, u defines a unitary in
$F(B,A_\infty )$
such that
$u\alpha _g(u^*)=v_g$
for all
$g\in G$
.
For a finite group G, we denote by
$e_{g,h}\in \mathcal {B}(l^2(G))$
the canonical matrix units defined by
$$ \begin{align*} e_{g,h}(f)(k)=\begin{cases} f(h),\ \text{if}\ k=g,\\ 0,\ \text{otherwise}, \end{cases} \end{align*} $$
for
$f\in l^2(G)$
. The proof of the next lemma is based on the proof of [Reference Jones29, Proposition 3.4.1].
Lemma 4.4 Let G be a finite group, and let A be a separable C
$^*$
-algebra such that
$A\cong A\otimes \mathbb {M}_{|G|^\infty }$
. Let
$(\alpha ,u)$
be an anomalous action with the Rokhlin property of G on A. Then there exists a G-equivariant unital embedding
$$ \begin{align*}(\mathbb{M}_{|G|^\infty}, s_G)\rightarrow (F(A),\alpha).\end{align*} $$
Proof To prove this, we inductively construct unital equivariant
$^*$
-homomorphisms
$\phi _n:(\mathcal {B}(l^2(G)),\mathrm {Ad}(\lambda _G))\rightarrow (F(A),\alpha )$
for
$n\in \mathbb {N}$
with commuting images. Then the map defined by
$a_1\otimes \dots \otimes a_n \otimes \dots \longmapsto \prod _{i\in \mathbb {N}}\phi _i(a_i)$
will induce an
$s_G$
to
$\alpha $
equivariant map into
$F(A)$
.
Suppose
$\phi _1,\phi _2,\dots ,\phi _n:(\mathcal {B}(l^2(G)),\mathrm {Ad}(\lambda _G))\rightarrow (F(A),\alpha )$
are equivariant maps with commuting images and let
$\psi _i:B(l^2(G))\rightarrow A_\infty \cap A'$
be linear lifts of
$\phi _i$
for
$1\leq i\leq n$
, then:
-
(i)
$\psi _i(m)\psi _j(m')a-\psi _j(m')\psi _i(m)a=0,\quad \forall a\in A, m,m'\in B(l^2(G)), 1\leq i\neq j\leq n$
, -
(ii)
$\alpha _g(\psi _i(m))a-\psi _i(\lambda _G(g)(m))a=0,\quad \forall a\in A, m\in B(l^2(G)), 1\leq i \leq n, g\in G$
, -
(iii)
$\psi _i(m)^*a-\psi _i(m^*)a=0,\quad \forall a\in A, m\in B(l^2(G)),1\leq i\leq n$
, -
(iv)
$\psi _i(1)a-a=0,\quad \forall \in A, 1\leq i\leq n$
.
Let
$$ \begin{align*}S=\{\psi_i(m)a: m\in B(l^2(G)),\ a\in A,\ 1\leq i\leq n\}.\end{align*} $$
Then S is separable,
$S=S^*$
, and S is
$(\alpha ,u)$
invariant. We check that
$u_{g,h}S\subset S$
for all
$g,h\in G$
, the remaining conditions follow similarly. For
$a\in A$
,
$m\in B(l^2(G))$
, and
$1\leq i \leq n$
, letting
$m'=\mathrm {Ad}(\lambda _G)_{h^{-1}g^{-1}}(m)$
, one has that
$$ \begin{align*} u_{g,h}\psi_i(m)a&=u_{g,h}\psi_i(\mathrm{Ad}(\lambda_{G})_{gh}(m'))a\\ &\overset{(ii)}{=}u_{g,h}\alpha_{gh}(\psi_i(m'))a\\ &=\alpha_g\alpha_h(\psi_i(m'))u_{g,h}a\\ &\overset{(ii)}{=}\psi_i(\mathrm{Ad}(\lambda_G)_{gh}(m'))u_{g,h}a \in S. \end{align*} $$
As
$A\cong A\otimes M_{|G|^\infty }$
, pick a unital embedding from
$B(l^2(G))$
into
$F(S,A_\infty )$
. (As
$A\otimes \mathbb {M}_{|G|^\infty }\cong A$
, there exists a unital embedding of
$B(l^2(G))$
into
$F(A)$
by [Reference Toms and Winter47, Theorem 2.2]. Moreover, by reindexing, one can also choose a homomorphism as stated.) It follows from Lemma 4.2 that there exists a unital embedding
$B(l^2(G))\rightarrow F(S,A_{\infty })^{\alpha }$
. Let
$(e^{\prime }_{g,h})_{g,h\in G}$
in
$F(S,A_{\infty })^\alpha $
be the images of
$e_{g,h}$
under this unital embedding. The permutation unitary
$v_g=\sum _{h\in G} e^{\prime }_{gh,h}$
gives a unitary representation of G on
$F(S,A_{\infty })^\alpha $
and as
$\alpha _g(v_h)=v_h$
it follows that
$v_g$
is an
$\alpha $
-cocycle. Therefore, by Lemma 4.3, there exists a unitary
$u\in F(S,A_{\infty })$
such that
$u\alpha _g(u^*)=v_g$
. Now,
$f_{g,h}=u^*e^{\prime }_{g,h}u$
for
$g,h\in G$
is a set of matrix units such that
$$ \begin{align*} \alpha_k(f_{g,h})&=\alpha_k(u^*)e^{\prime}_{g,h}\alpha_k(u)\\ &=u^*v_ke^{\prime}_{g,h}v_k^*u\\ &=u^*\left(\sum_{h^{\prime},h^{\prime\prime}\in G}e^{\prime}_{kh^{\prime},h^{\prime}}e^{\prime}_{g,h}e^{\prime}_{h^{\prime\prime},kh^{\prime\prime}}\right)u\\ &=u^*(e^{\prime}_{kg,kh})u\\ &=f_{kg,kh}. \end{align*} $$
Hence, the
$^*$
-homomorphism
$$ \begin{align*} \phi_{n+1}:\mathcal{B}(l^2(G))&\rightarrow F(S,A_{\infty}),\\ e_{g,h}&\mapsto f_{g,h} \end{align*} $$
defines an
$\mathrm {Ad}(\lambda _G)$
to
$\alpha $
equivariant
$^*$
-homomorphisms. Moreover, the image of
$\phi _{n+1}$
commutes with
$\phi _i$
for all
$1\leq i\leq n$
. Considering
$\phi _{n+1}$
as a unital equivariant homomorphism into
$A_{\infty }\cap A'/A_{\infty }\cap A^{\perp }$
, the induction argument is complete.
We have collected all the necessary ingredients to prove the model action absorption.
Proposition 4.5 Let G be a finite group, and let A be a separable C
$^*$
-algebra such that
$A \cong A\otimes \mathbb {M}_{|G|^\infty }$
. Let
$(\alpha ,u)$
be a
$(G,\omega )$
action on A with the Rokhlin property. Then
$(\alpha ,u)$
and
$(\alpha \otimes s_G,u\otimes 1_{\mathbb {M}_{|G|^\infty }})$
are cocycle conjugate through an isomorphism that is approximately unitarily equivalent to
$\operatorname {id}_A\otimes 1_{\mathbb {M}_{|G|^\infty }}$
.
5 Classification
We now discuss the abstract approach to bootstrapping the classification of group actions on a given class of C
$^*$
-algebras to a classification of anomalous actions. This method is a generalization of that used by Connes in [Reference Connes8, Section 6], a similar strategy was recently used in [Reference Izumi24] to classify G-kernels of poly-
$\mathbb {Z}$
groups on
$\mathcal {O}_2$
.
Before proceeding with the result, we set up notation. For a group G, we say “
$(\alpha ,u)$
is an anomalous G-action on A” and “
$(A,\alpha ,u)$
is an anomalous G-C
$^*$
-algebra” interchangeably. Let
$\Lambda $
be a functor whose domain category is the category of C
$^*$
-algebras (denoted C*alg). We say
$\Lambda $
is invariant under approximate unitary equivalence if
$\Lambda (\alpha )=\Lambda (\theta )$
whenever
$\alpha \approx _{a.u}\theta $
(see Definition 4.1 for notation). We also say that
$\Lambda $
restricted to a subcategory 
is full on isomorphisms, if whenever
$\Phi \in \operatorname {Hom}(\Lambda (A),\Lambda (B))$
is an isomorphism for
$A,B\in \mathcal {C}$
, then there exists an isomorphism
$\varphi :A\rightarrow B$
in
$\mathcal {C}$
with
$\Lambda (\varphi )=\Phi $
. The sort of functors with these properties are those used in the classification of C
$^*$
-algebras. For example, the functor K from the category of unital C
$^*$
-algebras into the category consisting of pairs of an abelian group and a pointed abelian group defined at the level of objects by
$A\mapsto K(A)=((K_0(A),[1_A]), K_1(A))$
is invariant under approximate unitary equivalence. The functor K is also full on isomorphisms when restricted to the category of unital Kirchberg algebras satisfying the UCT (see [Reference Phillips39]). Similarly, the functors
$KT_u$
and
$\underline {K}T_u$
of [Reference Carrión, Gabe, Schafhauser, Tikuisis and White5] are invariant under approximate unitary equivalence and are full on isomorphisms when restricted to classifiable C
$^*$
-algebras.
If
$\Lambda $
is invariant under unitary equivalence, an anomalous action
$(A,\alpha ,u)$
induces a G-action on
$\Lambda (A)$
through the automorphisms
$\Lambda (\alpha _g)$
. If
$(A,\alpha ,u)$
and
$(B,\beta ,v)$
are anomalous actions, we say that the induced actions
$\Lambda (\alpha _g)$
and
$\Lambda (\beta _g)$
are conjugate if there exists an isomorphism
$\Phi :\Lambda (A)\rightarrow \Lambda (B)$
with
$\Phi \Lambda (\alpha _g)\Phi ^{-1}=\Lambda (\beta _g)$
for all
$g\in G$
. We denote this by
$\Lambda (\alpha )\sim \Lambda (\beta )$
.
Let
$(A,\alpha ,u)$
and
$(A,\beta ,v)$
be two anomalous G-C
$^*$
-algebras. We write
$(\alpha ,u)\simeq _{\Lambda }(\beta ,v)$
if
$(\alpha ,u)\simeq (\beta ,v)$
through an automorphism
$\theta $
with
$\Lambda (\theta )=\operatorname {id}_{\Lambda (A)}$
. This notion recovers K-trivial cocycle conjugacy of Definition 2.5 when
$\Lambda $
is taken to be the functor consisting of
$K_0\oplus K_1$
. Finally, if
$\mathfrak {R}$
is a class of anomalous G-C
$^*$
-algebras, we will say
$\mathfrak {R}$
is closed under conjugacy, if whenever
$(A,\alpha ,u)\in \mathfrak {R}$
and
$\varphi :A\rightarrow B$
is an isomorphism in C*alg then
$(B,\varphi \alpha \varphi ^{-1},\varphi (u))\in \mathfrak {R}$
.
Lemma 5.1 Let G be a group,
$\mathcal {D}$
a strongly self-absorbing C
$^*$
-algebra, and
$\mathfrak {R}$
a class of anomalous G-C
$^*$
-algebras that is closed under conjugacy. Let
$\Lambda $
be a functor with domain category the category of C
$^*$
-algebras such that
$\Lambda $
is invariant under approximate unitary equivalence and is full on isomorphisms for C
$^*$
-algebras in
$\mathfrak {R}$
. Suppose further that:
-
(A1) there exists a G-action
$(\mathcal {D},s_G,1)$
such that if
$(A,\alpha ,u)\in \mathfrak {R}$
, then
$(A,\alpha ,u)\simeq (A\otimes \mathcal {D},\alpha \otimes s_G,u\otimes 1)$
through an isomorphism that is approximately unitarily equivalent to
$\operatorname {id}_A\otimes 1_{\mathcal {D}}$
; -
(A2) if there exists a
$(G,\omega )$
action in
$\mathfrak {R}$
for some
$\omega \in Z^3(G,\mathbb {T})$
, then there exist a
$(G,\omega )$
and
$(G,\overline {\omega })$
action
$(\mathcal {D},s_G^{\omega },u^\omega )$
and
$(\mathcal {D},s_G^{\overline {\omega }},u^{\overline {\omega }})$
, respectively, such that
$(\mathcal {D},s_G^{\overline {\omega }},u^{\overline {\omega }})\otimes (\mathcal {D},s_G^{\omega },u^\omega )\simeq (\mathcal {D},s_G,1)$
and for any
$(G,\omega )$
-action
$(A,\alpha ,u)\in \mathfrak {R}$
,
$(A,\alpha ,u)\otimes (\mathcal {D},s_G^{\overline {\omega }},u^{\overline {\omega }})\in \mathfrak {R}$
; -
(A3) for cocycle actions
$(A,\alpha ,u),(B,\beta ,v)\in \mathfrak {R}$
,
$\Lambda (\alpha )\sim \Lambda (\beta )$
if and only if
$\alpha \simeq \beta $
.
Then, if
$(A,\alpha ,u)$
and
$(B,\beta ,v)$
in
$\mathfrak {R}$
,
$(A,\alpha ,u)\simeq (B,\beta ,v)$
if and only if
$\Lambda (\alpha )\sim \Lambda (\beta )$
and
$o(\alpha ,u)=o(\beta ,v)$
.
With the same hypothesis but replacing (A3) with the condition that
-
(A3’) for cocycle actions
$(A,\alpha ,u)$
and
$(A,\beta ,v)$
in
$\mathfrak {R}$
,
$(A,\alpha ,u)\simeq _{\Lambda }(A,\beta ,v)$
if and only if
$\Lambda (\alpha _g)=\Lambda (\beta _g)$
for all
$g\in G$
,
then if
$(A,\alpha ,u)$
and
$(A,\beta ,v)$
in
$\mathfrak {R}$
,
$(A,\alpha ,u)\simeq _{\Lambda }(A,\beta ,v)$
if and only if
$o(\alpha ,u)=o(\beta ,v)$
and
$\Lambda (\alpha _g)=\Lambda (\beta _g)$
for every
$g\in G$
.
Proof First we show that if (A1)–(A3) hold and
$(A,\alpha ,u)$
,
$(B,\beta ,v)$
are anomalous actions in
$\mathfrak {R}$
, then
$(A,\alpha ,u)\simeq (B,\beta ,v)$
if and only if
$\Lambda (\alpha )\sim \Lambda (\beta )$
and
$o(\alpha ,u)=o(\beta ,v)$
. If
$(A,\alpha ,u)\simeq (B,\beta ,v)$
, it is clear that
$o(\alpha ,u)=o(\beta ,v)$
and also that
$\Lambda (\alpha )\sim \Lambda (\beta )$
as
$\Lambda $
is trivial when evaluated at inner automorphisms. We now turn to the converse. Suppose
$\Lambda (\alpha )\sim \Lambda (\beta )$
and
$o(\alpha ,u)=o(\beta ,v)$
. First, note that this implies that also
$\Lambda (\alpha \otimes \operatorname {id}_{\mathcal {D}})\sim \Lambda (\beta \otimes \operatorname {id}_{\mathcal {D}})$
. Indeed, by (A1), let
$\phi _A:A\rightarrow A\otimes \mathcal {D}$
and
$\phi _B:B\rightarrow B\otimes \mathcal {D}$
be isomorphisms which are approximately unitarily equivalent to the first factor embeddings and
$\Phi :\Lambda (A)\rightarrow \Lambda (B)$
be an isomorphism such that
$\Phi \Lambda (\alpha _g)\Phi ^{-1}=\Lambda (\beta _g)$
for
$g\in G$
. Note that
$\Lambda (\alpha _g\otimes \operatorname {id}_{\mathcal {D}})\Lambda (\phi _A)=\Lambda (\alpha _g\otimes \operatorname {id}_{\mathcal {D}})\Lambda (\operatorname {id}_A\otimes 1_{\mathcal {D}})=\Lambda (\alpha _g\otimes 1_{\mathcal {D}})=\Lambda (\phi _A)\Lambda (\alpha _g)$
(and similarly replacing A by B). Hence, we compute that
$$ \begin{align*} \Lambda(\alpha_g\otimes \operatorname{id}_{\mathcal{D}})\Lambda(\phi_A)\Phi \Lambda(\phi_B)^{-1}&=\Lambda(\phi_A)\Lambda(\alpha_g)\Phi \Lambda(\phi_B)^{-1}\\ &=\Lambda(\phi_A)\Phi \Lambda(\beta_g) \Lambda(\phi_B)^{-1}\\ &= \Lambda(\phi_A)\Phi \Lambda(\phi_B)^{-1} \Lambda(\beta_g\otimes \operatorname{id}_{\mathcal{D}}), \end{align*} $$
it follows that
$\Lambda (\phi _B)\Phi \Lambda (\phi _A)^{-1}$
conjugates
$\Lambda (\alpha _g\otimes \operatorname {id}_{\mathcal {D}})$
to
$\Lambda (\beta _g\otimes \operatorname {id}_{\mathcal {D}})$
for all
$g\in G$
. Now, by hypothesis, we have that
$$ \begin{align} (A,\alpha,u) \overset{(A1)}{\simeq}& (A\otimes \mathcal{D},\alpha\otimes s_G,u\otimes 1_{\mathcal{D}})\nonumber\\ \overset{(A2)}{\simeq}& (A\otimes(\mathcal{D}\otimes\mathcal{D}),\alpha\otimes(s_G^{\overline{\omega}}\otimes s_G^{\omega}),u\otimes (u^{\overline{\omega}}\otimes u^{\omega}))\nonumber\\ &=((A\otimes\mathcal{D})\otimes\mathcal{D},(\alpha\otimes s_G^{\overline{\omega}})\otimes s_G^{\omega},(u\otimes u^{\overline{\omega}})\otimes u^{\omega})\nonumber\\ \overset{(A3),(A2)}{\simeq}&((B\otimes\mathcal{D})\otimes\mathcal{D},(\beta\otimes s_G^{\overline{\omega}})\otimes s_G^{\omega},(v\otimes u^{\overline{\omega}})\otimes u^{\omega})\\ &=(B\otimes(\mathcal{D}\otimes\mathcal{D}),\beta\otimes(s_G^{\overline{\omega}}\otimes s_G^{\omega}),v\otimes (u^{\overline{\omega}}\otimes u^{\omega})\nonumber)\\ &\overset{(A2)}{\simeq} (B\otimes \mathcal{D},\beta\otimes s_G,v\otimes 1_{\mathcal{D}})\nonumber\\ &\overset{(A1)}{\simeq} (B,\beta,v).\nonumber \end{align} $$
Where in the third isomorphism we have used (A3) for the cocycle actions
$(A\otimes \mathcal {D},\alpha _g\otimes s_G^{\overline {\omega }},u\otimes u^{\overline {\omega }})$
and
$(B\otimes \mathcal {D},\beta _g\otimes s_G^{\overline {\omega }},v\otimes u^{\overline {\omega }})$
. The reason we may apply (A3) in this setting is that
$s_G^{\overline {\omega }}$
is approximately inner and hence our previous computation shows that
$\Lambda (\alpha _g\otimes s_G^{\overline {\omega }})=\Lambda (\alpha _g\otimes \operatorname {id}_{\mathcal {D}})\sim \Lambda (\beta _g\otimes \operatorname {id}_{\mathcal {D}})=\Lambda (\beta _g\otimes s_G^{\overline {\omega }})$
as required for the application of (A3).
Now suppose that we replace condition (A3) with (A3’). We will show that under the hypothesis of the lemma, (A3’) implies (A3). Therefore, the cocycle conjugacies in (5.1) still hold. Then we compute the isomorphisms that induce the cocycle conjugacies in (5.1) and show that their composition is the identity after applying
$\Lambda $
. Let
$(A,\alpha ,u)$
and
$(B,\beta ,v)$
be cocycle actions in
$\mathfrak {R}$
. Suppose
$\Lambda (\alpha )\sim \Lambda (\beta )$
. There exists an isomorphism
$\Phi \in \operatorname {Hom}(\Lambda (A),\Lambda (B))$
such that
$\Phi \Lambda (\beta _g)\Phi ^{-1}=\Lambda (\alpha _g)$
for all
$g\in G$
. As
$\Lambda $
is full on isomorphisms, there exists a
$^*$
-isomorphism
$\varphi :B\rightarrow A$
with
$\Lambda (\varphi )=\Phi $
. Therefore,
$\Lambda (\varphi \beta _g\varphi ^{-1})=\Lambda (\alpha _g)$
for all
$g\in G$
. By (A3’), one has that
$(A,\alpha ,u)\simeq _\Lambda (A,\varphi \beta \varphi ^{-1},\varphi (v))\simeq (B,\beta ,v)$
.
Set
$A=B$
in (5.1). Reading from top to bottom in (5.1), denote by
$\varphi _1$
,
$\varphi _2$
,
$\varphi _3$
,
$\varphi _4$
, and
$\varphi _5$
the isomorphisms inducing each of the conjugacies. Note that
$\varphi _5=\varphi _1^{-1}$
and
$\varphi _4=\varphi _2^{-1}$
. By (A1),
$\varphi _1\approx _{a.u} \operatorname {id}_A\otimes 1_{\mathcal {D}}$
. Moreover,
$\varphi _2\approx _{a.u} \operatorname {id}_A\otimes \operatorname {id}_{\mathcal {D}}\otimes 1_{\mathcal {D}}$
by [Reference Toms and Winter47, Corollary 1.12]. Denote by
$\varphi $
the isomorphism inducing the cocycle conjugacy from
$(A\otimes \mathcal {D},\alpha \otimes s_G^{\overline {\omega }},u\otimes u^{\overline {\omega }})$
to
$(A\otimes \mathcal {D},\beta \otimes s_G^{\overline {\omega }},u\otimes u^{\overline {\omega }})$
which satisfies
$\Lambda (\varphi )=\Lambda (\operatorname {id}_A\otimes \operatorname {id}_{\mathcal {D}})$
. We may use the functoriality of
$\Lambda $
and its invariance under approximate unitary equivalence to see that
$$ \begin{align*} \Lambda(\varphi_5\varphi_4\varphi_3\varphi_2\varphi_1)&=\Lambda(\operatorname{id}_A\otimes 1_{\mathcal{D}}\otimes 1_{\mathcal{D}})^{-1}\Lambda(\varphi\otimes\operatorname{id}_{\mathcal{D}})\Lambda(\operatorname{id}_A\otimes 1_{\mathcal{D}}\otimes 1_{\mathcal{D}})\\ &=\Lambda(\operatorname{id}_A\otimes 1_{\mathcal{D}}\otimes 1_{\mathcal{D}})^{-1}\Lambda(\operatorname{id}_A\otimes \operatorname{id}_{\mathcal{D}}\otimes \operatorname{id}_{\mathcal{D}})\Lambda(\operatorname{id}_A\otimes 1_{\mathcal{D}}\otimes 1_{\mathcal{D}})\\ &=\operatorname{id}_{\Lambda(A)}.\\[-34pt] \end{align*} $$
We now prove our classification theorems.
Theorem 5.2 Let G be a finite group and A be a unital Kirchberg algebra satisfying the UCT with
$A\cong A\otimes \mathbb {M}_{|G|^\infty }$
. If
$(\alpha ,u)$
,
$(\beta ,v)$
are anomalous actions of G on A with the Rokhlin property, then
$(\alpha ,u)\simeq _{K}(\beta ,v)$
if and only if
$o(\alpha ,u)=o(\beta ,v)$
and
$K_i(\alpha _g)=K_i(\beta _g)$
for all
$g\in G$
and
$i=0,1$
.
Proof We check that the hypothesis of Lemma 5.1 is satisfied. Let
$\mathcal {D}=\mathbb {M}_{|G|^\infty }$
,
$\Lambda $
be the functor given by the pointed
$K_0$
group direct sum the
$K_1$
group, i.e.,
$\Lambda (A)=((K_0(A),[1_A]),K_1(A))$
, and
$\mathfrak {R}$
the class of Rokhlin anomalous G-actions on unital Kirchberg algebras satisfying the UCT that absorb
$\mathbb {M}_{|G|^\infty }$
. That
$\Lambda $
is full on isomorphisms follows from [Reference Phillips39]. Condition (A1) follows from Proposition 4.5. For any
$\omega \in Z^3(G,\mathbb {T})$
, we have actions
$(\mathcal {D},s_G^{\omega },u^{\omega })$
as discussed in Section 3. That
$(\mathcal {D},s_G^{\overline {\omega }},u^{\overline {\omega }})\otimes (\mathcal {D},s_G^{\omega },u^\omega )\simeq (\mathcal {D},s_G,1)$
follows from [Reference Herman and Jones19, Theorem III.6] combined with [Reference Izumi22, Lemma 3.12] as the actions
$(\mathcal {D},s_G^{\omega },u^{\omega })$
have the Rokhlin property by Proposition 3.1. Therefore, (A2) is also satisfied. Finally, (A3’) is satisfied by Izumi’s classification result [Reference Izumi23, Theorem 4.2] and that every cocycle action with the Rokhlin property is a unitary perturbation of a group action [Reference Izumi23, Lemma 3.12].
Theorem 5.3 Let G be a finite group and A be a unital, simple, nuclear TAF-algebra in the UCT class such that
$A\cong A\otimes \mathbb {M}_{|G|^\infty }$
and
$(\alpha ,u),(\beta ,v)$
are anomalous actions on A with the Rokhlin property, then
$(\alpha ,u)\simeq _{K}(\beta ,v)$
if and only if
$o(\alpha ,u)=o(\beta ,v)$
and
$K_i(\alpha _g)=K_i(\beta _g)$
for all
$g\in G$
and
$i=0,1$
.
Proof We apply Lemma 5.1 with
$\mathcal {D}=\mathbb {M}_{|G|^\infty }$
,
$\mathfrak {R}$
the class of Rokhlin anomalous actions on
$\mathbb {M}_{|G|^\infty }$
-stable unital, simple, separable, nuclear TAF-algebras satisfying the UCT and
$\Lambda $
the functor consisting of the ordered, pointed
$K_0$
functor direct sum
$K_1$
. First,
$\Lambda $
is full on isomorphisms by [Reference Lin34]. (A1) holds by Proposition 4.5. (A2) holds for the same reason as in the proof of Theorem 5.2. Condition (A3’) follows from a combination of [Reference Izumi23, Theorem 4.3] and [Reference Izumi22, Lemma 3.12].
Similarly, one may classify anomalous actions with the Rokhlin property on the Razak–Jacelon algebra
$\mathcal {W}$
.
Theorem 5.4 Let G be a finite group and
$(\alpha ,u)$
,
$(\beta ,v)$
be anomalous G actions with the Rokhlin property on
$\mathcal {W}$
. Then
$(\alpha ,u)\simeq (\beta ,v)$
if and only if
$o(\alpha ,u)=o(\beta ,v)$
.
Proof We check the conditions of Lemma 5.1 with
$\mathcal {D}=\mathbb {M}_{|G|^\infty }$
,
$\mathfrak {R}$
the class of Rokhlin anomalous actions on
$\mathcal {W}$
and
$\Lambda $
the trivial functor. First, (A1) holds by Proposition 4.5. Moreover, (A2) holds as in the proof of Theorem 5.2. Finally, (A3) follows from [Reference Nawata35, Corollary 3.7] as every cocycle action of a finite group on
$\mathcal {W}$
is cocycle conjugate to a group action (this follows as
$\mathcal {W}\cong \mathcal {W}\otimes M_{|G|}$
and hence [Reference Gabe and Szabó15, Remark 1.5] applies).
In light of [Reference Carrión, Gabe, Schafhauser, Tikuisis and White5, Theorem B], it follows from [Reference Izumi22, Theorem 3.5] that all Rokhlin anomalous actions of G on classifiable
$\mathbb {M}_{|G|^\infty }$
-stable C
$^*$
-algebras are classified up to cocycle conjugacy by their induced action on the total invariant
$\underline {K}T_u$
(see [Reference Carrión, Gabe, Schafhauser, Tikuisis and White5, Section 3]) and their anomaly.
Corollary 5.5 Let G be a finite group. Let A be a unital, simple, separable, nuclear,
$\mathbb {M}_{|G|^\infty }$
-stable C
$^*$
-algebra satisfying the UCT and
$(\alpha ,u)$
,
$(\beta ,v)$
be anomalous G-actions with the Rokhlin property on A. Then
$(\alpha ,u)\simeq (\beta ,v)$
if and only if
$\underline {K}T_u(\alpha )\sim \underline {K}T_u(\beta )$
and
$o(\alpha ,u)=o(\beta ,v)$
.
Proof We apply Lemma 5.1 with
$\mathcal {D}=\mathbb {M}_{|G|^\infty }$
,
$\mathfrak {R}$
the class of Rokhlin anomalous actions on
$\mathbb {M}_{|G|^\infty }$
-stable unital, simple, separable, nuclear C
$^*$
-algebras satisfying the UCT and
$\Lambda =\underline {K}T_u$
. First,
$\Lambda $
is full on isomorphisms by [Reference Carrión, Gabe, Schafhauser, Tikuisis and White5, Theorem A]. (A1) holds by Proposition 4.5. (A2) holds as in the proof of Theorem 5.2. It remains to show (A3). By [Reference Izumi22, Lemma 3.12], it suffices to show that for any two Rokhlin G-actions
$(A,\alpha )$
and
$(B,\beta )$
such that
$\underline {K}T_u(\alpha )\sim \underline {K}T_u(\beta )$
then
$\alpha \simeq \beta $
. This has been shown for simple, unital AH-algebras in [Reference Gardella and Santiago16, Theorem 3.8]. With [Reference Carrión, Gabe, Schafhauser, Tikuisis and White5, Theorem B] in hand, this also follows for arbitrary unital, simple, separable, nuclear,
$\mathcal {Z}$
-stable C
$^*$
-algebras satisfying the UCT. Indeed, as
$\underline {K}T_u$
is full on isomorphisms, there exists an isomorphism
$\theta :A\rightarrow B$
such that
$\underline {K}T_u(\theta \alpha _g\theta ^{-1})=\underline {K}T_u(\beta _g)$
for all
$g\in G$
. Therefore, it follows from [Reference Carrión, Gabe, Schafhauser, Tikuisis and White5, Theorem B] that
$\theta \alpha _g\theta ^{-1}\approx _{a.u}\beta _g$
. Now, it follows immediately from [Reference Izumi22, Theorem 3.5] that
$\alpha \simeq \beta $
.
We illustrate another application of Lemma 5.1 to the classification of Rokhlin anomalous actions on a class of non-simple C
$^*$
-algebras. Precisely, we can classify Rokhlin anomalous actions on inductive limits of one-dimensional NCCW complexes with trivial
$K_1$
-groups as a consequence of the classification results of [Reference Gardella and Santiago16, Section 3.3.1].
Theorem 5.6 Let G be a finite group and A be a C
$^*$
-algebra that can be written as an inductive limit of one-dimensional NCCW complexes with trivial
$K_1$
groups satisfying
$A\cong A\otimes M_{|G|^\infty }$
. If
$(\alpha ,u)$
,
$(\beta ,v)$
are anomalous actions of G on A, then
$(\alpha ,u)\simeq (\beta ,v)$
through an automorphism that is approximately inner if and only if
$o(\alpha ,u)=o(\beta ,v)$
and
$\operatorname {Cu}^{\sim }(\alpha _g)=\operatorname {Cu}^{\sim }(\beta _g)$
for all
$g\in G$
.Footnote
8
Proof We apply Lemma 5.1 with
$\mathcal {D}=M_{|G|^\infty }$
,
$\mathfrak {R}$
the class of Rokhlin anomalous actions on
$M_{|G|^\infty }$
-stable C
$^*$
-algebras that can be written as an inductive limit of one-dimensional NCCW complexes with trivial
$K_1$
groups and
$\Lambda =\operatorname {Cu}^{\sim }$
. First,
$\Lambda $
is invariant under approximate unitary equivalence. Moreover, it is full on isomorphisms by [Reference Robert41, Theorem 1.0.1] (see also [Reference Robert41, Corollary 5.2.3]). Conditions (A1) and (A2) hold as in the proof of Theorem 5.2. Condition (A3’) holds as a consequence of [Reference Gardella and Santiago16, Theorem 3.6] (note also that
$M_{|G|}(A)\cong A$
so [Reference Gabe and Szabó15, Remark 1.5] applies). Now, it follows from Lemma 5.1 that any two Rokhlin anomalous actions
$(\alpha ,u)$
,
$(\beta ,v)$
of G on an inductive limit of one-dimensional NCCW complex satisfy
$(\alpha ,u)\simeq _{\operatorname {Cu}^{\sim }}(\beta ,v)$
. But any automorphism of an inductive limit of one-dimensional NCCW complexes with trivial
$K_1$
groups that is the identity under
$\operatorname {Cu}^{\sim }$
is approximately inner by [Reference Robert41, Theorem 1].
Remark 5.7 Note that, by [Reference Gardella and Santiago16, Theorem 5.2], the UHF-stability assumption in Theorem 5.6 is immediate for the following subclasses:
-
(i) unital C
$^*$
-algebras that can be written as inductive limits of one-dimensional NCCW-complexes; -
(ii) simple C
$^*$
-algebras with trivial
$K_0$
-groups that can be written as inductive limits of one-dimensional NCCW-complexes; -
(iii) C
$^*$
-algebras that can be written as inductive limits of punctured-tree algebras.
We have shown a classification of anomalous actions on some classes of simple C
$^*$
-algebras. Such a classification also implies a classification of G-kernels, we illustrate it by using Theorem 5.2, the same argument may also be used to rewrite the results of Theorem 5.4, Theorem 5.3, and Corollary 5.5. As in the case of group actions, we say two G-kernels
$\overline {\alpha }$
and
$\overline {\beta }$
on a C
$^*$
-algebra A are K trivially conjugate if there exists an automorphism
$\theta \in \mathrm {Aut}(A)$
with
$K_i(\theta )=\operatorname {id}_{K_i(A)}$
and
$\overline {\theta \alpha _g\theta ^{-1}}=\overline {\beta _g}$
for all
$g\in G$
.
Corollary 5.8 Let A be a unital Kirchberg algebra satisfying the UCT with
$A\cong A\otimes M_{|G|^\infty }$
and
$\overline {\alpha }$
,
$\overline {\beta }$
be G-kernels with the Rokhlin property on A. Then
$\overline {\alpha }$
and
$\overline {\beta }$
are K trivially conjugate if and only if
$\operatorname {ob}(\overline {\alpha })=\operatorname {ob}(\overline {\beta })$
and
$K_i(\overline {\alpha }_g)= K_i(\overline {\beta }_g)$
for all
$g\in G$
and
$i=0,1$
.
Proof The forward direction is clear. To show the reverse direction, pick lifts
$(\alpha ,u)$
of
$\overline {\alpha }$
and
$(\beta ,v)$
of
$\overline {\beta }$
such that
$o(\alpha ,u)=o(\beta ,v)$
. As
$(\alpha ,u)$
and
$(\beta ,v)$
satisfy the hypothesis of Theorem 5.2, it follows that
$(\alpha ,u)\simeq (\beta ,v)$
and so
$\overline {\alpha }$
and
$\overline {\beta }$
are conjugate.
6 Applications
We start this section by giving an alternative construction of a
$(G,\omega )$
action on the UHF algebra
$M_{|G|^\infty }$
which is visibly compatible with a Bratteli diagram of
$M_{|G|^\infty }$
. This action is an AF-action in the sense of [Reference Elliott and Su11] and [Reference Chen, Palomares and Jones6, Definition 4.8] (see also the discussion in [Reference Girón Pacheco17, Section 6.1]). The existence of an AF
$\omega $
-anomalous action on
$M_{|G|^\infty }$
follows from an adaptation of the Ocneanu compactness argument to the C
$^*$
-setting [Reference Ocneanu37]. We build it explicitly below. Before we do so, let us recall the definition of an AF anomalous action.
Definition 6.1 Let A be a unital AF-C
$^*$
-algebra and
$(\alpha ,u)$
be a
$(G,\omega )$
-action on A. We say
$(\alpha ,u)$
is an AF anomalous action if there exists an inductive limit
$(A_n,\varphi _n)$
consisting of finite-dimensional C
$^*$
-algebras
$A_n$
with unital connecting maps
$\varphi _n$
and
$(G,\omega )$
actions
$(\alpha _n,u_n)$
on
$A_n$
such that:
-
(1)
$\varphi _n\alpha _n=\alpha _{n+1}\varphi _n,\ \forall n\in \mathbb {N},$
-
(2)
$\varphi _n(u_n)=u_{n+1}, \forall n\in \mathbb {N},$
-
(3)
$(A,\alpha ,u)$
is cocycle conjugate to
$\varinjlim (A_n,\varphi _n,\alpha _n,u_n),$
where
$\varinjlim (A_n,\varphi _n,\alpha _n,u_n)$
is the C
$^*$
-algebra
$\varinjlim (A_n,\varphi _n)$
with the canonical anomalous action induced by the sequence
$(\alpha _n,u_n)$
(see [Reference Girón Pacheco17, Section 6.1] for more details).
Proposition 6.1 Let G be a finite group and
$\omega \in Z^3(G,\mathbb {T})$
, then there exists an AF
$\omega $
-anomalous G action with the Rokhlin property on
$M_{|G|^\infty }$
. We denote this action by
$\theta _G^{\omega }$
.
Proof In this proof, we will use the symbols
$g,h,k,x,y,x_i,y_i,s_i$
for
$i\in \mathbb {N}$
to denote elements of the group G. Let
$A_n=C(G)\otimes \bigotimes _{i=1}^{n-1}\mathcal {B}(l^2(G))$
for
$n\in \mathbb {N}$
, where by convention
$A_1=C(G)$
. For
$f\in C(G)$
, let
$M_f\in \mathcal {B}(l^2(G))$
be the multiplication operator by f. Consider the
$^*$
-homomorphisms
$\varphi _n:A_n\rightarrow A_{n+1}$
defined by
$\varphi _n(f\otimes T)=1\otimes M_f\otimes T$
for
$f\in C(G)$
and
$T\in \bigotimes _{i=1}^{n-1}\mathcal {B}(l^2(G))$
.
The inductive system
$(A_n,\varphi _n)$
has an inductive limit (we write the limit by A) which is known to be isomorphic to
$\mathbb {M}_{|G|^\infty }$
. Indeed, the Bratelli diagram of this AF-algebra is easily seen to be the complete bipartite graph on
$|G|$
-vertices, it is common knowledge that this coincides with the UHF-algebra of type
$|G|^{\infty }$
(see [Reference Davidson9, Example III.2.4] for the case
$|G|=2$
). We construct a
$(G,\omega )$
action on each finite-dimensional algebra
$A_n$
such that the actions commute with the inclusion maps
$\varphi _n$
. This will induce an AF
$\omega $
-anomalous G action on
$M_{|G|^\infty }$
by the universal property of the inductive limit (see [Reference Girón Pacheco17, Section 6.1]).
To be precise, we construct a family of maps
$\theta _n:G\rightarrow \mathrm {Aut}(A_n)$
and
$u_n:G\times G\rightarrow U(A_n) $
such that:
-
(1)
$\theta _n(g)\theta _n(h)=\mathrm {Ad}(u_n(g,h))\theta _n(gh)$
, -
(2)
$\omega _{g,h,k}=\theta _n(g)(u_n(h,k))u_n(g,hk)u_n(gh,k)^*u_n(g,h)^*$
, -
(3)
$\varphi _{n}(u_n(g,h))=u_{n+1}(g,h)$
, -
(4)
$\varphi _n \theta _n(g)=\theta _{n+1}(g)\varphi _n$
,
for all
$n\in \mathbb {N}$
. To build this, we will consider the group actions
$\theta _n':G\rightarrow \mathrm {Aut}(A_n)$
defined by
$\theta _n'(g)=\lambda _G(g)\otimes \bigotimes _{i=1}^{n-1} \mathrm {Ad}(\lambda _G)_g$
where
$\lambda _G$
is the left regular representation of G. Note that
$\varphi _n\theta _n'(g)=\theta _{n+1}'(g)\varphi _n$
. To take into account the anomaly, we will tweak
$\theta _n'$
by suitable diagonal operators
$d_n\in \mathrm {Aut}(A_n)$
and ensuring that (13) and (13) hold. To define
$d_n$
, we start by introducing some notation. Let
$\delta _k\in C(G)$
be the point mass at k, i.e.,
$$ \begin{align*} \delta_k(g)=\begin{cases} 1,\ \text{if}\ g=k,\\ 0,\ \text{otherwise}. \end{cases} \end{align*} $$
We now let
$$ \begin{align*} \theta_n(g)=d_n(g)\theta_n'(g) \end{align*} $$
with
$d_n(g)$
defined inductively
$$ \begin{align*} d_1(g)&=\operatorname{id}_{A_1},\\ d_2(g)(\delta_k\otimes e_{x_1,y_1})&=\omega_{x_1^{-1},g,g^{-1}k}\overline{\omega_{y_1^{-1},g,g^{-1}k}}(\delta_k\otimes e_{x_1,y_1}),\end{align*} $$
and
$$ \begin{align*} d_n(g)(\delta_k&\otimes e_{x_1,y_1}\otimes\dots\otimes e_{x_{n-1},y_{n-1}})\\ &=\omega_{x_{n-1}^{-1},g,g^{-1}x_{n-2}}\overline{\omega_{x_{n-3}^{-1},g,g^{-1}x_{n-2}}}\overline{\omega_{y_{n-1}^{-1},g,g^{-1}y_{n-2}}}\omega_{y_{n-3}^{-1},g,g^{-1}y_{n-2}}\\ & (d_{n-2}(g)(\delta_k\otimes e_{x_1,y_1}\dots\otimes e_{x_{n-3},y_{n-3}})\otimes e_{x_{n-2},y_{n-2}}\otimes e_{x_{n-1},y_{n-1}}) \end{align*} $$
for all
$n>2$
with the convention that
$x_{0}=y_{0}=k$
. As we have defined
$d_n(g)$
on a spanning set of
$A_n$
,
$d_n(g)$
extend to linear maps from
$A_n$
to itself. In fact, each
$d_n(g)$
is an endomorphism of
$A_n$
. First, it is clear that they preserve the
$^*$
-operation. To show the multiplicativity, it is sufficient to check on a spanning set. We show this by induction. For the case
$n=2$
, it is only nontrivial to check that
$$ \begin{align*}d_2(g)(\delta_k\otimes e_{x_1,y_1})d_2(g)(\delta_k\otimes e_{y_1,y_2})=d_2(g)(\delta_k\otimes e_{x_1,y_2}).\end{align*} $$
The left-hand side is given by
$$ \begin{align*} &d_2(g)(\delta_k\otimes e_{x_1,y_1})d_2(g)(\delta_k\otimes e_{y_1,y_2})\\ &=\omega_{x_1^{-1},g,g^{-1}k}\overline{\omega_{y_1^{-1},g,g^{-1}k}}\omega_{y_1^{-1},g,g^{-1}k}\overline{\omega_{y_2^{-1},g,g^{-1}k}}(\delta_k\otimes e_{x_1,y_2})\\ &=\omega_{x_1^{-1},g,g^{-1}k}\overline{\omega_{y_2^{-1},g,g^{-1}k}}(\delta_k\otimes e_{x_1,y_2}), \end{align*} $$
which coincides with the right-hand side. To show that
$d_n(g)$
is multiplicative, for
$n>2$
, it suffices to show that
$$ \begin{align*} d_n(g)(\delta_k\otimes &e_{x_1,y_1}\otimes \dots \otimes e_{x_{n-1},y_{n-1}})d_n(g)(\delta_k\otimes e_{y_1,s_1}\otimes....\otimes e_{y_{n-1},s_{n-1}})\\ &=d_n(g)(\delta_k\otimes e_{x_1,s_1}\otimes....\otimes e_{x_{n-1},s_{n-1}}). \end{align*} $$
This follows immediately from the induction hypothesis and a direct computation of the left-hand side (as in the case for
$n=2$
). Notice that each
$d_n(g)$
fixes elements of the form
$\delta _k\otimes e_{x_1,x_1}\otimes e_{x_2,x_2}\dots \otimes e_{x_{n-1},y_{n-1}}$
.
To construct a
$(G,\omega )$
action on the first stage
$A_1$
, we let
$u_1(g,h)(k)=\omega _{k^{-1},g,h}$
. That
$(\theta _1,u_1)$
defines a
$(G,\omega )$
action on
$C(G)$
is a straightforward computation (this is computed in [Reference Bouwknegt, Hannabuss and Mathai3, Section 4]). We proceed to extend this action on
$A_1$
to all of
$M_{|G|^\infty }$
through the inductive limit. Let
$u_n(g,h)=\varphi _{1,n}(u_1(g,h))$
and
$\theta _n(g)=d_n(g)\theta _n'(g)$
. For the remaining part of the proof, we check that
$(\theta _n,u_n)$
satisfy (13)–(13) for all
$n\in \mathbb {N}$
. We will repeatedly use the
$3$
-cocycle formula during the calculations, instead of commenting on this every time, we will instead color-code the parts of our equations to which we apply the
$3$
-cocycle formula.
We start by showing (13). First,
$$ \begin{align*} \theta_n(g)\theta_n(h)&=d_n(g)\theta_n'(g)d_n(h)\theta_n'(h)\\ &=d_n(g)\theta_n'(g)d_n(h)\theta_n'(g)^{-1}\theta_n'(gh)\\ &=d_n(g)[g\cdot d_n(h)]\theta_n'(gh), \end{align*} $$
denoting
$g\cdot d_n(h)=\theta _n'(g)d_n(h)\theta _n'(g)^{-1}$
. It is clear that (13) holds for all
$n\in \mathbb {N}$
if and only if
$d_n(g)[g\cdot d_n(h)]d_n(gh)^{-1}=\mathrm {Ad}(u_n(g,h))$
on
$A_n$
for all
$n\in \mathbb {N}$
. This holds trivially for
$n=1$
. For
$n=2$
, it follows from the
$3$
-cocycle formula that
We now proceed with an inductive argument for arbitrary n. We assume that (13) holds for
$n-2$
, preforming a similar computation to the case
$n=2$
:
For (13), it suffices to show that
$\varphi _nd_n(g)=d_{n+1}(g)\varphi _n$
. For
$n=1$
,
$$ \begin{align*} d_2(g)\varphi_1(\delta_k)&=\sum_{r\in G}d_2(g)(\delta_r\otimes e_{k,k})\\ &=(1\otimes e_{k,k})\\ &=\varphi_1d_1(g)(\delta_k) \end{align*} $$
as
$d_1$
is the identity map. The case
$n=2$
follows too as
$$ \begin{align*} d_3(g)\varphi_2(\delta_k\otimes e_{x,y})&=\sum_{r\in G}d_3(g)(\delta_r \otimes e_{k,k}\otimes e_{x,y})\\ &=(1\otimes e_{k,k}\otimes e_{x,y})\omega_{x^{-1},g,g^{-1}k}\overline{\omega_{y^{-1},g,g^{-1}k}}\\ &=\varphi_2d_2(g)(\delta_k\otimes e_{x,y}). \end{align*} $$
Assuming that the case
$n-2$
holds, we now argue, by induction,
$$ \begin{align*} &d_{n+1}(g)\varphi_n(\delta_k\otimes e_{x_1,y_1}\dots\otimes e_{x_{n-1},y_{n-1}})\\ &=d_{n+1}(g)(\varphi_{n-2}(\delta_k\otimes e_{x_1,y_1}\dots\otimes e_{x_{n-3},y_{n-3}})\otimes e_{x_{n-2},y_{n-2}}\otimes e_{x_{n-1},y_{n-1}})\\ &=(d_{n-1}(g)\varphi_{n-2}(\delta_k\otimes e_{x_1,y_1}\dots\otimes e_{x_{n-3},y_{n-3}})\otimes e_{x_{n-2},y_{n-2}}\otimes e_{x_{n-1},y_{n-1}})\\ &\omega_{x_{n-1}^{-1},g,g^{-1}x_{n-2}}\overline{\omega_{x_{n-3}^{-1},g,g^{-1}x_{n-2}}}\overline{\omega_{y_{n-1}^{-1},g,g^{-1}y_{n-2}}}\omega_{y_{n-3}^{-1},g,g^{-1}y_{n-2}}\\ &=(\varphi_{n-2}d_{n-2}(g)(\delta_k\otimes e_{x_1,y_1}\dots\otimes e_{x_{n-3},y_{n-3}})\otimes e_{x_{n-2},y_{n-2}}\otimes e_{x_{n-1},y_{n-1}})\\ &\omega_{x_{n-1}^{-1},g,g^{-1}x_{n-2}}\overline{\omega_{x_{n-3}^{-1},g,g^{-1}x_{n-2}}}\overline{\omega_{y_{n-1}^{-1},g,g^{-1}y_{n-2}}}\omega_{y_{n-3}^{-1},g,g^{-1}y_{n-2}}\\ &=\varphi_n d_{n}(g)(\delta_k\otimes e_{x_1,y_1}\dots\otimes e_{x_{n-1},y_{n-1}}). \end{align*} $$
Condition (13) is immediate. It remains to show that (13) holds for arbitrary n. This follows from (13) for the case
$n=1$
and from (13). For
$n\in \mathbb {N}$
,
$$ \begin{align*} &\theta_n(g)(u_n(h,k))u_n(g,hk)u_n(gh,k)^*u_n(g,h)^*\\ &=\theta_n(g)(\varphi_{1,n}(u_1(h,k)))\varphi_{1,n}(u_1(g,hk))\varphi_{1,n}(u_1(gh,k)^*)\varphi_{1,n}(u_1(g,h)^*)\\ &=\varphi_{1,n}(\theta_1(g)(u_1(h,k))u_1(gh,k)u_1(g,hk)^*u_1(g,h)^*)\\ &=\omega_{g,h,k}\varphi_{1,n}(1_{A_1})\\ &=\omega_{g,h,k}. \end{align*} $$
This completes the construction of the AF anomalous action
$\theta _G^{\omega }$
.
To show that
$\theta _G^\omega $
has the Rokhlin property, we construct a family of Rokhlin projections. The projections
$\delta _g\otimes \operatorname {id}_{\mathcal {B}(l^2(G))^{\otimes n-1}}\in Z(A_n)$
satisfy
$\theta _n(g)(\delta _h\otimes \operatorname {id}_{\mathcal {B}(l^2(G))^{\otimes n-1}})=\delta _{gh}\otimes \operatorname {id}_{\mathcal {B}(l^2(G))^{\otimes n-1}}$
and also
$\sum _{g\in G}\delta _g\otimes \operatorname {id}_{\mathcal {B}(l^2(G))^{\otimes n-1}}=\operatorname {id}_{A_n}$
. Therefore, the projections
$p_g\in A_{\infty }$
with the nth coordinate given by
$\varphi _{n,\infty }(\delta _g\otimes \operatorname {id}_{\mathcal {B}(l^2(G))^{\otimes n-1}})$
for
$g\in G$
satisfy the conditions of Definition 2.3.
Remark 6.2 In the case that
$\omega =1$
, the construction in Proposition 6.1 greatly simplifies. Indeed,
$d_n(g)$
is the identity automorphism and
$u_n(g,h)$
is the unit for all
$g,h\in G$
and
$n\in \mathbb {N}$
. Therefore,
$\theta _G^{1}$
restricts to the group action
$\theta _n=\lambda _G\otimes \bigotimes _{i=0}^{n-1}\mathrm {Ad}(\lambda _G)$
on each
$A_n$
with
$\lambda _G$
the left regular representation. This action coincides with the infinite tensor product action
$s_G$
(see Section 4). To see this, consider the inductive system
$(B_n,\phi _n)$
with
$B_{2n-1}=A_n$
,
$B_{2n}=\bigotimes _{i=0}^nB(l^2(G))$
and
$\phi _{2n-1}(f\otimes T)=M_f\otimes T$
,
$\phi _{2n}(S)=1\otimes S$
for all
$n\in N, f\otimes T\in A_n$
, and
$S\in B_{2n}$
. The even terms of the inductive system
$(B_{2n},\phi _{2n+1}\circ \phi _{2n})$
coincide with the inductive limit
$(\bigotimes _{i=1}^n B(l^2(G)),M\mapsto \ \operatorname {id}_{B(l^2(G))}\otimes M)$
. The odd terms
$(B_{2n-1},\phi _{2n}\circ \phi _{2n-1})$
coincide with the inductive system
$(A_n,\varphi _n)$
from the proof of Proposition 6.1. This allows to interpolate between
$(\bigotimes _{i=1}^n B(l^2(G)),M\mapsto \ \operatorname {id}_{B(l^2(G))}\otimes M)$
and
$(A_n,\varphi _n)$
. It is immediate that
$\theta _G$
and
$s_G$
are conjugate. Moreover, it follows from Theorem 5.3 that
$\theta _G^{\omega }$
is cocycle conjugate to
$s_G^{\omega }$
for any
$\omega \in Z^3(G,\mathbb {T})$
.
We end this paper by studying to what extent Rokhlin anomalous actions on AF-algebras are AF-actions and vice versa. To do this, we will require results of [Reference Chen, Palomares and Jones6]. In [Reference Chen, Palomares and Jones6], the authors associate an invariant with any AF-action F, of a fusion category
$\mathcal {C}$
, on an AF-algebra A. Vaguely, this invariant consists of the
$K_0$
-groups of all Q-system extensions of A by F and all natural maps between these extensions. The authors also show that any two AF-actions on AF-algebras A and B are equivalent if and only if their invariants are isomorphic. As observed in [Reference Chen, Palomares and Jones6, Section 5.1], if the acting category
$\mathcal {C}$
is torsion-free (see [Reference Arano and De Commer1, Definition 3.7]), the invariant of [Reference Chen, Palomares and Jones6] simplifies to just the module structure of
$K_0(A)$
under the action of the fusion ring of
$\mathcal {C}$
. We apply this when the acting category is
$\boldsymbol {\mathrm {Hilb}}(G,\omega )$
and the action is induced by an anomalous action
$(\alpha ,u)$
as explained in [Reference Evington and Girón Pacheco13, Proposition 5.6]. The fusion ring of
$\boldsymbol {\mathrm {Hilb}}(G,\omega )$
is
$\mathbb {Z}[G]$
, and the module structure of
$K_0(A)$
is given by
$K_0(\alpha _g)$
.
Corollary 6.3 Let G be a finite group and A a simple, unital AF-algebra such that
$A\cong A\otimes M_{|G|^\infty }$
. Let
$(\alpha ,u)$
be a
$(G,\omega )$
-action on A such that
$K_0(\alpha _g)=\operatorname {id}_A$
for all
$g\in G$
. If
$(\alpha ,u)$
has the Rokhlin property, then
$(\alpha ,u)$
is an AF-action. Moreover, if
$[\omega |_H]\neq 0$
for any subgroup
$H<G$
, then the converse holds.
Proof If
$(\alpha ,u)$
is a
$(G,\omega )$
-action with the Rokhlin property on an AF-algebra A, then by Theorem 5.3 it is cocycle conjugate to the AF
$\omega $
-anomalous G-action
$\operatorname {id}_A\otimes \ \theta _G^{\omega }$
on A. Therefore,
$(\alpha ,u)$
is AF.
We now consider the converse statement. An AF
$\omega $
-anomalous G action
$(\alpha ,u)$
induces an AF-action of the fusion category
$\boldsymbol {\mathrm {Hilb}}(G,\omega )$
in the sense of [Reference Chen, Palomares and Jones6] (to see how a
$(G,\omega )$
-action induces a
$\boldsymbol {\mathrm {Hilb}}(G,\omega )$
action, see [Reference Evington and Girón Pacheco13, Proposition 5.6], that this is AF is discussed [Reference Girón Pacheco17, Remark 6.1.7]). By the hypothesis on
$\omega $
, the fusion category
$\boldsymbol {\mathrm {Hilb}}(G,\omega )$
is torsion-free, so as
$K_0(\alpha _g)=\operatorname {id}_A$
and
$K_0(\operatorname {id}_A\otimes \ \theta _G^{\omega })=\operatorname {id}_A$
, then [Reference Chen, Palomares and Jones6, Theorem A] yields that the AF
$\omega $
-anomalous G actions induced by
$(\alpha ,u)$
and
$\operatorname {id}_A\otimes \ \theta _G^\omega $
are cocycle conjugate. So
$(\alpha ,u)$
has the Rokhlin property.
Remark 6.4 One may drop the hypothesis that
$A\cong A\otimes M_{|G|^\infty }$
in Corollary 6.3 if one instead assumes that the anomaly
$\omega $
of
$(\alpha ,u)$
is such that
$[\omega ]$
has order
$|G|$
. Indeed, it follows from [Reference Girón Pacheco17, Corollary 5.4.4] that in this case A will automatically absorb
$M_{|G|^\infty }$
. Also, note that under this assumption on
$[\omega ]$
, it is automatic that
$[\omega |_H]\neq 0$
for any subgroup
$H<G$
.
The behavior observed in the converse of Corollary 6.3 is quite different from the behavior of group actions. It was already observed in [Reference Fack and Maréchal14] that there exist AF-actions of
$\mathbb {Z}_2$
on
$M_{2^\infty }$
which do not have the Rokhlin property.
Acknowledgements
The author would like to thank Samuel Evington, Eusebio Gardella, André Henriques, Corey Jones, Ulrich Pennig, and Stuart White for comments and discussions that have been useful for this paper. An initial version of this work forms part of the author’s Ph.D. thesis [Reference Girón Pacheco17].
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