2. Preliminaries

2.1 Monotone complete C*-algebras

As made abundantly clear in the introduction, the injective envelopes I(A) and $I_G(A)$ (reviewed in §2.2) played a crucial role in all recent ideal-structure results for $A \rtimes_\lambda G$ . One of the most important facts about injective algebras that make them especially convenient to work with is that they are what is known as monotone complete. This means that every bounded increasing net of self-adjoint elements admits a supremum.

Every von Neumann algebra is monotone complete, and while the converse is false, such algebras will still in many ways behave as if they were von Neumann algebras. Quite important to us is the fact that polar decompositions still work inside monotone complete C*-algebras (for a more general version, we refer to [Reference BerberianBer72, §21]).

Proposition 2.1 ([Reference YenYen55, Lemma 2.1] and [Reference SasakiSas55, Lemma 4.2]). Let A be a monotone complete C*-algebra, and let $a \in A$ . There exists a unique projection $p\in A$ , called the right projection of a, such that $ap = a$ , and $ab = 0$ if and only if $pb = 0$ , for every $b\in A$ . This projection is denoted by $\textrm{RP}(a)$ . The left projection of a is defined similarly, and is denoted by $\textrm{LP}(a)$ . Moreover, there is a unique partial isometry $u \in A$ with the property that $a = u \left| a \right| $ and $u^*u = \textrm{RP}(a)$ .

It is important to recognize a subtle point, and that is that the left and right projections $\textrm{LP}(a)$ and $\textrm{RP}(a)$ were defined intrinsically in terms of the monotone complete C*-algebra A, and not in terms of any representation $A \subseteq B(H)$ . In the setting of von Neumann algebras $M \subseteq B(H)$ , the polar decomposition $x = u \left| x \right| $ of an element $x \in M$ is typically done with requiring that the support projection of u is the projection onto $\overline{\operatorname{ran}} \left| x \right| $ . In this setting, this projection lies in M and coincides with $\textrm{RP}(x)$ , but in the general monotone complete setting, there is no reason to expect them to coincide.

Also worth mentioning is the fact that, just like von Neumann algebras canonically admit a weak*-topology, monotone complete C*-algebras admit notions of order convergence, with the one of interest to us given by Hamana in [Reference HamanaHam82b, §1]. Note that Hamana defines a notion of convergence for nets, without necessarily showing that it arises out of a topology. While we will not actually need to use this notion of convergence most of the time, it is still worth keeping its existence in mind, as in particular it is a very necessary part of the definition of Hamana’s monotone complete crossed product, which we make heavy use of.

Many times, certain properties of a non-monotone complete C*-algebra can be studied more easily by embedding it into a certain monotone completion. This is done, for example, in [Reference HamanaHam81], where the regular monotone completion $\overline{A}$ of A is constructed. This object admits nice abstract properties that describe it uniquely, but is perhaps a bit difficult to get a concrete handle on. If we instead consider a crossed product C*-algebra $A \rtimes_\lambda G$ , then, under certain conditions, there is another monotone complete C*-algebra that it embeds into, and is far easier to explicitly write down. In [Reference HamanaHam82c, §3], the monotone complete crossed product is defined as follows.

Assume G is a discrete group acting on a unital C*-algebra $A \subseteq B(H)$ , and recall that a reduced crossed product $A \rtimes_\lambda G$ can be viewed as bounded operators acting on the Hilbert space $H \otimes \ell^2(G)$ . One may concretely view every operator $T \in B(H \otimes \ell^2(G))$ as a matrix $T = [T_{r,s}]_{r,s \in G}$ over B(H). With respect to this representation, every finitely supported element $\sum_{t \in G} a_t \lambda_t\in A\rtimes_\lambda G$ embeds as the matrix $[r^{-1} \cdot a_{rs^{-1}}]_{r,s \in G}$ . Using the fact that such symmetry is still present after taking limits, every element $a\in A\rtimes_\lambda G$ can be written as a formal sum $\sum_{t\in G}a_t\lambda_t$ , in a unique way. However, there is nothing stopping us from instead considering the following, perfectly valid, operator subsystem of $B(H \otimes \ell^2(G))$ :

\[M(A,G) := \bigg\{ \sum_{t \in G} a_t \lambda_t \text{ (formal sum)} \,\, \bigg| \,\, [r^{-1} \cdot a_{rs^{-1}}]_{r,s \in G}\in B(H \otimes \ell^2(G)) \bigg\} . \]

It is immediately clear that the coefficients $(a_t)_{t\in G}$ uniquely determine the elements of M(A,G). What is not immediately clear is why M(A,G) is, in any way, closed under multiplication. We summarize below results from [Reference HamanaHam82b, §6] and [Reference HamanaHam82c, §3] (see also [Reference HamanaHam85, §3]), which show that things mostly still behave how one would expect, as long as we specify the correct multiplication structure to use.

Proposition 2.2. Let G be a discrete group acting on a monotone complete C*-algebra A. Then M(A,G) is a C*-algebra, when inheriting the involution and Banach space structure from $B(H \otimes \ell^2(G))$ , but equipped with the new multiplication

\[ [ x_{r,s} ]_{r,s\in G} \cdot [ y_{r,s} ]_{r,s\in G} := \bigg[ O-\sum_{t \in G} x_{r,t} y_{t,s} \bigg]_{r,s\in G}, \]

where $[x_{r,s}]_{r,s \in G}$ and $[y_{r,s}]_{r,s \in G}$ are the matrix representations with respect to the Hilbert space $H \otimes \ell^2(G)$ , and $O-\sum_{t \in G} x_{r,t} y_{t,s}$ denotes the order limit in A of the finite sums. The multiplication of the formal sums is reflected in the following way:

\[ \bigg( \sum_{g \in G} a_g \lambda_g \bigg) \cdot \bigg( \sum_{g \in G} b_g \lambda_g \bigg) = \sum_{g \in G} \bigg( O-\sum_{t \in G} a_{t^{-1}} (t^{-1} \cdot b_{tg}) \bigg) \lambda_g. \]

Moreover, M(A,G) is monotone complete.

Remark 2.3. As remarked in [Reference HamanaHam85, §3], the above multiplication does not necessarily coincide with the usual multiplication in $B(H \otimes \ell^2(G))$ , due to the fact that we are taking an order-limit in A instead of, say, a strong limit in B(H).

Most of the time, we will not be multiplying arbitrary elements in M(A,G) together, and thus the reader should not worry too much about order-convergent sums. However, one fact that we absolutely will be making heavy use of is the fact that M(A,G) is itself monotone complete, whenever A is.

Remark 2.4. Just as for reduced crossed products, there is an equivariant faithful conditional expectation $E \colon M(A,G) \to A$ , extending the canonical conditional expectation on $A \rtimes_\lambda G$ . Every $x \in M(A,G)$ is given by the formal sum $x=\sum_{g\in G}E(x\lambda_g^*)\lambda_g$ .

Moving back to the more basic theory of monotone complete C*-algebras, they admit fairly canonical monotone complete subalgebras. It is an easy exercise to show that the fixed point set $A^G$ of a group G acting on a monotone complete C*-algebra A by *-automorphisms is itself monotone complete, and in particular, we obtain the following result.

Proposition 2.5. Let A be a monotone complete C*-algebra. Then the center Z(A), is automatically monotone complete.

We will often be interested in working more hands-on with the center of a monotone complete C*-algebra. It is a well-known result (we include a more modern citation) that, in the commutative setting, monotone complete C*-algebras are determined by a topological condition on their spectra.

Theorem 2.6 [Reference Saitô and Maitland WrightSW15, Theorem 2.3.7] Let X be a compact Hausdorff space. The C*-algebra C(X) is monotone complete if and only if X is extremally disconnected, i.e. the closure of any open set $U \subseteq X$ is in fact clopen.

2.2 Injective envelopes

The theory of injective envelopes of C*-algebras was introduced by Hamana in the 1970s and 1980s, originally in [Reference HamanaHam79a] and [Reference HamanaHam79b], along with an equivariant version in [Reference HamanaHam85]. When dealing with Hamana’s theory, the right category to work in is the category of G-operator systems. That is, the category where objects are operator systems with an action of a discrete group G by unital complete order isomorphisms (these automatically become *-isomorphisms whenever the operator system G is acting upon is a C*-algebra). The non-equivariant version can be obtained by letting $G = \{e\}$ . However, the objects we deal with in this paper will always end up being C*-algebras. Therefore, for convenience, we recall some of Hamana’s theory here in a way that requires only very basic theory of operator systems (see [Reference PaulsenPau03] for an extensive introduction).

Definition 2.7. Let S and T be operator systems. We say that $\phi\colon S\to T$ is a complete order isomorphism if $\phi$ is a unital completely positive linear isomorphism with completely positive inverse.

When two operators systems S and T are equipped with an action of a discrete group G by complete order isomorphisms, we say that S embeds equivariantly into T, and write $(S,G)\subseteq (T,G)$ , if there is a G-equivariant unital completely positive map $\phi: S\to T$ which is a complete order isomorphism onto its range. Such a map is called an equivariant embedding.

Theorem 2.8. Let G be a discrete group acting on a unital C*-algebra A. Then there exists a unique G-C*-algebra, denoted $I_G(A)$ , with the following properties.

1. (i) We have $(A,G)\subseteq (I_G(A),G)$ (inclusion of C*-algebras).

2. (ii) Let S and T be G-operator systems such that $(S,G)\subseteq (T,G)$ . Then every unital G-equivariant completely positive map $\phi\colon S\to I_G(A)$ extends to a unital G-equivariant completely positive map $\tilde{\phi}\colon T\to I_G(A)$ .

3. (iii) Let S be a G-operator system and assume that $\phi\colon I_G(A)\to S$ is a G-equivariant unital completely positive map which is restricted to an embedding on A. Then $\phi$ is an embedding.

When $G=\{e\}$ , one obtains the injective envelope of A, which is denoted by I(A).

Definition 2.9. Property (ii) expresses the fact that $I_G(A)$ is a G-injective object in the category of G-operator systems. Property (iii) is known as G-essentiality of the inclusion $A\subseteq I_G(A)$ . Moreover, the inclusion $A\subseteq I_G(A)$ is G-rigid. Namely, the identity map on $I_G(A)$ is the only G-equivariant unital completely positive map $\phi\colon I_G(A)\to I_G(A)$ which is restricted to the identity map on A (see [Reference HamanaHam85, Lemma 2.4]). If we set $G = \{e\}$ , we obtain the non-equivariant versions of these properties, and we simply call them injectivity, essentiality, and rigidity.

Very importantly, we may apply all of our theory of monotone complete C*-algebras to the theory of injective envelopes, as the following well-known proposition shows.

Proposition 2.10. An injective C*-algebra is automatically monotone complete, and in particular this applies to I(A) for any unital C*-algebra A. Moreover, a G-injective C*-algebra is automatically injective, and is therefore monotone complete as well. In particular, this applies to $I_G(A)$ for any unital C*-algebra A.

As remarked in [Reference HamanaHam85, Remark 2.3], $I_G(A)$ is always an injective C*-algebra. However, as I(A) is an essential extension of A, it is not hard to construct the following chain of inclusions, where the embeddings are operator system embeddings:

\[ A\subseteq I(A)\subseteq I_G(A). \]

The next proposition shows that we may in fact obtain such inclusions in a G-C*-algebra sense.

Proposition 2.11 ([Reference HamanaHam85, §3], [Reference HamanaHam79a, Corollary 4.3], and [Reference HamanaHam85, Lemma 6.2]). Let G be a discrete group acting on a unital C*-algebra A. The action of G on A extends uniquely to an action of G on I(A), and there is a G-equivariant injective *-homomorphism from I(A) to $I_G(A)$ restricting to the identity map on A. In other words, we may view

\[ (A,G) \subseteq (I(A),G) \subseteq (I_G(A),G). \]

These inclusions automatically restrict to inclusions on the center of each algebra. That is

\[ (Z(A),G) \subseteq (Z(I(A)),G) \subseteq (Z(I_G(A)),G). \]

A slightly surprising version of Proposition 2.11 for crossed products is the following theorem by Hamana, which will be instrumental in passing from $I_G(A)$ to I(A).

Theorem 2.12 [Reference HamanaHam85, Theorem 3.4]. Let G be a discrete group acting on a unital C*-algebra A. We have a *-homomorphic embedding $I_G(A) \rtimes_\lambda G \hookrightarrow I(A \rtimes_\lambda G)$ that is restricted to the identity on the copy of $A \rtimes_\lambda G$ in both algebras. Consequently, we may view

\[ A \rtimes_\lambda G \subseteq I(A) \rtimes_\lambda G \subseteq I_G(A) \rtimes_\lambda G \subseteq I(A \rtimes_\lambda G). \]

Keeping the above in mind, the following result is a very easy but very important observation, which follows from the uniqueness of the injective envelope. It will let us transfer properties between $A \rtimes_\lambda G$ , $I(A) \rtimes_\lambda G$ , and $I_G(A) \rtimes_\lambda G$ .

Proposition 2.13. Let A and B be unital C*-algebras such that $A \subseteq B \subseteq I(A)$ . Then there is a *-isomorphism $I(B) \cong I(A)$ which is restricted to the identity on B.

Hamana proves in [Reference HamanaHam81, Theorem 7.1] the equivalence between a C*-algebra B being prime, and the regular monotone completion $\overline{B}$ being a factor. This is also true when considering the injective envelope I(B) instead, which can be proven using the exact same proof, or the fact that $Z(\overline{B}) = Z(I(B))$ (see [Reference HamanaHam81, Theorem 6.3]). Together with Theorem 2.12 and Proposition 2.13, we obtain the following proposition.

Proposition 2.14. Let B be a unital C*-algebra. Then B is prime if and only if I(B) is a factor. In particular, for a discrete group G acting on a unital C*-algebra A, the following are equivalent:

1. (i) $A\rtimes_\lambda G$ is prime;

2. (ii) $I(A)\rtimes_\lambda G$ is prime;

3. (iii) $I_G(A)\rtimes_\lambda G$ is prime.

Although $I(A)\rtimes_\lambda G$ and $I_G(A)\rtimes_\lambda G$ embed into the monotone complete C*-algebra $I(A\rtimes_\lambda G)$ , which is also their common injective envelope, we may alternatively consider the embeddings of $I(A)\rtimes_\lambda G$ and $I_G(A)\rtimes_\lambda G$ into the monotone complete C*-algebras M(I(A),G) and $M(I_G(A),G)$ , respectively. This is usually more helpful, because it allows us to work with elements that still admit well-behaved series $x = \sum_{t \in G} x_t \lambda_t$ , with $x_t$ inside I(A) or $I_G(A)$ .

We will also need the following easy observation.

Proposition 2.15. Let G be a discrete group acting minimally on a unital C*-algebra A. The induced actions on the C*-algebras I(A), $I_G(A)$ , Z(A), Z(I(A)), and $Z(I_G(A))$ are all minimal as well.

Proof. Assume that $J \subseteq I_G(A)$ is a non-trivial G-invariant ideal of $I_G(A)$ . The quotient map $q \colon I_G(A) \to I_G(A) / J$ is necessarily injective on A by minimality, but is not injective on $I_G(A)$ . This contradicts G-essentiality of the inclusion $A\subseteq I_G(A)$ .

The exact same proof will work for I(A), since the inclusion $A\subseteq I(A)$ is essential and, in particular, G-essential.

Since A was arbitrary, if we show that (Z(A),G) is minimal, this will automatically hold for I(A) and $I_G(A)$ as well. To see this, identify Z(A) with C(X), for some compact space X, and assume that $U \subseteq X$ is a non-zero proper invariant open subset for the induced action on X.

Choose $x\in X\setminus U$ , and extend the evaluation map $\delta_x\colon Z(A)\to\mathbb{C}$ to a state $\rho\colon A\to \mathbb{C}$ . Then $\rho$ vanishes on the ideal generated by $C_0(U)$ in A, namely $\overline{\operatorname{span}} (C_0(U) \cdot A)$ , since Z(A) lies in the multiplicative domain (see [Reference Brown and OzawaBO08, Proposition 1.5.7] and [Reference Brown and OzawaBO08, Definition 1.5.8]) of $\rho$ . (As a side note, the ideal $\overline{\operatorname{span}} (C_0(U) \cdot A)$ is just $C_0(U) \cdot A$ by the Cohen-Hewitt factorization theorem). In any case, as $\rho$ is non-trivial, this ideal (which is clearly G-invariant and non-zero) cannot be all of A, contradicting minimality of the action.

We also briefly mentioned in the introduction of this paper that a large part of the difficulty in characterizing simplicity of $A \rtimes_\lambda G$ is the difficulty in passing from results on $I_G(A)$ to results on I(A). The following proposition, which we prove here for convenience, was observed by Hamana in [Reference HamanaHam85, Remark 3.8].

Proposition 2.16. Let G be a discrete amenable group and let M be an injective von Neumann algebra. Then M is automatically G-injective.

This is highly contingent on M being a von Neumann algebra, thus having the unit ball be compact under a nice topology (and, for example, being able to apply fixed point theorems for amenable groups acting on compact convex sets). In general, I(A) is only monotone complete, and thus the closest analog is a notion of order convergence. See, for example, the discussion near the start of [Reference HamanaHam82b, §1]. It is unlikely that the unit ball of I(A) is compact under any order topology, and thus this crucial piece of the puzzle is missing in the above proof. In other words, one expects $I_G(A) \neq I(A)$ . With that being said, this does not rule out the existence of some other deep reason for the equality $I_G(A) = I(A)$ to still hold, and such a result would indeed trivialize half of our work.

3. Pseudo-expectations and central elements

Recall that there is always a canonical conditional expectation $E\colon A \rtimes_\lambda G \to A$ , determined by mapping a finitely supported element $\sum_{t \in G} x_t \lambda_t\in A\rtimes_\lambda G$ to $x_e$ . All of the existing ideal-structure results for crossed products from the last few years pass through the machinery of what are known as pseudo-expectations, which is an analog obtained by expanding the codomain to the G-injective envelope $I_G(A)$ . We recall the precise definition below.

Pseudo-expectations were first introduced in the non-equivariant setting by Pitts in [Reference PittsPit17], and even in that setting were found to be helpful for understanding the ideal structure of $A\rtimes_\lambda G$ (see [Reference ZarikianZar19]). However, we will focus on equivariant pseudo-expectations, following the work of Kennedy and Schafhauser [Reference Kennedy and SchafhauserKS19].

Before we proceed further, the following was not explicitly mentioned in [Reference Kennedy and SchafhauserKS19], but is a very convenient result that is used implicitly. We prove it for convenience, and will certainly make use of it as well.

There are many reasons why pseudo-expectations are useful for studying simplicity of $A \rtimes_\lambda G$ . One piece of evidence is the following proposition, which directly characterizes simplicity, or more generally the intersection property, in terms of pseudo-expectations.

This requirement that every equivariant pseudo-expectation be faithful is perhaps a bit mysterious. If A is commutative, it follows from a combination of Proposition 3.3, [Reference KawabeKaw17, Theorem 3.4] and [Reference Kennedy and SchafhauserKS19, Theorem 6.4] that this is in fact equivalent to having a unique equivariant pseudo-expectation, namely the canonical conditional expectation. (See also [Reference Pitts and ZarikianPZ15, Theorem 4.6] for the non-equivariant setting.) In the noncommutative setting, it is not known whether they are equivalent. This is one of the obstructions to obtaining a nice characterization of simplicity in the noncommutative case, as precisely what makes a pseudo-expectation non-faithful is poorly understood, while in contrast we have a better idea of what arbitrary pseudo-expectations look like. Proposition 3.4 paints a picture on the latter (see also [Reference UrsuUrs21], where similar ideas are used by the second coauthor in the context of traces on crossed products). We would also like to remark that in an unpublished work of Zarikian it was proven that, in the non-equivariant setting, having all pseudo-expectations $A \rtimes_\lambda G\to I(A)$ faithful implies that there is a unique pseudo-expectation.

We now turn to proving each individual property of the coefficients. It is clear that $x_e = 1$ . Now let $y \in I_G(A)$ and $t\in G$ be given, and using multiplicative domain again, observe that

\[ x_t y = F(\lambda_t) y = F(\lambda_t y) = F((t \cdot y) \lambda_t) = (t \cdot y) F(\lambda_t) = (t \cdot y) x_t. \]

Furthermore, $s \cdot x_t = x_{sts^{-1}}$ is due to G-equivariance of F. Finally, the last positivity condition is due to the fact that

\begin{align*} F^{(n)} \left( \begin{bmatrix} \lambda_{s_1} \\ \vdots \\ \lambda_{s_n} \end{bmatrix} \begin{bmatrix} \lambda_{s_1} \\ \vdots \\ \lambda_{s_n} \end{bmatrix}^* \right) = \left[x_{s_i s_j^{-1}}\right]_{i,j = 1, \dots, n}.\end{align*}

The following highlights another, seemingly unrelated, area in which such coefficients $(x_t)_{t \in G}$ show up. This will be crucial for us when constructing non-trivial central elements.

Proof. Given $a\in A$ , it is not hard to verify that

\[ a\cdot x=\sum_{t\in G}a x_t^*\lambda_{t} \quad \text{and} \quad x\cdot a=\sum_{t\in G} x_t^*(t\cdot a)\lambda_{t}. \]

Using the fact that elements in $A\rtimes_\lambda G$ are uniquely determined by their coefficients, we see that x commutes with the copy of A inside $A\rtimes_\lambda G$ if and only if condition (i) of the proposition is satisfied.

Now given $g\in G$ , we also have

\[ \lambda_s\cdot x=\sum_{t\in G}(s \cdot x_{s^{-1}t}^*)\lambda_t \quad \text{and} \quad x\cdot \lambda_s = \sum_{t\in G} x_{ts^{-1}}^*\lambda_t. \]

Comparing coefficients, we see that x commutes with the set of unitaries $ \left\{ \lambda_s \,\,\middle| \,\, s \in G \right\} $ if and only if condition (ii) of the proposition is satisfied. The result then follows.

The following closely related proposition was proven by Hamana.

The equivalence between properties (ii) and (iii) will be the one which is relevant for us, and its proof is similar to the proof of Proposition 3.6. The equivalence to condition (i) will not be used later, but it is interesting nonetheless.

For convenience, we will sometimes want to go from a subset of pairs $(p_t,u_t)_{t\in C}$ indexed over a conjugacy class $C\subseteq G$ , and satisfying the conditions above, to a single pair (p,u), and vice versa.

To show that this map is surjective, we need to check that the elements $p_{srs^{-1}} := s \cdot p$ and $u_{srs^{-1}} := s \cdot u$ , for $s\in G$ , are well defined and satisfy the required properties. Assume that $s_1 r s_1^{-1} = s_2 r s_2^{-1}$ , for some $s_1,s_2\in G$ . This is equivalent to the requirement that $s_2^{-1} s_1$ belongs to $C_G(r)$ , and so $s_2^{-1} s_1 \cdot u = u$ , or, in other words, $s_1 \cdot u = s_2 \cdot u$ . Thus, the aforementioned elements give rise to well-defined pairs $\{(p_t,u_t)\}_{t\in C}$ . Observe that we indeed have $p_r = p$ and $u_r = u$ . It is straightforward to verify that each $p_t$ is a t-invariant central projection in A, that $u_t\in U(Ap_t)$ , and that $s\cdot u_t=u_{sts^{-1}}$ , for all $s\in G$ and all $t\in C$ . Finally, we check that the action of t is given by $\operatorname{Ad} u_t$ on the corner $Ap_t$ , for every $t\in C$ . Given $srs^{-1} \in C$ and $x \in A p_{srs^{-1}}$ we have that $s^{-1} x \in A p$ , and so

\begin{align*} srs^{-1} \cdot x = sr \cdot (s^{-1} \cdot x) = s \cdot (u (s^{-1} \cdot x) u^*) = (s \cdot u) x (s \cdot u)^* = u_{srs^{-1}} x u_{srs^{-1}}^*.\\[-30pt] \end{align*}

4. The intersection property for crossed products by FC-groups

The goal of this section is to prove the first half of Theorem A, which characterizes the intersection property for the crossed product $A \rtimes_\lambda G$ in terms of the dynamics of G on I(A). Specifically, the lack of the intersection property is equivalent to the existence of a triple (p,u,r), where:

• – $r \in G \setminus \{e\}$ ;

• – p is an r-invariant central projection in I(A);

• – u is a unitary in I(A)p such that r acts by $\operatorname{Ad} u$ on I(A)p;

• – u is $C_G(r)$ -invariant.

We would first like to give some brief intuition and sketch the proof of this theorem in the minimal setting. The setting of the intersection property relies on some technical results that make perhaps not as much intuitive sense without the appropriate context.

Assume that the action of G on A is minimal, and assume that $A \rtimes_\lambda G$ is not simple. Through the technology of pseudo-expectations (see §3), it is possible to write down a non-trivial element of $Z(I_G(A) \rtimes_\lambda G)$ , which in particular implies that $I_G(A) \rtimes_\lambda G$ is not prime, and, by Proposition 2.14, neither is $I(A) \rtimes_\lambda G$ . From here, letting I and J be two non-trivial ideals of $I(A) \rtimes_\lambda G$ with $I\cdot J = 0$ , we may take a supremum of the positive elements $x\in I$ with $ \left\| x \right\| <1$ inside of the monotone complete crossed product M(I(A),G). Denote this supremum by q. We will see in Proposition 4.1 that this element must commute with the copy of $I(A) \rtimes_\lambda G$ , and it cannot be a scalar (since it must still be orthogonal to J; see Proposition 4.3). By Proposition 3.7, the coefficients of $q= \sum_{r \in G} x_r \lambda_r\in M(I(A),G)$ satisfy certain properties that we are after, and moreover, there is at least one $r\in G\setminus\{e\}$ for which $x_r\neq 0$ (otherwise, q would be a non-trivial element of $Z(I(A))^G$ , and this is impossible due to minimality; see Proposition 2.15). Following the procedure carried out in Propositions 3.8 and 3.9, we deduce that the polar decomposition of $x_r$ inside I(A) gives rise to a triple (p,u,r) that has the desired properties.

In order to deal with the general case, we will at some point need to define a suitable variant of being prime (see Lemma 4.5), one that is appropriate for dealing with the intersection property when the action is not minimal.

We would now like to also motivate the converse direction of the Theorem, relative to the ideas of Kennedy and Schafhauser in [Reference Kennedy and SchafhauserKS19] (specifically for the case of amenable groups, which slightly simplifies their proof). The existence of a triple (p,u,r) as in the theorem is clearly stronger than requiring the action on I(A) (equivalently, on A) to be non-properly outer. However, in [Reference Kennedy and SchafhauserKS19, Theorem 9.5], it is instead assumed that the action on A is not properly outer and that the system (A,G) has vanishing obstruction. We now briefly explain what this means and how they use it in order to prove the existence of a non-faithful pseudo-expectation on $A\rtimes_\lambda G$ (which then violates the intersection property for (A,G), by Proposition 3.3). Let $u_t \in U(I(A)p_t)$ be the unitaries arising out of the decompositions $I(A) = I(A) p_t \oplus I(A) (1-p_t)$ into inner and properly outer parts, as described in Theorem 2.18. If it were possible to ‘untwist’ these unitaries in such a way so that the map $t \mapsto u_t$ behaves essentially like a group homomorphism (or, more precisely, if this map is a partial representation of G, see [Reference ExelExe17, Definition 9.1] or [Reference Kennedy and SchafhauserKS19, Definition 8.2]), then the proofs in the commutative case can be mimicked, and a new non-faithful equivariant pseudo-expectation $F'\colon A \rtimes_\lambda G \to I_G(A)$ can be defined by $F'(a \lambda_t) = a u_t$ . (One can also start with proper outerness on $I_G(A)$ , and untwisting the resulting unitaries from the decomposition of $I_G(A)$ instead. In the case of non-amenable groups, doing this becomes mandatory, due to coefficients from I(A) only defining pseudo-expectations on the universal crossed product $A \rtimes G$ , and not the reduced $A\rtimes_\lambda G$ . See the proof of [Reference Kennedy and SchafhauserKS19, Theorem 9.1].) This ability to untwist the unitaries (vanishing obstruction) trivially holds when A is commutative, but as mentioned before is a very strong property that frequently fails in the noncommutative case. Likewise, even if these unitaries $u_t$ arise out of the polar decomposition of elements $x_t \in I_G(A)$ satisfying the positive-definiteness condition mentioned in Proposition 3.4 (i.e. $[x_{st^{-1}}]_{s,t \in \mathcal{F}} \geq 0$ for all finite subsets $\mathcal{F} \subseteq G$ ), there is no guarantee that the matrices $[u_{st^{-1}}]_{s,t \in \mathcal{F}}$ are positive, a necessary condition for defining the aforementioned pseudo-expectation.

Let us now explain the different route taken in our proof: starting with a triple (p,u,r) as in the theorem, we obtain a set of pairs $\{(p_t,u_t)\}_{t\in C}$ as in Proposition 3.9, where C denotes the (finite) conjugacy class of r. These do not necessarily allow us to construct a non-faithful pseudo-expectation for $A\rtimes_\lambda G$ , as explained above and done in [Reference Kennedy and SchafhauserKS19]. However, a direct application of Proposition 3.6 shows that the element $\sum_{t\in C}u_t^*\lambda_t$ lies in the center of $I(A)\rtimes_\lambda G$ , but does not belong to the copy of I(A). Using this fact, we then prove that $I(A)\rtimes_\lambda G$ is not prime in a strong way (see Lemma 4.5 for the precise statement), which allows us to deduce that (I(A),G) does not have the intersection property, and so neither does (A,G) [Reference BryderBry22, Theorem 3.2].

We now start building up the technical results that are necessary in order to give a complete proof. The following proposition is very similar to [Reference HamanaHam82a, Lemma 1.1] (see also the proof of [Reference HamanaHam81, Theorem 7.1]), which is done in the context of having B below being either $\overline{A}$ , Hamana’s regular monotone completion, or I(A). For our purposes, we will usually consider $A \rtimes_\lambda G$ and M(A,G) as the monotone complete algebra, which is in general neither isomorphic to $\overline{A \rtimes_\lambda G}$ nor to $I(A \rtimes_\lambda G)$ (this is easy to see in the case of $A = \mathbb{C}$ and G abelian).

Proof. For convenience, we will denote $\sup^B I^+_1$ by p. First, we show that it commutes with all of A. To see this, note that any *-isomorphism $\alpha \colon B \to C$ between monotone complete C*-algebras will preserve suprema. In other words, if $(b_\lambda)_\lambda$ is a bounded increasing net of self-adjoint elements in B, then $(\alpha(b_\lambda))_\lambda$ is such a net in C, and

\[ \sup_\lambda \alpha(b_\lambda) = \alpha\!\left(\vphantom{{}^{1}_{2}}\right.\! \sup_\lambda b_\lambda\!{\kern-1pt}\left.\vphantom{{}^{1}_{2}}\right)\!. \]

In particular, if we let $u \in U(A)$ be any unitary element, then the *-automorphism $\alpha = \operatorname{Ad} u$ will also preserve the supremum of $I^+_1$ . However, $x \mapsto uxu^*$ is a bijection on this set. Thus,

\[ upu^* = u ( \sup{^B} I^+_1 ) u^* = \sup{^B} ( u I^+_1 u^* ) = \sup{^B} I^+_1 = p. \]

Since U(A) spans A, we conclude that $p \in A' \cap B$ .

To see that p is a projection, note that for any $x \in I^+_1$ , we have that $x^{1/2} \in I^+_1$ as well, and so $p \geq x^{1/2}$ . Since p and x commute, we also have $p^2\geq x$ . In other words, $p^2$ is an upper bound to $I^+_1$ . But since $p \leq 1$ , we have that $p^2 \leq p$ . Given that p was the least upper bound, it follows that $p^2 = p$ .

Finally, we show that p acts on I like a unit. Let $j\in I_+^1$ . Since $j\leq p$ , we have

\[(1-p)j(1-p)\leq (1-p)p(1-p)=0. \]

This implies $(1-p)j=0$ . Since $I^+_1$ spans I, we have $pj=j$ , for every $j\in I$ .

It is interesting to note that, in [Reference HamanaHam81, Theorem 7.1], Hamana does not take a supremum of $I^+_1$ directly, but rather a supremum of the left projections of the elements in this set. It is possible that he wished for the result to be a projection, but at least in this earlier paper, overlooked the fact that $\sup I^+_1$ is itself always a projection. In [Reference HamanaHam82a, Lemma 1.1], approximate units are used directly.

Lemma 4.2 [Reference HamanaHam81, Lemma 1.9 and Corollary 4.10].0020Let B be any C*-algebra, and let $\mathcal{F} \subseteq B$ be a bounded set of self-adjoint elements that admits a supremum. Then, for every $b\in B$ , $\sup ( b \mathcal{F} b^* )$ exists, and

\[b(\sup \mathcal{F}) b^* = \sup (b \mathcal{F} b^*).\]

Proof. Let $j \in J$ be any positive element. Using Lemma 4.2 and the fact that p is a projection that commutes with A (see Proposition 4.1), we have that

\[ jp = j^{1/2} p j^{1/2} = j^{1/2} (\sup{^B} I^+_1) j^{1/2} = \sup{^B} (j^{1/2} I^+_1 j^{1/2}) = 0. \]

Since positive elements span a C*-algebra, we have that $p\cdot J=0$ . Similarly, $q\cdot I=0$ .

Applying Lemma 4.2 once again, we have

\[ pqp = p (\sup{^B} J^+_1) p = \sup{^B} (p J^+_1 p) = 0, \]

and thus $pq=0$ .

One of Hamana’s goals with these sorts of results was to prove the equivalence between a C*-algebra B being prime, and the regular monotone completion $\overline{B}$ being a factor, although the same proof directly works with the injective envelope I(B) instead. One direction is clear from the above result. Orthogonal ideals I and J in B give orthogonal projections in I(B) that commute with B, and therefore lie in Z(I(B)) (see [Reference HamanaHam79a, Corollary 4.3]).

We will also need the following concrete computation that shows we may move back and forth between projections in I(B) and ideals in B, in a certain sense. This is essentially just part of [Reference HamanaHam82a, Lemma 1.3(i)], but Hamana phrases it in terms of injective envelopes of non-unital C*-algebras. To avoid referencing this theory just yet (and leave it contained to §6), we give a short re-proof here.

As mentioned in the beginning of this section, in the minimal setting, the proof of the main theorem of this section (Theorem 4.7) involves showing that $I_G(A) \rtimes_\lambda G$ is not prime and conclude that $I(A) \rtimes_\lambda G$ is not prime, since they share the same injective envelope. In the context of the intersection property, we need a stronger notion of ‘not prime’ that plays well relative to the subalgebras A, I(A), and $I_G(A)$ .

Note that $E(z_1)\in C(X)$ , since the canonical conditional expectation $E\colon I_G(A) \rtimes_\lambda G\to I_G(A)$ maps $Z(I_G(A) \rtimes_\lambda G)$ onto $Z(I_G(A))^G$ . Moreover, $Z(I_G(A))^G$ is a monotone complete C*-algebra, and so X is an extremally disconnected space (see the discussion at the end of §2.1). This implies that

\[ \overline{\textrm{supp} (E(z_1))} = \overline{ \left\{ x \in X \,\, \middle| \,\, E(z_1)(x) \neq 0 \right\} }\]

is a clopen subset of X. Let $p_1\in C(X)$ be the characteristic function of $\overline{\textrm{supp}(E(z_1))}$ (in other words, $p_1$ is the support projection of $E(z_1)$ ). Set $w_1:= (1-p_1)+z_1$ and $w_2:= z_2p_1$ . We will show the following properties:

1. (i) $w_1, w_2\in Z(I_G(A) \rtimes_\lambda G)$ are non-zero orthogonal positive contractions;

2. (ii) if $w_1a=0$ for some $a\in I_G(A)$ , then $a=0$ .

It is immediately clear that $w_1, w_2\in Z(I_G(A) \rtimes_\lambda G)$ are positive contractions and that $w_1\neq 0$ . We have to show that $w_2\neq 0$ . For this, we first observe that $p_1z_1=z_1$ . This is due to the fact that $E((1-p_1)z_1)=(1-p_1)E(z_1)=0$ , and since E is faithful, we get $(1-p_1)z_1=0$ . By definition, we have that $z_1(y_1)=1$ , and thus we must have that $p_1(y_1)=1$ . As $p_1\in C(X)$ and $q(y_1)=q(y_2)$ , we see that $p_1(y_1)=p_1(y_2)=1$ under the embedding $C(X) \subseteq C(Y)$ . Now, $w_2(y_2)=z_2(y_2)p_1(y_2)=1$ , so $w_2\neq 0$ . Orthogonality of $w_1$ and $w_2$ follows from orthogonality of $z_1$ and $z_2$ .

Finally, consider the set

\[I = \{ a\in I_G(A)\,\,|\,\,w_1a=0 \}.\]

Using that $w_1\in Z(I_G(A) \rtimes_\lambda G)$ , it is easy to check that I is a closed two-sided G-invariant ideal of $I_G(A)$ . Let $q=\sup^{I_G(A)}I_1^+$ . By Proposition 4.1, we know that q is a projection inside $Z(I_G(A))$ , which is also G-invariant since

\[ g \cdot q = g \cdot (\sup I_1^+) = \sup(g \cdot I_1^+) = \sup I_1^+ = q. \]

By Lemma 4.2, we have $w_1qw_1=w_1(\sup I_1^+)w_1=\sup(w_1I_1^+w_1)=0$ . Thus $w_1q=0$ . This in turn implies that $E(w_1q)=0$ , or, equivalently, $E(w_1)q=0$ , which means that $((1-p_1)+E(z_1))q=0$ inside $Z(I_G(A))^G = C(X)$ . However, by definition of $p_1$ , we see that the function $(1-p_1)+E(z_1)$ is non-vanishing on

\[\textrm{supp}(E(z_1)) \cup (X \setminus \overline{\textrm{supp}(E(z_1))}),\]

which is a dense subset of X. This forces q to be the zero element, and consequently, $I=0$ as required.

Next, recalling that $I(I_G(A)\rtimes_\lambda G)=I(A\rtimes_\lambda G)$ (see Theorem 2.12 and Proposition 2.13) and that $Z(B)\subseteq Z(I(B))$ for every unital C*-algebra B [Reference HamanaHam79a, Corollary 4.3], we have that $Z(I_G(A)\rtimes_\lambda G) \subseteq Z(I(A\rtimes_\lambda G))$ . This allows us to view $w_1,w_2$ as elements of $Z(I(A\rtimes_\lambda G))$ , which is again monotone complete. Denote by $r_1$ and $r_2$ the right (or equivalently left) projections of $w_1$ and $w_2$ , respectively, inside of this commutative and monotone complete C*-algebra (recall the definition of these projections from Proposition 2.1). The following properties hold:

1. (i) $r_1,r_2\in Z(I(A\rtimes_\lambda G))$ are non-zero orthogonal projections;

2. (ii) if $r_1a=0$ for some $a\in I(A)\subseteq I(A\rtimes_\lambda G)$ , then $a=0$ .

The first property follows from the fact that, because $w_1 w_2 = 0$ , then $w_1 r_2 = 0$ , and so $r_1 r_2 = 0$ as well. The second property is also not hard to see, as if $r_1 a = 0$ for some $a \in I(A)$ , then

\[ a^* w_1^* w_1 a \leq a^* r_1 a = 0, \]

and so $w_1 a = 0$ , which in turn we know implies $a = 0$ .

Now we may finally construct the ideals that we are after. Define

\begin{align*} J & := (r_1 \cdot I(A \rtimes_\lambda G)) \cap (I(A) \rtimes_\lambda G), \\ K &:= (r_2 \cdot I(A \rtimes_\lambda G)) \cap (I(A) \rtimes_\lambda G). \end{align*}

By essentiality of the inclusion $I(A)\rtimes_\lambda G\subseteq I(A\rtimes_\lambda G)$ , it follows that J and K are non-zero ideals inside $I(A) \rtimes_\lambda G$ , and they are clearly orthogonal as well. Let $a \in I(A)$ be given, and assume that $ja=0$ for every $j\in J$ . By Proposition 4.4, we have that $\sup^{I(A\rtimes_\lambda G)} J_1^+ = r_1$ , and so $a^*r_1a=0$ by Lemma 4.2, and so $r_1 a = 0$ . Due to condition (ii) above, we obtain $a=0$ . In summary, if $J\cdot a=0$ for some $a\in I(A)$ , then $a=0$ .

Remark 4.6. Consider the statement of Lemma 4.5, and assume that instead of requiring the inclusion $Z(I_G(A))^G \subseteq Z(I_G(A) \rtimes_\lambda G)$ to be proper, we required $Z(I(A))^G \subseteq Z(I(A) \rtimes_\lambda G)$ to be proper. This would automatically imply that the former inclusion is proper as well. Indeed, assume $x \in Z(I(A) \rtimes_\lambda G) \setminus Z(I(A))^G$ . It is clear that $x \notin Z(I_G(A))^G$ . Moreover, if we consider the inclusions

\[ A \rtimes_\lambda G \subseteq I(A) \rtimes_\lambda G \subseteq I_G(A) \rtimes_\lambda G \subseteq I(A \rtimes_\lambda G),\]

from Theorem 2.12, then we see that any element of $Z(I(A) \rtimes_\lambda G)$ in particular commutes with $A \rtimes_\lambda G$ , and therefore with all of $I(A \rtimes_\lambda G)$ by [Reference HamanaHam79a, Corollary 4.3]. Hence, it must lie in $Z(I_G(A) \rtimes_\lambda G)$ .

Letting $u_g=0$ for every $g\in G\setminus C$ , we check that the elements $(u_g)_{g\in G}$ satisfy the conditions stated in Proposition 3.6. Indeed, for every $g\in G\setminus C$ and every $y\in I(A)$ , it is clear that $u_gy=(g\cdot y)u_g$ . However, if $g\in C$ , we have that

\[ u_gy =u_gp_gy = (u_g(p_gy)u_g^*)u_g = (g \cdot (p_g y)) u_g = p_g (g\cdot y) u_g = (g\cdot y)u_g, \]

and so condition (i) of Proposition 3.6 is satisfied, and condition (ii) is immediate by construction. Therefore

\[z:=\sum\limits_{c\in C} u_c^*\lambda_c\in Z(I(A)\rtimes_\lambda G). \]

However, $z\notin Z(I(A))^G$ , since $u_r=u$ and $r\neq e$ . Hence, $Z(I(A))^G\subseteq Z(I(A)\rtimes_\lambda G)$ is a proper inclusion and we can apply Lemma 4.5 together with Remark 4.6 in order to obtain non-trivial orthogonal ideals $J,K\subseteq I(A)\rtimes_\lambda G$ with the property that $J^\perp\cap I(A)=\{0\}$ . In particular, K is a non-trivial ideal of $I(A)\rtimes_\lambda G$ which intersects I(A) trivially. This violates the intersection property for (I(A),G), and thus for (A,G) as well by [Reference BryderBry22, Theorem 3.2].

Conversely, assume that (A,G) does not have the intersection property, and let $F\colon A\rtimes_\lambda G\to I_G(A)$ be a non-canonical pseudo expectation, which exists by Proposition 3.3. The map F gives us non-trivial coefficients $(x_t)_{t \in G}$ in $I_G(A)$ by letting $x_t:= F(\lambda_t)$ , for every $t\in G$ . These satisfy the properties listed in Proposition 3.4. Let $r\in G \setminus \{e\}$ be such that $x_r \neq 0$ and denote by $C \subseteq G$ its conjugacy class. We obtain that $\sum_{t \in C} x_t^* \lambda_t$ lies in $Z(I_G(A) \rtimes_\lambda G)$ by Proposition 3.6, and so we conclude that the canonical inclusion

\[Z(I_G(A))^G\subseteq Z(I_G(A) \rtimes_\lambda G) \]

is proper.

Applying Lemma 4.5, we obtain non-trivial orthogonal ideals $J,K\subseteq I(A) \rtimes_\lambda G$ with the property that whenever $J\cdot a=0$ , for some $a\in I(A)$ , then $a=0$ . Denote $q := \sup^{M(I(A),G)}K_1^+$ . By Proposition 4.1, we have that q is a non-zero projection which commutes with the copy of $I(A) \rtimes_\lambda G \subseteq M(I(A),G)$ . Moreover, $J\cdot q=0$ , by Proposition 4.3, which implies that $q\notin I(A)$ .

Let $(q_t)_{t\in G}$ denote the coefficients in I(A) of $q = \sum_{t \in G} q_t \lambda_t \in M(I(A),G)$ , and let $z_t := q_t^*$ , for every $t\in G$ . By Proposition 3.7, we have that $z_t y =(t\cdot y) z_t$ , for all $t \in G$ and all $y \in I(A)$ , and moreover that $s\cdot z_t=z_{sts^{-1}}$ , for all $s,t\in G$ .

Given that $q \notin I(A)$ , at least one coefficient $z_r$ is non-zero for some $r \in G \setminus \{e\}$ . Letting $z_r=u \left| z_r \right| $ be the polar decomposition of this element, and $p=u^*u$ , we conclude by Proposition 3.8 and Proposition 3.9 that the triple (p,u,r) satisfies the desired properties.

Since, under minimality of the action of G on A, the intersection property is equivalent to simplicity of the associated crossed product, the theorem above also gives a characterization of simplicity of crossed products by FC-groups. Moreover, in general (not assuming the action is minimal), we almost immediately obtain a characterization for when crossed products by FC-groups are prime, and we state it separately below. This is in slight contrast to what is done in §5, which characterizes exactly when a crossed product $A \rtimes_\lambda G$ is prime regardless of what G is, but assuming the action is minimal.

Recall that a C*-algebra A equipped with a G-action is called G-prime if whenever $I,J\subseteq A$ are non-trivial G-invariant ideals in A, their product $I\cdot J$ is non-trivial.

Conversely, assume that $A\rtimes_\lambda G$ is not prime. We claim that the intersection property for (A,G) cannot hold. Indeed, since $A\rtimes_\lambda G$ is not prime, we can find two non-trivial orthogonal ideals $J,K\subseteq A\rtimes_\lambda G$ . As $J\cap A$ and $K\cap A$ are G-invariant orthogonal ideals in A, and A is G-prime, we have that at least one of them must be the zero ideal. In other words, at least one of J or K violates the intersection property for (A,G). By Theorem 4.7, we obtain a triple (p,u,r) which satisfies the desired properties.

5. Primality for minimal actions and simplicity for FC-hypercentral groups

This section is essentially an observation that most of the arguments in the previous section, which deal with characterizing when a reduced crossed product has the intersection property (a generalization of simplicity), actually also apply in the context of characterizing when a reduced crossed product is prime (Theorem C), for all minimal actions of any discrete groups. Afterwards, we show via an application of a result of Echterhoff from [Reference EchterhoffEch90a] that Theorem B, which characterizes simplicity of $A \rtimes_\lambda G$ in the case of FC-hypercentral G, immediately follows.

First, we require a seemingly arbitrary lemma with quite a convoluted proof, and it will have its usefulness become clear near the end of the proof of the main theorem. The proof of the lemma makes use of the basic properties of the Furstenberg boundary $\partial_F G$ . This was originally introduced by Furstenberg several decades ago in [Reference Furstenberg and MooreFur73] (see also [Reference FurstenbergFur63]), as a topological boundary to be used largely in the study of Lie groups. It was more recently studied in the C*-simplicity results [Reference Kalantar and KennedyKK17] and [Reference Breuillard, Kalantar, Kennedy and OzawaBKKO17], where in particular it was observed that for discrete groups, $I_G(\mathbb{C}) \cong C(\partial_F G)$ as G-C*-algebras. If G is a discrete group, the Furstenberg boundary $\partial_F G$ is the universal compact Hausdorff G-space with the following properties:

1. (i) it is minimal;

2. (ii) it is strongly proximal, in the sense that for any probability measure $\nu \in P(\partial_F G)$ , there is a net of group elements $(g_\lambda)_\lambda$ such that $\lim_\lambda g_\lambda \nu = \delta_x$ , for some $x \in X$ .

As noted in Proposition 2.10, $I_G(\mathbb{C})$ is monotone complete, and hence $\partial_F G$ is an extremally disconnected space (see, for example, [Reference Saitô and Maitland WrightSW15, Theorem 2.3.7]).

Before we proceed with the proof of the lemma, it might also interest the reader to realize that in the case of amenable groups, we are guaranteed the existence of an invariant state on any unital C*-algebra, and the proof of the lemma becomes extremely easy. The significant difficulty in the non-amenable case is that now, we are only guaranteed the existence of G-equivariant unital completely positive maps into $I_G(\mathbb{C})$ .

Lemma 5.1. Let G be a discrete group acting minimally on a compact Hausdorff space X, and let $f \in C(X)$ be a non-zero positive function. Consider the set

\[ \{ g \cdot f \,\, | \,\, g \in G \} \]

without counting repetition among the elements. If this set is infinite, then the sum of its elements cannot be uniformly bounded from above, in the sense that there cannot exist some scalar $k \geq 0$ such that every sum of finitely many elements is bounded by k from above.

For convenience, let $W \subseteq G$ be a choice of elements of G so that

\[ \{ g \cdot f \,\, | \,\, g \in W \} = \{ g \cdot f \,\, | \,\, g \in G \} \]

and $g \cdot f$ are distinct for distinct values of $g \in W$ . We will do a proof by contradiction, and start with the assumption that $\sum_{g \in W} g \cdot f$ is uniformly bounded.

Since $I_G(\mathbb{C})$ is G-injective and isomorphic to $C(\partial_F G)$ , we know that there is at least one G-equivariant unital completely positive map $\phi \colon C(X) \to C(\partial_F G)$ . We also know that the left-ideal

\[ \{ h \in C(X) \,\, | \,\, \phi(h^*h) = 0 \}\]

is a two-sided G-invariant ideal in C(X). Since 1 does not lie in this ideal, by minimality, this ideal must necessarily be zero. In other words, the map $\phi$ is faithful.

We know that $\phi(f) \neq 0$ , by faithfulness. We now divide the proof into two cases.

If it were the case that $\phi(f) \geq \delta$ for some $\delta > 0$ , then the sum $\sum_{g \in W} g \cdot f$ could not be bounded, as neither could $\sum_{g \in W} \phi(g \cdot f)$ .

Assume therefore that $\phi(f)$ is not bounded from below. From here, choose some fixed $\delta > 0$ so that $\phi(f)(y) > \delta$ for some $y \in \partial_F G$ . Let $U := \phi(f)^{-1}(\delta,\infty)$ . Given that $\partial_F G$ is extremally disconnected, we have that $\overline{U}$ is a clopen subset, and $\phi(f)(z) \geq \delta$ for every $z\in \overline{U}$ , while $\phi(f)(z) \leq \delta$ for every $z\in \partial_FG\setminus \overline{U}$ . Note that, by assumption of $\phi(f)$ not being bounded below by any strictly positive scalar, we have $\overline{U} \neq \partial_FG$ . For convenience, we will also let $p \in C(\partial_F G)$ be the non-trivial projection coming from this clopen subset, so that $\phi(f) \geq \delta \cdot p$ .

Another observation we make is that, if $g_1 \cdot p \neq g_2 \cdot p$ , this implies that $g_1 U \neq g_2 U$ , which are exactly the set of points on which $g_1 \cdot \phi(f)$ and $g_2 \cdot \phi(f)$ are strictly greater than $\delta$ , respectively. Consequently, it must be the case that $g_1 \cdot \phi(f) \neq g_2 \cdot \phi(f)$ . In other words, distinct translates of p give rise to distinct translates of $\phi(f)$ . Thus, if we show that the sum of the distinct translates of p is unbounded, then, since $\phi(f) \geq \delta \cdot p$ , this will automatically imply the same for the distinct translates of $\phi(f)$ .

Since G is countable, the Furstenberg boundary $\partial_F G$ is separable. This is a direct consequence of minimality of the space. Let $(y_n)_{n=1}^\infty$ be a dense subset of this space. Observe that there cannot exist any $g \in G$ with the property that $g \cdot y_n \in \overline{U}$ for all $n\in\mathbb{N}$ . Otherwise, we would have $y_n \in g^{-1} \overline{U}$ for all $n\in\mathbb{N}$ , implying $g^{-1} \overline{U} = \partial_F G$ by density. This is impossible, since we have already shown $\overline{U} \neq \partial_F G$ .

Thus, if we construct a probability measure $\omega \in P(\partial_F G)$ by $\omega = \sum_{n=1}^\infty ({1}/{2^n}) \delta_{y_n}$ (which we will also view as a state on $C(\partial_F G)$ ), the previous paragraph essentially tells us that $(g\cdot \omega)(p)$ is never exactly equal to 1 for any $g \in G$ . However, choosing any $z \in \overline{U}$ , we know that since $\partial_F G$ is a strongly proximal, there is a net $(g_\lambda)_\lambda$ in G such that $\lim_\lambda g_\lambda \cdot \omega = \delta_z$ . In other words, even though $(g\cdot \omega)(p) \neq 1$ for any $g\in G$ , we can still find infinitely many elements $g\in G$ for which $(g\cdot \omega)(p) = \omega(g^{-1} \cdot p)$ is arbitrarily close to 1 and, say, greater than $({1}/{2})$ . Thus, we have infinitely many $g\in G$ for which $g^{-1} p$ are distinct, and $\omega(g^{-1} \cdot p) \geq ({1}/{2})$ .

Consequently, the sum of the distinct translates of p cannot be bounded. By what was mentioned earlier, the same must be true for the corresponding distinct translates of $\phi(f)$ , and thus the distinct translates of $f \in C(X)$ .

The case of uncountable groups is not that hard to deduce from the countable case. Assume G is uncountable, and consider our starting function $f \in C(X)$ . Let $U \subseteq X$ be a nonempty open set such that $f(x) > 0$ for all $x \in U$ . As $\cup_{g\in G} g\cdot U$ is an open invariant set, it must cover X by minimality. Compactness then tells us that there is a finite collection $g_1 U, \dots, g_n U$ covering X. In terms of our function translates, this tells us that $g_1 \cdot f, \dots, g_n \cdot f$ have the property that for any $x \in X$ , there is some $i\in \{1,\ldots,n\}$ with $(g_i \cdot f)(x) > 0$ .

By assumption, we may also choose countably many distinct group elements $(k_i)_{i=1}^\infty$ such that the translates $k_i \cdot f$ are all distinct. Now let H be the necessarily countable subgroup of G generated by $g_1, \dots, g_n$ and $(k_i)_{i=1}^\infty$ . Although the restricted action of H on X is not necessarily minimal, we can pass to a minimal H-subsystem $Y\subseteq X$ by Zorn’s lemma. Consider the restriction map $\pi \colon C(X) \to C(Y)$ . The function $\pi(f)$ is non-zero, by construction.

If the set $ \{ h \cdot \pi(f) \,\, | \,\, h \in H \} $ is finite, then by the pigeonhole principle, because $ \{ h \cdot f \,\, | \,\, h \in H \} $ is infinite, there are infinitely many distinct translates $h \cdot f$ mapping to $h_0 \cdot \pi(f)$ , for some $h_0\in H$ . Thus, the sum of the distinct translates $h \cdot f$ mapping to this value is unbounded, as under the image of $\pi$ , the sum of infinitely many copies of $h_0 \cdot \pi(f)$ is certainly unbounded.

If the set $ \{ h \cdot \pi(f) \,\, | \,\, h \in H \} $ is infinite, then we may simply apply the lemma we have already proven in the countable setting to deduce that the sum of the infinitely many distinct translates $h \cdot \pi(f)$ in C(Y) is unbounded. Thus, the sum of the infinitely many distinct translates in the original space C(X), i.e. the sum of the distinct $h \cdot f$ , is certainly unbounded as well.

Proof. To see this, let $E : M(I(A),G) \to I(A)$ be the canonical conditional expectation, and consider

\[ E(zz^*) = O-\sum_{t \in G} z_t^* z_t, \]

so that the net of finite sums is order convergent to a concrete element in I(A). The precise details of order convergence are not important. What is important is that this in particular implies that

\[\sum_{t \in \mathcal{F}} z_t^* z_t\leq E(zz^*)\]

for every finite subset $\mathcal{F}\subseteq G$ (see [Reference HamanaHam82b, Lemma 1.2.(iv)]).

We recall from Proposition 3.7 that $z_t y = (t \cdot y) z_t$ , for all $y \in I(A)$ . Consequently,

\[ z_t^* z_t y = z_t^* (t \cdot y) x_t = y z_t^* z_t \]

for all $y \in I(A)$ , or, in other words, $z_t^*z_t$ lies in Z(I(A)). Proposition 3.7 also tells us that $s \cdot z_t = z_{sts^{-1}}$ , and so $s \cdot (z_t^*z_t) = z_{sts^{-1}}^*z_{sts^{-1}}$ for all $s,t\in G$ .

Let us restrict our attention to an infinite conjugacy class $C \subseteq G$ , and show that the coefficients $z_c$ must be zero for every $c\in C$ . Assume otherwise, so that one, hence all, of these coefficients are non-zero. If $ \{ z_c^*z_c \,\, | \,\, c \in C \} $ is a finite set, then by the pigeonhole principle, there are infinitely many $c \in C$ for which $z_c^*z_c$ are all the same value. Given that $z_c^*z_c \neq 0$ , this clearly contradicts the assumption that the sum $\sum_{t \in G} z_t^*z_t$ is uniformly bounded above. Hence, we are left to conclude that $ \{ z_c^*z_c \,\, | \,\, c \in C \} $ is an infinite set. Equivalently, if we choose a fixed $c_0 \in C$ , the set

\[\{g \cdot (z_{c_0}^*z_{c_0})\,\,|\,\,g \in G \}\]

has infinitely many distinct elements in Z(I(A)). By Lemma 5.1, these once again cannot have a bounded sum, as Z(I(A)) is minimal by Proposition 2.15. Therefore $z_c = 0$ .

Conversely, assume that $A \rtimes_\lambda G$ is not prime. Hence, neither is $I(A) \rtimes_\lambda G$ (by Proposition 2.14). Choose non-trivial orthogonal ideals $J,K \subseteq I(A) \rtimes_\lambda G$ , and let

\[q:= \sup {}^{M(I(A),G)}K_1^+.\]

By Propositions 4.1 and 4.3, we have that q is a non-zero projection which commutes with the copy of $I(A) \rtimes_\lambda G \subseteq M(I(A),G)$ , and satisfies $J\cdot q=0$ . Moreover, $q\notin I(A)$ , as otherwise q would be an element of $Z(I(A))^G$ . This is impossible, since the induced system (Z(I(A)),G) is minimal (see Proposition 2.15), and so $Z(I(A))^G=\mathbb{C}$ .

Letting $q = \sum_{t\in G}z_t^*\lambda_t$ as an element of M(I(A),G), we know that there exists $r\in G\setminus \{e\}$ so that $z_r\neq 0$ . By Lemma 5.2, r necessarily has finite conjugacy class. The proof now proceeds exactly as in Theorem 4.7. More precisely, if $z_r=u | z_r | $ is the polar decomposition of $z_r$ , with $p=u^*u$ , then the triple (p, u, r) has the desired properties.

Now we state the following immediate, and somewhat surprising, corollary to Theorem 5.3.

We remark that the hypotheses that the action is minimal and G is ICC are far from being necessary. For example, by [Reference Matui and RørdamMR15, Proposition 2.8], the action of $\mathbb{Z}$ on its one-point compactification gives rise to a prime crossed product.

Finally, we want to take the ideas from §4, and show that at least in the minimal setting, they can be generalized to the case of FC-hypercentral groups. After the first public release of this preprint, Siegfried Echterhoff kindly pointed out to us that Theorem 5.6 immediately follows from a combination of the main result of his paper [Reference EchterhoffEch90a] together with Theorem 5.3. Previously, this was done using different techniques inspired by [Reference Bédos and OmlandBO16], generalized to the context of pseudo-expectations of crossed products.

6. From the injective envelope to the original C*-algebra

The first dynamical characterizations of simplicity and primality that we obtain in our paper are initially written in terms of the dynamics of G on I(A), the injective envelope of A. While this gives the most elegant characterizations from a theory perspective, the injective envelope is still a somewhat mysterious object that is not that easy to describe concretely in many cases. Strictly speaking, it was also possible in all of our results to simply use Hamana’s regular monotone completion $\overline{A}$ , which is significantly smaller in many cases, but at the end of the day it suffers from the same problem of still being relatively difficult to write down concretely. It would be desirable to have results which relate back to the dynamics on the original C*-algebra A.

It was mentioned in the introduction that the notion of a properly outer automorphism plays an important role in the study of simplicity of crossed products, and indeed, our results are an equivariant version of this notion. Recall the classical, non-equivariant notion first, which was discussed in §2.3 (especially Theorem 2.21, which we state again for convenience). Let A be a unital C*-algebra, and let $\alpha \in \operatorname{Aut}(A)$ , with its unique extension to I(A) also denoted by $\alpha$ . Consider the following conditions:

1. (i) there is a non-zero $\alpha$ -invariant ideal $J \subseteq A$ such that $\Gamma_B(\alpha|_J) = \{e\}$ ;

2. (ii) there is a non-zero $\alpha$ -invariant central projection $p \in I(A)$ , and a unitary $u \in U(I(A)p)$ such that $\alpha$ acts by $\operatorname{Ad} u$ on I(A)p;

3. (iii) there is a non-zero $\alpha$ -invariant ideal $J \subseteq A$ and a unitary $u \in M(J)$ , such that $ \| \alpha|_J - (\operatorname{Ad} u)|_J \| < 2$ ;

4. (iv) given any $\varepsilon\in (0,2)$ , there exists a non-zero $\alpha$ -invariant ideal $J \subseteq A$ and a unitary $u \in M(J)$ such that $ \| \alpha|_J - (\operatorname{Ad} u)|_J \| < \varepsilon$ .

We have that (i) and (ii) are equivalent, and both are implied by (iii) and (iv). If A is separable, all four are equivalent.

We will ultimately require some notion of invariance when it comes to all of the pieces involved in the above theorem. In particular, our characterization of simplicity, or lack thereof, is a modified version of condition (ii), where we require the unitary u corresponding to some automorphism $\alpha_r$ (where $r \in \text{FC}(G) \setminus \{e\}$ ) to be $C_G(r)$ -invariant. Note that this is in contrast to the usual definition of proper outerness, where the automorphisms are considered individually without any regard to the rest of the group action.

Let us briefly investigate the feasibility and outcome of generalizing each characterization of proper outerness on the original C*-algebra A. First, consider (i). The Borchers spectrum is normally defined for actions of abelian groups (and for single automorphisms $\alpha$ , it secretly considers the corresponding $\mathbb{Z}$ -action). In the setting of abelian groups, [Reference HamanaHam85, Theorem 7.3] essentially gives us the exact result we are after, and makes use of the Borchers spectrum for the action of the entire group (as opposed to a single automorphism). Note, however, that the definition of the Borchers spectrum involves the dual group $\widehat{G}$ . This strongly hints to the fact that any invariant Borchers-type characterization that is meant to work in the non-abelian setting will involve the use of the non-abelian dual group. Such an item is borderline impossible to get a concrete handle on, in practice, for most infinite groups. Moreover, the definition of the Borchers spectrum also involves considering all possible hereditary C*-subalgebras, and these are already non-trivial enough in general as-is. Thus, while it could in theory be possible to obtain an appropriate generalization, we highly doubt it would be one that people would wish to use in practice.

We also mention that [Reference Olesen and PedersenOP82, Theorem 6.6] contains several other characterizations of proper outerness. However, the vast majority (with a couple of exceptions) also involve considering either all possible hereditary C*-subalgebras of A, or at least the invariant ones. Again, this would make for conditions that are perhaps mysterious and hard to check in practice, and so we choose to also skip those. It is worth noting though that all of these conditions are very likely still possible to generalize.

The obvious characterization remaining is Elliott’s characterization, i.e. condition (iii). While it requires separability of the underlying C*-algebra A to be a true characterization, most C*-algebras that people are interested in end up being separable anyways. Moreover, it is only necessary to consider the space of invariant ideals of A, as opposed to the set of all invariant hereditary subalgebras. Thus, in our opinion, it is the most worthwhile characterization to generalize.

We would like to first give an initial and incorrect guess as to how the generalization would proceed. One might guess that ‘invariant’ proper outerness of $\alpha_t$ would mean that for any $C_G(t)$ -invariant ideal $I \subseteq A$ , and any $C_G(t)$ -invariant unitary $u \in M(I)$ , we have $ \| \alpha|_I - (\operatorname{Ad} u)|_I \| = 2$ . However, this turns out to be just slightly too weak of a notion, as there is simply no reason to expect enough suitable invariant ideals $I \subseteq A$ and invariant unitaries $u \in M(I)$ for approximating the automorphism. As it turns out, the correct notion to use is approximately invariant ideals and unitaries, in the appropriate sense.

Before proceeding further, we remark that all of the theory of injective envelopes that we use was in the setting of unital C*-algebras. In [Reference HamanaHam82b, §6], Hamana defines the injective envelope of a non-unital C*-algebra as follows (and studies it further in [Reference HamanaHam82a]).

We also state the following result of Hamana, which gives a correspondence between ideals of a C*-algebra A and central projections in I(A). This, and other results, usually work in the setting of A being non-unital, but to avoid unnecessary subtleties, we will just stick with the unital setting unless necessary.

This now hints at how the proof of the aforementioned equivalences would work in general. Any automorphism $\alpha \in \operatorname{Aut}(A)$ that is ‘close enough’ to being inner on an $\alpha$ -invariant ideal $J \subseteq A$ will give something that is genuinely inner on I(J), which is a central corner of I(A).

We may also realize the multiplier algebra of a C*-algebra A as an idealizer of A inside its injective envelope I(A).

Proposition 6.3 [Reference HamanaHam82a, §1]. Assume A is a not necessarily unital C*-algebra, and denote by M(A) its multiplier algebra. We have

\[ M(A) = \{ x \in I(A) \,\, | \,\, xa \in A \text{ and } ax \in A \text{ for all } a \in A \} . \]

Recall that an ideal K is a C*-algebra A is essential if and only if whenever $K \cdot a = 0$ for some $a\in A$ (which is equivalent to $a \cdot K = 0$ ), then $a=0$ . As mentioned earlier, we will also be dealing with ideals that are ‘almost invariant’ in the appropriate sense. The following observation allows us to establish what this means rigorously.

For the second statement, assume that $p=\sup^{I(A)}J_1^+=\sup^{I(A)}K_1^+$ and let $k\in K$ be an element which is orthogonal to J. Then $kp=0$ by Proposition 4.2. However, by Proposition 4.1, p acts as the identity on K, and we conclude that $k=0$ , as desired.

The following two observations are well known, but we recall them nevertheless.

Before proceeding, we mention what our suitable replacement for having an ideal $J \subseteq A$ invariant on the nose. Such an ideal will be considered ‘almost invariant’ with respect to an action of a group H on A if, while we do not necessarily have $h \cdot J = J$ , we at least have that $J \cap h \cdot J$ is essential in both J and $h \cdot J$ .

We will also require a stronger version of Elliott’s proper outerness characterization than the one presented in Theorem 2.21. There, it can be observed that if $\alpha$ is an automorphism of a (not necessarily unital) separable C*-algebra A such that $\alpha$ extends to an inner automorphism on all of I(A) (such automorphisms are called quasi-inner), then given any $\varepsilon > 0$ , there is some ideal $J \subseteq A$ and unitary $u \in M(J)$ such that $ \| \alpha|_J - (\operatorname{Ad} u)|_J \| < \varepsilon$ . However, [Reference Olesen and PedersenOP82, Corollary 6.7] more or less observes that in the case of quasi-inner automorphisms, the ideal J can be required to be essential. Proposition 6.9 uses essentially the same argument, but we recall two lemmas before proving it.

Observe that $J := \overline{\operatorname{span}} J_\lambda$ is a new $\alpha$ -invariant ideal in A, which is isomorphic to $\oplus_\lambda J_\lambda$ by Lemma 6.8. We then know by Lemma 6.7 that its multiplier algebra is the $\ell^\infty$ -direct sum $\prod_\lambda M(J_\lambda)$ . In particular, we are free to set the individual coordinates to anything bounded with no other restriction, and so we have that $u := (u_\lambda)_{\lambda\in\Lambda}$ is an element of the multiplier algebra M(J) satisfying

\[\|\alpha|_J - (\operatorname{Ad} u)|_J \| \leq \varepsilon.\]

Maximality of the family $(J_\lambda,u_\lambda)_{\lambda\in\Lambda}$ gives us the final result we are after, namely that J above is essential. If it were not, then $J^{\perp}$ would be $\alpha$ -invariant and orthogonal to every $J_\lambda$ . Moreover, using the fact that $I(J^{\perp})$ is a corner of I(A) by Proposition 6.2, we have that $\alpha$ is inner on $I(J^{\perp})$ . Applying Theorem 2.21 to the unitization $(J^\perp)^+$ , there at least exists some non-trivial $\alpha$ -invariant ideal $L \subseteq (J^\perp)^+$ (and $L \cap J^+$ is a non-trivial $\alpha$ -invariant ideal of A), and some unitary $v \in M(L) \subseteq M(L \cap J^+)$ with the property that $ \| \alpha|_L - (\operatorname{Ad} v)|_L \| < \varepsilon$ . The pair $(L \cap J^+,v)$ could then be added to our maximal family from before, contradicting the fact that it is actually maximal.

Unlike the proof of the equivalence between Elliott’s characterization of proper outerness (condition (iii) in Theorem 2.21) and the injective envelope version (condition (ii) in Theorem 2.21), our approach will completely avoid the theory of derivations, and instead take a different path (one which genuinely appears to be necessary for the equivalence between the equivariant notions of proper outerness). Essentially, what was shown in the proof of Theorem 2.21 is that if $ \| \beta - \operatorname{id} \| < 2$ on an ideal $J\subseteq A$ for some automorphism $\beta\in\operatorname{Aut}(J)$ , then we have that $\beta$ extends to an inner automorphism on I(J). What we need is some control over the unitary itself, in I(J). Intuitively, if $\beta$ is ‘close’ to the identity, then the resulting unitary should be ‘close’ to 1. In the context of von Neumann algebras at least, this intuition is indeed true, and the following lemma can be found in an important paper of Kadison and Ringrose.

Lemma 6.10 [Reference Kadison and RingroseKR67, Lemma 5]. Let $\alpha \in \operatorname{Aut}(M)$ be an automorphism of a von Neumann algebra M such that $ \| \alpha - \operatorname{id} \| < 2$ . Then there is a unitary $u \in M$ such that $\alpha = \operatorname{Ad} u$ , and moreover u has the following restriction on its spectrum:

\[ \sigma(u) \subseteq \bigl\{ z \in \mathbb{T} \,\, \left|\right. \,\, \operatorname{Re} z \geq \tfrac{1}{2} \sqrt{4 - \| \alpha - \operatorname{id} \| ^2} \bigr\}. \]

Note that we indeed have that if $ \| \alpha - \operatorname{id} \| $ gets closer to zero, then $ \| u - 1 \| $ indeed gets closer to zero as well. We suspect that the above lemma might work for general monotone complete C*-algebras. However, we work around this in a sneaky manner and can derive the result on just injective envelopes quite easily. For the proof, we will need the following lemma. We would also like to apologize to Zarikian, as when we had originally released the first preprint of our paper, we did not notice that this same exact result could be found in one of his papers.

Lemma 6.12. Let $\alpha \in \operatorname{Aut}(A)$ be an automorphism of a not necessarily unital C*-algebra A, such that $ \| \alpha - \operatorname{id} \| < 2$ . Then the unique extension of $\alpha$ to I(A) (obtained by first extending canonically to $A^+$ ) is inner and of the form $\operatorname{Ad} u$ , where u is a unitary in I(A) which can be required to satisfy the following restriction on its spectrum:

\[\sigma(u) \subseteq \bigl\{z \in \mathbb{T} \,\,\left|\right.\,\, \operatorname{Re} z \geq \tfrac{1}{2} \sqrt{4 - \| \alpha - \operatorname{id} \| ^2} \bigr\}.\]

In any case, let M and $\widetilde{\alpha} \in \operatorname{Aut}(M)$ be as above, and observe that $ \| \widetilde{\alpha} - \operatorname{id}_M \| = \| \alpha - \operatorname{id}_A \| $ , via a straightforward application of weak* continuity of $\widetilde{\alpha}$ and Kaplansky’s density theorem. Let $u \in M$ be a unitary obtained from Lemma 6.10. In other words, $\widetilde{\alpha} = \operatorname{Ad} u$ , and

\[ \sigma(u) \subseteq \bigl\{ z \in \mathbb{T} \,\, \left|\right. \,\, \operatorname{Re} z \geq \tfrac{1}{2} \sqrt{4 - \| \alpha - \operatorname{id} \| ^2} \bigr\} . \]

For convenience, denote the bound $({1}/{2}) \sqrt{4 - \| \alpha - \operatorname{id} \| ^2}$ by r. We have $\operatorname{Re} \sigma(u) \subseteq [r,1]$ , and so by the continuous functional calculus, we genuinely have $\operatorname{Re} u \geq r$ , where $\operatorname{Re} u = ({1}/{2})(u + u^*)$ .

We proceed to construct a similar element in I(A) using the only method we know. We know that $A^+ \subseteq M$ , and by injectivity, there is a unital and completely positive map $E \colon M \to I(A)$ such that it is the identity map on $A^+$ . Observe that, because $\operatorname{Re} u \geq r$ with $r > 0$ , we have that

\[ \operatorname{Re} E(u) = E(\operatorname{Re} u) \geq E(r) = r. \]

In particular, we have that E(u) is non-zero. Moreover, observe that for any $b \in A^+$ (which lies in the multiplicative domain of E), we have

\[ E(u) b = E(ub) = E(\alpha(b) u) = \alpha(b) E(u). \]

By Proposition 6.11 this automatically implies that $E(u) z = \alpha(z) E(u)$ for all $z \in I(A)$ .

For convenience, write $y = E(u)$ . We have no reason to expect that y is in any way a unitary element. However, this is not too hard to patch up. Let $y = v | y | $ be the polar decomposition of y in I(A), with $p = v^*v$ being the support projection of v. Recall from Theorem 2.19 that:

• – p is an $\alpha$ -invariant central projection in I(A);

• – v is a unitary in I(A)p, and on this corner, we in fact have $\alpha|_{I(A)p} = \operatorname{Ad} v$ .

It was also shown in the proof of Theorem 2.19 that $y^*y$ (and therefore also $ | y | $ ) will always lie in the center of I(A). Given that v is a unitary on its respective corner, it follows that $C^*(1,v, | y | )$ is a commutative C*-algebra of the form C(X).

Writing $y = v | y | $ inside of C(X), and using that $\operatorname{Re} y \geq r$ for $r > 0$ , we note that y is invertible in C(X). Consequently, so is $ | y | $ , and it follows that the support projection p of v was in fact 1. In other words, v is a unitary in I(A) implementing $\alpha$ . Finally, given any $x \in X$ , we have $\operatorname{Re} y(x) \geq r$ , and $0 < | y | (x) \leq 1$ (which follows from $ \| y \| = \| E(u) \| \leq 1$ ). Thus, $v(x) = {y(x)}/({ | y | (x)})$ also satisfies $\operatorname{Re} v(x) \geq r$ for all $x\in X$ . As the spectrum of v as an element of C(X) is given by $ \{ v(x) \,\, | \,\, x \in X \} $ , and coincides with its spectrum as an element of A, this finishes the proof.

Clearly, if u is a unitary in any unital C*-algebra with $\operatorname{Re} \sigma(u) \subseteq [r,1]$ for r close to 1, then it is the case that u is close to 1 as well by the continuous functional calculus. The following lemma establishes the precise norm estimate.

Lemma 6.13. Let $k \in [0,2]$ , and let $u \in U(A)$ be a unitary in a unital C*-algebra A with $\operatorname{Re} \sigma(u) \subseteq [({1}/{2}) \sqrt{4 - k^2},1]$ . Then we have

\[ \| u - 1 \| \leq \sqrt{2 - \sqrt{4-k^2}}. \]

Proof. This is essentially an exercise in Euclidean geometry. Consider Figure 1. It depicts the unit circle $\mathbb{T}$ , and the spectrum of u lies on the arc of the circle that is to the right of the dotted line. Some unknown side lengths are labeled as x, y, and z. It is clear that $x = 1 - ({1}/{2}) \sqrt{4 - k^2}$ , and by the Pythagorean theorem that $y = ({1}/{2}) k$ . One more application of the Pythagorean theorem gives us that

\[ z = \sqrt{2 - \sqrt{4 - k^2}}. \]

The value of z represents the maximum distance of any given point in $\sigma(u)$ from 1, and so by the continuous functional calculus, it must be the case that $ \| u - 1 \| \leq z$ , our desired result.

Figure 1. Estimate of $ \| u - 1 \| $ based on estimate of $\operatorname{Re} \sigma(u)$ .

A similar and even easier estimate characterizes when $\sigma(u)$ has a positive lower bound as well.

Proof. Again, we perform basic Euclidean geometry. In Figure 2, we again need $\sigma(u)$ to lie strictly to the right of the dotted line, which is equivalent to any point in the spectrum being distance strictly less than distance $\sqrt{2}$ away from 1. By continuous functional calculus, this is equivalent to $ \| u-1 \| < \sqrt{2}$ .

Figure 2. Estimate of $ \| u-1 \| $ based on $\operatorname{Re} \sigma(u) > 0$ and vice versa.

Now we are ready to state the main result.