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Separability idempotents in C*-algebras

In this paper, we study the notion of a separability idempotent in the C*-algebra framework. This is analogous to the notion in the purely algebraic setting, typically considered in the case of (finite-dimensional) algebras with identity, then later also considered in the multiplier algebra framework by the second-named author. The current work was motivated by the appearance of such objects in the authors' ongoing work on locally compact quantum groupoids.


Introduction.
Consider a groupoid G over the set of units G (0) , together with the maps s : G → G (0) and t : G → G (0) (the "source map" and the "target map"). This means that there is a set of "composable pairs" G (2) = (p, q) ∈ G×G : s(p) = t(q) , on which the product pq in G is defined. This product is assumed to be associative, in an appropriate sense. The set of units, G (0) , may be naturally regarded as a subset of G. There exists also the inverse map p → p −1 (so that (p −1 ) −1 = p), for which we have s(p −1 ) = t(p), t(p −1 ) = s(p), and satisfying some natural conditions. For a more detailed discussion on the definition and the basic theory of groupoids, refer to [5], [9]. The groupoid notion can be further extended to incorporate locally compact topology, which is the notion of a locally compact groupoid. For this, refer to [20], [19].
Suppose G is a groupoid and consider A = K(G), the set of all complexvalued functions on G having finite support. For the time being, let us disregard any topology on G. Under the pointwise multiplication, A becomes a commutative algebra.
In particular, if G is a finite groupoid, the algebra A becomes unital (1 ∈ A). In that case, it is known that A can be given a structure of a weak Hopf algebra, together with the map ∆ from A into A ⊗ A (algebraic tensor product), defined by The ∆ map is referred to as the "comultiplication" (or "coproduct"). A weak Hopf algebra is in general noncommutative, but includes the above example as a fundamental case. For more discussion on weak Hopf algebras, refer to [2], [3]. In a sense, a weak Hopf algebra is a finite quantum groupoid (see also [29], [18], [23]). Recently, at the purely algebraic level, this notion has been further generalized to include the case of non-finite groupoids (so the algebra A is non-unital), which is the notion of a weak multiplier Hopf algebra developed by one of us (Van Daele) and Wang. Refer to [26], [28]. If (A, ∆) is a weak multiplier Hopf algebra, there exists a certain canonical idempotent element E, playing the role of "∆(1)". Among the main properties of E is the fact that it is a separability idempotent. Namely, there exist algebras B and C such that E is an idempotent element contained in the multiplier algebra M (B ⊗ C), and it satisfies some number of conditions. The case that is relevant to us is when A is a * -algebra and is "regular" (see Section 4 of [28]). As a consequence, it turns out that there exist bijective anti-automorphisms S B : B → C and S C : C → B such that In the special (commutative) case of the weak multiplier Hopf algebra A = K(G) for a groupoid G, let B be the subalgebra of M (A) given by the pull-back of the algebra K(G (0) ) via the source map s : G → G (0) , and C ⊆ M (A) the pull-back of the algebra K(G (0) ) via the target map t : G → G (0) . Then we will have E ∈ M (B ⊗ C). The maps S B , S C would come from the antipode map of the weak multiplier Hopf algebra.
One of the main reasons for studying separability idempotents lies in the fact that they play an important role in the theory of weak multiplier Hopf algebras. In a sense, any separability idempotent arises from a weak multiplier Hopf algebra, and having an appropriate separability idempotent element can guide us to construct an example of a weak multiplier Hopf algebra: See Proposition 3.2 of [27]. See [24] (refer to both versions v1 and v2, as they are substantially different), where a systematic discussion is given on separability idempotents in the setting of multiplier algebras, together with some examples. Separability idempotents also arise naturally from the theory of discrete quantum groups (See Section 3 of [24]).
Let us now turn our attention to the case when the groupoid G is equipped with a compatible locally compact topology. A natural question to ask is whether we can formulate a C * -algebraic theory similar to that of a weak multiplier Hopf algebra, together with a suitable separability idempotent element, keeping all the topological aspects.
The aim of the current paper is to define, develop, and clarify the notion of a separability idempotent in the setting of C * -algebras. While the setting may be more general, this makes things a little restrictive as well, because we need the element to be compatible with the C * -structures on B and C, unlike in the purely algebraic case. On the other hand, not every property from the purely algebraic case will carry over (for instance, the maps S B and S C will only have to be densely defined). However, it is worth noting that we are still able to formulate below a reasonable notion.
Such separability idempotents turn out to be quite useful in developing a C * -algebraic counterpart to the weak multiplier Hopf algebra theory, which would provide us with a (sub)class of locally compact quantum groupoids. This is discussed in the authors' upcoming work [12], [10], [11].
In the purely algebraic theory of separability idempotents (see [24]), the defining conditions imply the existence of two distinguished linear functionals, denoted ϕ C and ϕ B , on the algebras C and B respectively. In our formulation below (in the C * -algebra setting), we assume instead the existence of two faithful KMS weights µ (on C) and ν (on B), from which the other properties follow, including the densely defined anti-homomorphisms. This approach is somewhat similar in philosophy to the general theory of locally compact quantum groups as given in [15], [16], [25].
We will see that the separability idempotent E ∈ M (B ⊗C) is more or less determined by the C * -algebra B and the weight ν, where ⊗ is now the C *algebra (spatial) tensor product. Meanwhile, the existence of E means that the pair (B, ν) cannot be arbitrarily chosen. It is an interesting question to explore about the nature of the C * -algebra B, and we will do this towards the end of the paper.
The paper is organized as follows. In Section 1, we gather some basic results concerning C * -algebra weights. The purpose here is to set the terminologies and notations to be used in later sections. Our main definition and the properties are given in Sections 2 and 3. We introduce the notion of a separability triple and a separability idempotent in Section 2. As a consequence, we obtain the (densely-defined) anti-homomorphisms γ B and γ C (These are the maps corresponding to S B and S C in the purely algebraic case.). More results and properties are collected in Section 3, including the fact that a separability idempotent is "full".
In Section 4, some examples are considered. In addition to the natural example coming from a groupoid or a weak multiplier Hopf algebra, we included an example arising from a certain action groupoid and another example when the base C * -algebra is the algebra of compact operators.
In Section 5, the separability idempotent notion is considered in the von Neumann algebra setting. We show that the C * -algebra approach and the von Neumann algebra approach are equivalent, in the sense that one can start from the C * -algebra framework, then construct a separability idempotent in the von Neumann algebra framework, then from this, one can recover the idempotent and the C * -algebra we began with.
Finally, in Section 6, we explore the nature of the base C * -algebra B, which would allow the existence of a separability idempotent. We claim that B has to be a postliminal C * -algebra. grateful to his coauthor (Alfons Van Daele) and the mathematics department at University of Leuven for their warm support and hospitality during his stay.

Preliminaries on C * -algebra weights
In this section, we review some basic notations and results concerning the weights on C * -algebras, which will be useful later. For standard terminologies and for a more complete treatment on C * -algebra weights, refer to [6], [21], [22]. See also a nice survey given in [14].
Let A be a C * -algebra. A function ψ : for all x, y ∈ A + and ψ(λx) = λψ(x) for all x ∈ A + and λ ∈ R + , with the convention that 0 · ∞ = 0.
Given a weight ψ on A, we can consider the following subspaces: ψ , which turns out to be its positive part. It is possible to naturally extend ψ on M + ψ to a map from M ψ into C, which we will still denote by ψ.
A weight ψ is said to be "faithful", if ψ(a) = 0, a ∈ A + , implies a = 0. We say a weight ψ is densely-defined (or "semi-finite"), if N ψ is dense in A. Throughout this paper, a weight (on a C * -algebra A) will always be assumed to be faithful, semi-finite, and also lower semi-continuous.
It is useful to consider the following sets, first considered by Combes [6]: Here A * denotes the norm dual of A. Note that on F ψ , one can give a natural order inherited from A * + . Meanwhile, G ψ is a directed subset of F ψ . That is, for every ω 1 , ω 2 ∈ G ψ , there exists an element ω ∈ G ψ such that ω 1 ≤ ω, ω 2 ≤ ω. Because of this, G ψ is often used as an index set (of a net). Since ψ is lower semi-continuous, we would have: One can associate to ψ a GNS-construction (H ψ , π ψ , Λ ψ ). Here, H ψ is a Hilbert space, Λ ψ : N ψ → H ψ is a linear map such that Λ ψ (N ψ ) is dense in H ψ , and Λ ψ (a), Λ ψ (b) = ψ(b * a) for a, b ∈ N ψ . Since ψ is assumed to be lower semi-continuous, we further have that Λ ψ is closed. And, π ψ is the GNS representation of A on H ψ , given by π ψ (a)Λ ψ (b) = Λ ψ (ab) for a ∈ A, b ∈ N ψ . The GNS representation is non-degenerate, and the GNS construction is unique up to a unitary transformation.
Every ω ∈ A * has a unique extension to the level of the multiplier algebra M (A), which we may still denote by ω. From this fact, it follows easily that any proper weight ψ on A has a natural extension to the weightψ on M (A). For convenience, we will use the notations N ψ = Nψ and M ψ = Mψ. Also, the GNS construction for a proper weight ψ on A has a natural extension to the GNS construction forψ on M (A), with Hψ = H ψ .
To give somewhat of a control over the non-commutativity of A, one introduces the notion of KMS weights (see [6] and Chapter VIII of [22]). The notion as defined below (due to Kustermans [13]) is slightly different from the original one given by Combes, but equivalent. Definition 1.1. Let ψ be a faithful, semi-finite, lower semi-continuous weight. It is called a KMS weight, if there exists a norm-continuous oneparameter group of automorphisms (σ t ) t∈R of A such that ψ • σ t = ψ for all t ∈ R, and ψ(a * a) = ψ σ i/2 (a)σ i/2 (a) * for all a ∈ D(σ i/2 ).
Here, σ i/2 is the analytic generator of the one-parameter group (σ t ) at z = i/2, and D(σ i/2 ) is its domain. In general, with the domain properly defined, it is known that σ z , z ∈ C, is a closed map.
The one-parameter group (σ t ) is called the "modular automorphism group" for ψ. It is uniquely determined, as ψ is faithful. In the special case when ψ is a trace, that is, ψ(a * a) = ψ(aa * ) for a ∈ N ψ , it is clear that ψ is KMS, with the modular automorphism group being trivial (σ ≡ Id).
It is known (see [22], [14]) that when we lift a KMS weight to the level of von Neumann algebra π ψ (A) ′′ in an evident way, we obtain a normal, semifinite, faithful ("n.s.f.") weight ψ. In addition, the modular automorphism group (σ ψ t ) for the n.s.f. weight ψ leaves the C * -algebra B invariant, and the restriction of (σ ψ t ) to B coincides with our (σ t ). The operators J, ∇ above are none other than the restrictions of the corresponding operators arising in the standard Tomita-Takesaki theory of von Neumann algebra weights.
In Tomita-Takesaki theory, a useful role is played by the Tomita * -algebra. In our case, consider the n.s.f. weight ψ on the von Neumann algebra π ψ (A) ′′ . Then the Tomita * -algebra for ψ, denoted by T ψ , is as follows: It is a strongly * -dense subalgebra in π ψ (A) ′′ , and it satisfies many useful properties. Refer to the standard textbooks on the modular theory, for instance [22], [21]. Furthermore, it is known that T ψ := T ψ ∩ A is (norm)dense in the C * -algebra A, so turns out to be useful as we work with the weight ψ in the C * -algebra setting. In particular, it is easy to see that for every z ∈ C, the domains D(σ z ), which contain T ψ , are dense in A.

Definition of a separability idempotent
In this section, we wish to give the proper definition of a separability idempotent element in the C * -algebra framework. Let us begin by considering a triple (E, B, ν), where • B is a C * -algebra, • ν is a (faithful) KMS weight on B, together with its associated normcontinuous automorphism group (σ ν t ), • E ∈ M (B ⊗ C) is a self-adjoint idempotent element (so a projection, satisfying E 2 = E = E * ), for some C * -algebra C such that there exists a * -anti-isomorphism R : B → C.
Remark. In the above, we can see immediately that C ∼ = B op as C * -algebras. Even so, there is no reason to assume that C and B op to be exactly same. Going forward, our triple (E, B, ν) will be understood in such a way that the C * -algebra C and the * -anti-isomorphism R : B → C are implicitly fixed, with C possibly different from B op .
The weight theory says N ν is only a left ideal in B, but we have below a useful result involving B 0 = D(σ ν i/2 ), which is dense in B. Proposition 2.1. For any x ∈ N ν and b ∈ B 0 , we have: xb ∈ N ν . As a consequence, we have: As for the next two results, use the fact that M ν = N * ν N ν . Consider the triple (E, B, ν) as above, with C and R understood. We now give the definition of the separability triple: Definition 2.2. We will say that (E, B, ν) is a separability triple, if the following conditions hold: . If (E, B, ν) forms a separability triple, then we say E is a separability idempotent.
By Proposition 2.1, we know (id ⊗ω)(E) b ∈ M ν , ∀ω ∈ C * and ∀b ∈ B 0 . What condition (2) of the definition is saying is that we further require The following result provides sort of a uniqueness result for the separability idempotent E. Proof. From Definition 2.2 and Remark following it, we know that This shows that E is completely determined by the data (B, ν, R).
As we regard a separability triple (E, B, ν) to implicitly fix the C * -algebra C and the anti-isomorphism R (see an earlier remark), we can see from the above result that the E is more or less characterized by the pair (B, ν). However, the element E ∈ M (B ⊗ C) that is determined by (B, ν) via equation (2.1) does not necessarily have to be a projection. In other words, the existence of the separability idempotent E is really a condition on the pair (B, ν). In Section 6 below, we will discuss a little about the nature of the C * -algebra B that could allow this to happen.
Let us now gather some basic properties of the separability triple (E, B, ν) and the separability idempotent E ∈ M (B ⊗ C). For convenience, let us write γ B (b) := (R • σ ν i/2 )(b), for b ∈ B 0 , which determines a map γ B : B 0 → C. Since σ ν i/2 is a closed map (see comment given in Definition 1.1), we see that γ B is also a closed map.
Proposition 2.4. Considered as a map from B to C, the aforementioned map γ B is a closed, densely-defined, injective map, whose range is dense in C. Moreover, for b ∈ B 0 , we have: We have already seen that it is closed. The injectivity of γ B is easy to see, because R is an anti-isomorphism and σ ν t is an automorphism for all t. By the property of the Tomita algebra, we know that T ν ⊆ B 0 and σν i/2 (Tν) = Tν. So we have: σ ν i/2 (T ν ) = σ ν i/2 (Tν ∩ B) = Tν ∩ B = T ν , which is dense in B. In addition, since R is an anti-isomorphism between B and C, we see that Ran(γ B ) = R σ ν i/2 (B 0 ) ⊇ R(T ν ), which is dense in C. The last result is just re-writing (2) of Definition 2.2.
For convenience, write C 0 = Ran(γ B ), which is dense in C. By the above proposition, we can consider γ −1 B : C 0 → B. Viewed as a map from C to B, it is clear that the map γ −1 B is also a closed, densely-defined injective map, having a dense range. In fact, we will have: is closed under multiplication, and γ B is an anti-homomorphism.
Being analytic elements, we have bb ′ ∈ B 0 . Meanwhile, since γ B = R • σ ν i/2 , with σ ν i/2 being an automorphism and R an anti-isomorphism, we have Here is a property that relates the C * -algebras B and C, at the level of their dense subspaces: Proposition 2.6.
(1) For b ∈ B 0 , we have: In the second equation, we are using (2) of Definition 2.2. The third equation holds because γ B is an anti-homomorphism. Since ν is faithful, and since the result is true for all b ′ ∈ B 0 , which is dense in B, we conclude that . This is an immediate consequence of (1), because C 0 = Ran(γ B ) and , take the adjoint. Knowing that R is a * -anti-isomorphism, we have: . Or, put another way, we have: . So, by the same argument as in the case of the map γ B , it is clear that is closed and densely-defined on B, injective, and has a dense range in C. Also, its inverse map c → (σ ν i/2 • R)(c) is closed and densely-defined in C, injective, and has a dense range in B. So let us define the maps γ C : C → B and γ −1 C : B → C as follows: The map γ C is closed and densely-defined on C, injective, and has a dense range in B.
C is closed and densely-defined on B, injective, and has a dense range in C.
(1) and (2) were already observed in the previous paragraph. Proof of (3) is done exactly as in Proposition 2.5. The anti-homomorphism property is needed in the proof of the results (4) and (5), which is done using a similar argument as in Proposition 2.6.
Remark. The results of Proposition 2.6 and Proposition 2.7 suggest us that E behaves very much like a separability idempotent in the purely algebraic setting [24], justifying our definition given in Definition 2.2. Our maps γ B and γ C correspond to the maps S B and S C in [24]. Indeed, our Proposition 2.6 (1) and Proposition 2.7 (4) are exactly the defining axioms in the algebraic case (see Definition 1.4 of [24]), for S B and S C . The only difference is that in the algebraic case, the maps S B and S C are bijections, while our γ B , γ C maps are densely-defined with dense ranges.
Compared to the purely algebraic setting, we have here the * -structure. The next result is about the relationship between the * -structure and the maps γ B , γ C . Observe that in general, the maps γ B , γ C are not necessarily * -maps.
(2). Similar argument will show that: Notice the properties of the maps γ B , γ C obtained so far, namely, being closed and densely-defined anti-homomorphisms satisfying the result like Proposition 2.8 for the * -structure. One may observe that the behavior of the maps γ B , γ C resemble that of the antipode map for a locally compact quantum group. Indeed, in a certain context, it turns out that the maps γ B , γ C are indeed the antipode map of a locally compact quantum groupoid (A, ∆), restricted to the subalgebras B and C, which are essentially the "source algebra" and the "target algebra". And, our anti-isomorphism R would be the restriction to B of the "unitary antipode" R A : A → A. More systematic treatment on this direction will be given in our upcoming works [12], [10], [11].

Properties of the separability idempotent
Suppose we have a separability triple (E, B, ν) in the sense of the previous section, with C and R understood. We have already observed that E is uniquely determined by the pair (B, ν). Now, using the * -anti-isomorphism R : B → C, we can also define a faithful weight µ on C, by The second characterization is true because It is not difficult to show that µ is also a KMS weight on the C * -algebra C, together with its modular automorphism group (σ µ t ) t∈R , given by It turns out that the pair (C, µ) behaves a lot like (B, ν). Proposition 3.1. Let µ be as above. Then we have: Proof. As in Remark following Definition 2.2, the above equation means that (θ ⊗ id)(E) ∈ M µ for all θ ∈ B * (naturally extended to the multiplier algebra level), and that µ (θ ⊗ id)(E) = θ(1).
We can verify this for θ = ν( · b) ∈ B * , where b ∈ D(γ B ). Such functionals are dense in B * , so that will prove the proposition. To see this, consider an arbitrary b ∈ D(γ B ). Then by Definition 2.2 (2), we have (ν ⊗id) . Apply here µ. Then we have: , where, we used the fact that µ = ν •R −1 and that ν is σ ν -invariant. Observe that this equation can be re-written as proving the claim.
Corollary. As a consequence of the previous proposition, we have the following: We also have the following alternative descriptions for the maps γ B , γ −1 B , γ C , γ −1 C , which will be useful later: Proposition 3.2. We have: The definitions involving the weight ν have been already observed in the previous section. As for the descriptions involving the weight µ, we just use the fact that σ µ In the below is the result that is usually assumed as part of the definition of a separability idempotent in the purely algebraic setting: For all b ∈ B and all c ∈ C, we have: Here, ⊗ is the (spatial) C * -tensor product.
But the elements γ B (b 0 )c, for b 0 ∈ D(γ B ), c ∈ C, are dense in C, because Ran(γ B ) is dense in C and C 2 = C. It follows that E(1 ⊗ c) ∈ B ⊗ C for all c ∈ C. Similar argument holds for each of the other three cases.
Corollary. As a consequence of the proposition, we have: Then, we know from the above proposition that We already knew that (id ⊗ω)(E) ∈ M (B), but we now see that it is actually contained in B. Moreover, we see below that such elements form a dense subspace in B. Similar for C.
Proof. By Corollary following Proposition 3.3, we are assured that containment statement for each set is true. So we just need to show that they are dense subsets.
, which is dense in C because γ B was shown to have a dense range in C. This proves the first statement. Proofs for the other statements are very much similar, knowing that the maps Here is a result that is related to the fact that E is full. While it is possible to give the proof using the above proposition, we instead chose to give a direct proof, which seems simpler.
Since Ran(γ B ) is dense in C, this means that c = 0.
Remark. The "fullness" of E ∈ M (B ⊗ C), as given in Proposition 3.4, means that the left leg of E is B and the right leg of E is C. In the purely algebraic setting, the fullness of E was part of the definition of E being a separability idempotent [24]. There, from the defining axioms, one obtains the existence of certain "distinguished linear functionals", ϕ B and ϕ C . In our setting, however, we begin first with the weights µ and ν, then obtain the results on E, including its fullness, the maps γ B , γ C , and the like.
In the next proposition, we see that the analytic generators σ µ −i and σ ν −i can be characterized in terms of the maps γ B and γ C .
Proposition 3.6. We have: Remark. The significance of Proposition 3.6 is that the maps γ B • γ C and γ −1 B • γ −1 C provide "modular automorphisms" for the weights µ and ν, respectively. To be more precise, recall Lemma 1.2. Our result says that for . Similar also for the weight ν. We will conclude this section by showing a few different characterizations of the idempotent E.
be arbitrary, and compute: In the first equality, we are using the fact that σ ν t is an automorphism. In the second equality, we used ν • σ ν t = ν. This is true for any b ∈ D(σ ν i/2 ). By the uniqueness property observed in Proposition 2.3, we conclude that (σ ν t ⊗ σ µ −t )(E) = E, for any t ∈ R. Write σ to denote the flip map on M (B ⊗ C). So we will have σE ∈ M (C ⊗B). In the below, we wish to show that (γ C ⊗γ B )(σE) = E. However, as of now, we do not know if the expression (γ C ⊗ γ B )(σE) even makes sense as a bounded element. To make sense of all this, and anticipating other future applications, we prove first the following lemma: Lemma 3.8. Suppose b ∈ T ν and c ∈ T µ . We have: . This result is true for any b ∈ T ν , c ∈ T µ , which are dense in B and C, respectively. Moreover, we know that the weights ν, µ are faithful. It follows that , proving the claim. Other cases can be proved similarly.
By Lemma 3.8, we can now prove the following result: Proof. As written, we do not know whether (γ C ⊗ γ B )(σE) is bounded. However, for b ∈ T ν (dense in B) and c ∈ T µ (dense in C), we know from Lemma 3.8 that it can be made sense as follows: as a left and right multiplier map, on a dense subset of B ⊗ C. Since E is bounded, this implies that (γ C ⊗ γ B )(σE) can be canonically extended to a left and right multiplier map on all of B ⊗ C, which would mean that (γ C ⊗ γ B )(σE) ∈ M (B ⊗ C), and that (γ C ⊗ γ B )(σE) = E. By taking the flip map, we also have: Corollary. We have: Proof. From the proposition, we know: σE = (γ B ⊗ γ C )(E) ∈ M (C ⊗ B). Apply R −1 ⊗ R to both sides. By Proposition 3.2, we know γ B = R • σ ν i/2 For the last equality, we used the result of Proposition 3.7.

Examples and special cases
4.1. Groupoids. As indicated in the Introduction, the theory of weak multiplier Hopf algebras is motivated by attempts to generalize the notion of a groupoid. Our theory is essentially a C * -algebraic counterpart to the weak multiplier Hopf algebra theory. Therefore, while it is true that this is not fully general due to the fact that the existence condition for the separability idempotent element is rather strong to be compatible with some topological aspects, it remains the case that a certain subclass of groupoids provide us with typical examples. See below. Let Consider next the counting measure on G (0) , which naturally determines a weight on C 0 (G (0) ). Using the pull-back maps, we can define weights ν on B and µ on C. They are faithful, tracial weights (so KMS).
To describe the R map, consider It is not difficult to see that (E, B, ν) forms a separability triple in the sense of Definition 2.2, and therefore E is a separability idempotent.
Example 4.B. Consider again a discrete groupoid G. Denote by K(G) the space of all complex functions on G having finite support. In particular, . Note that any function f ∈ K(G) can be expressed in the form f = p∈G f (p)λ p . We can give K(G) a * -algebra structure by letting λ p λ q = λ pq , valid only when pq is defined and 0 otherwise; while at the same time letting λ * p = λ p −1 . This is none other than the convolution * -algebra structure.
Similar to the group case, one can consider the (left) regular representation of K(G) on l 2 (G), given by the counting measure on G. The regular representation extends K(G) to a C * -algebra A = C * r (G). Unless G (0) is a finite set, the algebra A is in general non-unital. However, at the level of the multiplier algebra, the unit element is 1 = λ e , where the sum is taken over all e ∈ G (0) . We will skip the details (left Haar system, and the like) and refer to the standard textbooks ( [20], [19]), as the issue at hand is more about the separability idempotents than about groupoid algebras. More discussion on the (quantum) groupoid aspects will be given in the authors' upcoming papers [10], [11].
Note that if e ∈ G (0) , we have s G (e) = t G (e) = e. Because of this, the base algebras B and C, which are generated by {λ s G (p) : p ∈ G} and {λ t G (p) : p ∈ G} respectively, will turn out to be isomorphic to C 0 (G (0) ). The counting measure on G (0) would provide the weights ν and µ. Being faithful and tracial, they are KMS weights. Let R : B → C be given by λ s G (p) → λ t G (p −1 ) . It is a well-defined map, which is clearly an (anti)isomorphism.
Now consider E := λ e ⊗ λ e , where the sum is taken over all e ∈ G (0) . It is not difficult to show that it becomes a separability idempotent.
Remark. (1). Since G is discrete, the topology does not play much significant role here. At the *-algebra level, the two examples above can be described in the framework of weak multiplier Hopf algebras [26], [28]. For a general weak multiplier Hopf algebra, the discussion about E being a separability idempotent in terms of its "distinguished linear functionals" (µ and ν in our case) can be found in section 4 of [24].
(2). There also exists the notion of a measured quantum groupoid by Lesieur and Enock, in the von Neumann algebra framework [17], [8]. It generalizes the notion of a groupoid, so in some special cases (for instance, when the base algebra is finite-dimensional), they provide a separability idempotent. But we will postpone the discussion on all these cases related to "quantum groupoids" to our future papers [10], [11].

4.2.
An example coming from an action groupoid. Consider the following specific example: The group (Z, +) acts on Z := Z ∪ {∞} by translation and leaving ∞ fixed, and let Z×Z be the corresponding action groupoid (or, transformation group groupoid). It can be considered as a locally compact groupoid, together with the discrete topology on Z, and Z being the compactification to ∞. Then consider G := (Z × Z)| N , the restriction of the action groupoid to N ⊂ Z.
G is also a groupoid (the "Cuntz groupoid", see [20], [19], [4]), where G (0) = N and equipped with the target map t : (m, p) → m and the source map s : (m, p) → m + p. We have: (m, p) · (n, q) = δ m+p,n (m, p + q). It is a locally compact groupoid, whose topology is inherited from that of the action groupoidZ × Z. The (left) Haar system is given by the discrete measure on each t-fiber.
We can certainly consider, as in Example 4.B above, a natural separability idempotent E 0 := m∈N λ m,0 ⊗ λ m,0 , which is characterized by B 0 ∼ = C 0 (G (0) ) and ν 0 , the counting measure on B 0 , namely, ν 0 (λ m,0 ) = 1, ∀m ∈ N. However, for this next example, we instead wish to consider all of CG as the base algebra for our (soon to be constructed) separability idempotent.
Consider B := C * (G) = CG. As in Example 4.B, it can be realized as a completion of the convolution algebra C c (G) given by the convolution product λ m,p λ n,q = δ m+p,n λ m,p+q and the involution λ * m,p = λ m+p,−p . As before, we are regarding the generators λ m,p as functions on G, given by λ m,p (n, q) = δ m,n δ p,q . It is known that B is isomorphic to the C * -algebra generated by the unilateral shift operator on l 2 (N), but we do not need this fact for our purposes (see [20]).   The second equality is using the fact that n = p, and the third equality is because p∈N e −p (1 − e − ) = 1.
Let C = C * (G), which will be considered as an opposite C * -algebra of B, via the map R : B → C, defined on the generators by R(λ n,q ) = λ n+q,−q . Coming from the involution, it is clear that R is indeed a * -anti-isomorphism.
Next, define the weight ν on B, whose values on the generators are  (1) ν , together with the one-parameter group (σ t ) t∈R , is a KMS weight on B.
By Definition 2.2, we conclude from Proposition 4.2 that (E, B, ν ) forms a separability triple, and therefore, E is a separability idempotent. We may have an occasion in the future to explore this example further.

4.3.
A case when B = B 0 (H). We wish to study an example that generalizes the example given in section 4 of [24]. Consider a Hilbert space H, possibly infinite-dimensional. Fix an orthonormal basis (ξ j ) j∈J for H, where J is an index set. For any v ∈ H, we may write v = j∈J v j ξ j , where v j = v, ξ j . Consider also Tr( · ), the canonical trace on B(H).
Define E 0 := i,j∈J e ij ⊗ e ij , where the e ij ∈ B(H) are the matrix units, given by (2) For a ∈ B(H), we can consider a T ∈ B(H), the "transpose" of a, defined by a T ij = a ji = aξ i , ξ j , for i, j ∈ J. We will have: Tr(a) = j∈J aξ j , ξ j = j∈J a jj = Tr(a T ).
Proof. For v, w ∈ H, by using Lemma 4.3, we have:

This is true for all
Consider r ∈ B(H), which is invertible and Tr(r * r) = 1. So r is a compact operator of Hilbert-Schmidt type. This means that r −1 is unbounded in general, unless H is finite-dimensional. Next, define a new operator E, by E := (r ⊗ 1)E 0 (r * ⊗ 1). (

4.4)
To learn more about the property of E, we will first consider a lemma.
In the last line, we used the assumption that Tr(r * r) = 1. It follows that We see that E is a self-adjoint idempotent, so bounded projection in B(H ⊗ H).
Next, define B := B 0 (H), the C * -algebra of compact operators on H. Let us consider a suitable weight on B: Proposition 4.7. Consider the map ν := Tr(r −1 · r * −1 ) = Tr p ·), where p = r * −1 r −1 = (rr * ) −1 . It is a KMS weight on the C * -algebra B. The associated norm-continuous automorphism group, the "modular automorphism group" for ν, is (σ ν t ) t∈R , defined by σ ν t (b) = p it bp −it . Proof. From the properties of Tr( · ), which is itself a proper weight, it is easy to see that ν is a lower semi-continuous weight and is densely-defined. It is also faithful, because if ν(b * b) = Tr(r −1 b * br * −1 ) = 0, it necessarily means br * −1 =0, so b = 0.
With the definition of the (σ ν t ) given above and for b ∈ M ν , observe that This shows the invariance of ν under the (σ ν t ). While, for b ∈ D(σ ν i/2 ), we have: . By the conditions given in Definition 1.1, we see that ν is a KMS weight. With the definition given in Equation (4.4), we showed in Proposition 4.6 that E is self-adjoint and idempotent such that E ∈ B(H ⊗ H) = M (B ⊗ C). In the next proposition, we see that (E, B, ν) indeed forms a separability triple.
Proposition 4.8. Consider the triple (E, B, ν) as given above. Then . By Definition 2.2, we see that (E, B, ν) forms a separability triple, and therefore, E is a separability idempotent.
(2). We can check the result directly, by using the characterization given in (1) and by imitating the proof of Proposition 4.7. Or, we can compute the following: In this way, we can see that The next proposition describes the (densely-defined) anti-homomorphism maps γ B , γ −1 B , γ C , γ −1 C that we saw in Sections 2 and 3. Proposition 4.10.

Separability idempotent in the von Neumann algebra setting.
It is possible to consider the notion of a separability idempotent in the setting of von Neumann algebras. We will do this here. Later in the section, we will also explore how the C * -algebra approach and the von Neumann algebra approach are related. We begin by considering a triple (E, N,ν), where • N is a von Neumann algebra; •ν is an n.s.f. weight on N , together with its associated modular automorphism group (σν t ); • E is a self-adjoint idempotent (so a projection) element contained in the von Neumann algebra tensor product N ⊗ L, where L is a von Neumann algebra such that there exists a * -anti-isomorphism R : N → L.
Consider the triple (E, N,ν) as above. As in Section 2, we consider L and R : N → L to be implicitly understood. We now give the definition of the separability triple: Definition 5.1. We will say that (E, N,ν) is a separability triple, if the following conditions hold: If (E, N,ν) forms a separability triple, then we say E is a separability idempotent.
Let (E, N,ν) be a separability triple. Then, using the * -anti-isomorphism R : N → L, we can define the n.s.f. weightμ on L, given byμ :=ν • R −1 . The arguments given in Sections 2 and 3 will all go through, including the existence of the densely-defined anti-homomorphisms γ N , γ L and γ −1 N , γ −1 L : They are closed and densely-defined, injective anti-homomorphisms, having dense ranges. We can also consider γ −1 N and γ −1 L , having similar properties. In addition, we have: (1) For b ∈ D(σν i/2 ), we have: (2) For c ∈ D(σμ i/2 ), we have: We can also show that E is full. The proof is done in essentially the same way as in Proposition 3.4. The only difference is that while the sets were norm dense in the C * -algebra setting, now the sets are strongly * -dense in the current setting: Other results in Sections 2 and 3 also have corresponding results, including the following: Proposition 5.4. Let the notation be as above, and as before, denote by σ the flip map on N ⊗ L. Then Proof. See Propositions 3.7 and 3.9.

5.2.
Comparison with the C * -algebraic notion. The results above seem to indicate that the C * -algebra approach and the von Neumann algebra approach to the separability idempotent are equivalent. To prove this, let us first begin with the C * -algebra framework.
Consider N := π ν (B) ′′ . Denote byν the weight on N lifted from ν. Since ν is a KMS weight, we know thatν should be an n.s.f. weight on N . Similarly, define L := π µ (C) ′′ . We can perform the lifting of µ and obtaiñ µ, an n.s.f. weight on L. We expect that our separability idempotent will be described at the level of these von Neumann algebras. First, let us show that the * -anti-isomorphism R : B → C extends toR : N → L, also a * -anti-isomorphism.
Lemma 5.5. We have: Proof. First statement is a consequence of E being "full" (Proposition 3.4), together with the fact that N ν is dense in B. Note also that the functionals of the form ν(b * 1 · b 2 ), b 1 , b 2 ∈ N ν , are dense in B * . Second statement is also a consequence of E being full.
Proof. From Lemma 5.5, we know the elements (id ⊗µ)((1 ⊗ c * 1 )E(1 ⊗ c 2 )), c 1 , c 2 ∈ N µ , generate B. Under R, since µ = ν • R −1 , we have: using the fact that (R ⊗ R −1 )(E) = σE (Corollary of Proposition 3.9). Similarly, we also have: In view of Lemma 5.6, looking at the generators of N and L, it is now clear that the R map extends to an injective mapR from N onto L. The preservation of the * -structure and the anti-multiplicativity ofR are all straightforward because of the corresponding properties of R.
Remark. For convenience, we may still write R instead ofR, with its inverse R −1 : L → N also a * -anti-isomorphism. It is evident that we will havẽ µ =ν • R −1 .
Finally, observe that E ∈ M (B ⊗ C) ⊆ N ⊗ L. Since B, C are dense in N , L, and sinceν,μ naturally extend ν, µ, it follows that (E, N,ν) forms a separability triple in the sense of Definition 5.1.
Let us next consider the problem of going back. So we begin with a separability triple (E, N,ν) in the von Neumann algebra setting, again with the von Neumann algebra L and the anti-isomorphism R : N → L understood. Also recall thatμ =ν • R −1 .

Corollary.
(1) B is a C * -subalgebra of N , and N is the von Neumann algebra closure of B.
(2) C is a C * -subalgebra of L, and L is the von Neumann algebra closure of C.
Proof. For B, being a norm-closed * -subalgebra of N , that is strongly *dense in N , the result follows immediately. Similar for C in L.
In the next proposition, we show that E ∈ M (B ⊗ C) and that E is a separability idempotent: Proposition 5.9. Let E ∈ N ⊗ L be a separability idempotent in the von Neumann algebraic sense, and consider the C * -subalgebras B of N and C of L, as defined above. Denote by ν the weight on B, restricted fromν on N , and similarly, consider µ on C, restricted fromμ on L. Then: (1) The σν t , t ∈ R, leaves B invariant. So we can consider σ ν t := σν t | B , for t ∈ R. Similar for (σ µ t ), on C. Proof. (1). Consider (id ⊗ω)(E) ∈ B, where ω ∈ B(Hμ) * . For any t ∈ R, we know from Proposition 5.4 that (σν −t ⊗ σμ t )(E) = E. So we have: Since B = (id ⊗ω)(E) : ω ∈ B(Hμ) * , this means that σν t (B) = B, for all t ∈ R. By a similar argument, we can show that σμ t (C) = C, for all t ∈ R.
Meanwhile, since σν t (B) = B, we may write σ ν t = σν t | B , for t ∈ R. With t → σν t being strongly continuous, and by Equation (5.4), we have that for b ∈ B, the function t → σ ν t (b) is norm-continuous. From the properties of the modular automorphism group (σν t ) t∈R forν, we can show easily the analogous properties for (σ ν t ) t∈R , and this means that ν is a KMS weight on the C * -algebra B. By construction, it is clear that the W * -lift of ν will be justν on N .
Similarly, if we let µ =μ| C , then it is a KMS weight on the C * -algebra C, together with the one-parameter group of automorphisms (σ µ t ) t∈R , where σ µ t = σμ t | C . The W * -lift of µ isμ on L.
(3). We already know that E is a self-adjoint idempotent contained in N ⊗ L. Therefore, by Proposition 5.3, we have for any ε > 0 and for any ξ ∈ Hν, η ∈ Hμ, we can find a finite number of elements p 1 , p 2 , . . . , p n ∈ B of the form (id ⊗ω)(E), ω ∈ B(Hμ) * and q 1 , q 2 , . . . , q n ∈ C of the form (θ ⊗ id)(E), θ ∈ B(Hν) * , such that Since N (also B) acts on Hν in a non-degenerate way, and similarly for L (also C) on Hμ, without loss of generality we may let ξ = Λ ν (b 0 ) = Λν(b 0 ) and η = Λ µ (c 0 ) = Λμ(c 0 ), where b 0 ∈ N ν , c 0 ∈ N µ . Then: Since b 0 , c 0 are arbitrary, while N ν is dense in B and N µ is dense in C, we can see that E is contained in the closure of B ⊗ C under the strict topology. In other words, E ∈ M (B ⊗ C).
(4). From (3), we saw that E ∈ M (B ⊗ C). From (1), (2), we saw that ν is a KMS weight on B, together with its associated norm-continuous automorphism group (σ ν t ). Meanwhile, note that for (id ⊗ω)(E) ∈ B (⊆ N ), by Proposition 5.4 (3), we have: This shows that the * -anti-isomorphism R : N → L restricts to R : B → C. This will be also a * -anti-isomorphism. Since R : B → C is a restriction of R : N → L and since σ ν t = σν t | B , the conditions of Definition 5.1 for (E, N,ν) immediately give rise to the conditions of Definition 2.2. This means that (E, B, ν) is a separability triple in the C * -algebraic sense. And, the associated γ B , γ C maps are none other than the restrictions of γ N and γ L to B and C, respectively, which have already appeared in (1), (3) of Proposition 5.8.
6. The nature of the base C * -algebra B The requirements for (E, B, ν) being a separability triple means that some possible restrictions will be needed on the pair (B, ν). This question was considered in the algebraic setting (see [24]), and the conclusion in that framework was that B has to be a direct sum of matrix algebras. While the current situation is different, we do expect that some similar conditions on the base C * -algebra B will exist. We wish to explore this question in this section.
To begin, assume that B is a C * -algebra, ν a KMS weight on B, such that the pair (B, ν) satisfies the relevant conditions that give us a separability idempotent E ∈ M (B ⊗ C). Here C is the opposite C * -algebra of B.
6.1. The subalgebra BbB for a fixed b ∈ B. Let us fix b ∈ B. By Proposition 3.3, we know that (b⊗1)E is contained in the C * -tensor product B ⊗ C. So, for any k ∈ N, we can find a finite number of elements b Next, letb ∈ B be arbitrary and let ε > 0. By Proposition 3.4 (since E is "full"), we can findc ∈ C,ω ∈ C * , such that b − (id ⊗ω)(E(1 ⊗c)) < ε 2 b . Then we have: Consider k ∈ N such that 1 k < ε 2 ω c , and as above, find corresponding Combine this with the earlier inequality to obtain (by triangle inequality): In the above, eachω(c (k) ic ) ∈ C. So we have shown that any bb is within ε-distance from span(b (k) i : k ∈ N, i = 1, 2, . . . , N k ). And, this result is true for arbitraryb ∈ B and any ε > 0. We thus have: From the countable generating set {b (k) i } for bB, consider a maximal linearly independent (basis) subset {b λ } λ∈Λ , indexed by a countable set Λ. We obtain the following result: Proposition 6.1.
(2) For any k ∈ N, we can find a finite set F k ⊂ Λ and a finite number of elements {c λ } λ∈F k in C, such that (3) For any z ∈ bB and any ε > 0, we can find a finite setF ⊂ Λ and a finite number of elements {c λ } λ∈F in C, such that for bB may be expressed as a linear combination of the {b λ }. So the result is clear.
(2). For any finite sum of the form k,i b where the c λ are appropriate elements in C. In particular, for a fixed k ∈ N, we can find a finite set F k ⊂ Λ and suitable elements Without loss of generality, we may assume thatb ∈ D(γ −1 , then by (2) we have a finite subset F = F k ⊂ Λ such that: By the triangle inequality, we have: (z ⊗ 1)E − λ∈F b λ ⊗c λ < ε.
Next, for a typical generator b λ (for a fixed λ ∈ Λ), we wish to explore the space Bb λ . Since E(b λ ⊗ 1) is contained in the C * -tensor product B ⊗ C, for any m ∈ N we can find a finite number of elements b From the set {b (m) λ,j : m ∈ N, j = 1, 2, . . . , N m }, find a maximal linearly independent subset {b λ,q } q∈Q λ , indexed by a countable set Q λ . Then we will have similar results as before: (1) Bb λ ⊆ span(b λ,q : q ∈ Q λ ) . (2) For any y ∈ Bb λ and any ε > 0, we can find a finite setF ⊂ Q λ and a finite number of elements {c λ } λ∈F in C, such that Proof. Proof is similar to that of Proposition 6.1.
Proposition 6.2 is valid for any b λ , for λ ∈ Λ. Meanwhile, it is evident that BbB ⊆ ∪ λ∈Λ Bb λ . So BbB ⊆ span(b λ,q : λ ∈ Λ, q ∈ Q λ ) . Again, similarly as above, find {b p } p∈P , indexed by a countable set P and is a maximal linearly independent subset of the {b λ,q }. Then we have the following result: (1) For a fixed b ∈ B, we can find a countable set of elements {b p } p∈P in B, such that BbB ⊆ span(b p : p ∈ P ) .
(2) For any x ∈ BbB and any ε > 0, we can find a finite setF ⊂ P and a finite number of elements {c p } p∈F in C, such that Proof. (1). This is a consequence of the previous paragraph. (2). Let x ∈ BbB and ε > 0 be arbitrary. We can find a finite number of Since γ −1 C is densely-defined in B, we may as well take each b ′′ i ∈ D(γ −1 C ).
We know from Proposition 6.1 (2) that the expression (b ⊗ 1)E can be approximated arbitrarily closely by a finite sum of the form λ∈F k b λ ⊗ c λ . This means that for each i, the expression (b ′ i bb ′′ i ⊗ 1)E can be approximated arbitrarily closely by a finite sum of the form λ∈F Meanwhile, by Proposition 6.2 and part (1) above that each b ′ i b λ can be expressed as a finite linear combination in terms of the {b p } p∈P . So we can find a finite set . . , n. We are able to find a finite setF ⊂ P 1 ∪ P 2 ∪ · · · ∪ P n ⊂ P and suitable elements Remark. The choice of the generators {b p } p∈P is determined by b, but of course not unique. Even so, to avoid confusion, we will from now on fix the choice of the {b p } as the generators for BbB. Without loss of generality, we may assume that each generator b p is a self-adjoint element. In addition, since N ν is dense in B, we may as well take each b p ∈ N ν , or even b p ∈ T ν (Tomita subalgebra). This would in turn means that each b p is contained in the domains of the maps γ B and γ −1 C .
In the below is a lemma concerning the elementsc p ∈ C: Lemma 6.4. In Theorem 6.3 (2) above, as we approximate (x ⊗ 1)E by the finite sums of the form b p ⊗c p , it is possible to choose eachc p to be contained in γ −1 C (BbB).
Proof. Since D(γ B ) is dense in B, we may, without loss of generality, assume that x ∈ D(γ B ) ∩ BbB. Let ε > 0. As in (2) above, we can find a finite subsetF ⊂ P and the elements {c ′ Consider now the subspace H b := span Λ(b p ) : p ∈ P H ⊆ H, where the b p ∈ T ν are as in Theorem 6.3. Under the representation π, the C *algebra BbB leaves the subspace H b invariant. In this way, we obtain a subrepresentation π b of BbB, acting on H b . Meanwhile, by Gram-Schmidt process, without loss of generality, we may further assume that ν(b q b p ) = Λ(b p ), Λ(b q ) = 0 if p, q ∈ P is such that p = q, while ν(b p b p ) = 1, ∀p ∈ P . (Recall that we are assuming each b p to be self-adjoint).
Definition 6.5. For any p, q ∈ P , denote by b p , b q E the operator acting on H b such that b p , b q E ξ := ξ, Λ(b q ) Λ(b p ). Clearly, these are finite-rank (actually, rank one) operators contained in B(H b ). Proposition 6.6. Any operator in π b (BbB) can be approximated by a linear combination of the operators b p , b q E , p, q ∈ P .
Let ε > 0 be arbitrary. By Theorem 6.3 (2), we can find a finite subset F ⊂ P and {c p } p∈F in C, such that (x ⊗ 1)E − p∈F b p ⊗c p < ε 2 b . By Lemma 6.4, we may assume that eachc p ∈ γ −1 C (BbB). Observe that: We know µ = ν • γ C . So µ(c p γ B (b)) = ν (γ C • γ B )(b)γ C (c p ) . And, using the fact that γ C • γ B = σ ν i (see Proposition 3.6), we see that µ(c p γ B (b)) = ν γ C (c p )b . As γ C (c p ) ∈ BbB, we can approximate it arbitrarily closely by a finite linear combination of the b q , for q ∈ F (p) ⊂ P . This can be done for each p ∈ F . Therefore, it is evident that we can find a finite linear combination of the expressions ν(b qb )b p = Λ(b), Λ(b q ) b p , that is within ε 2 -distance from p∈F µ(c p γ B (b))b p .
It follows that xb can be approximated within ε-distance by a finite linear combination of the Λ(b), Λ(b q ) b p . Applying the GNS-map Λ, we see that π b (x)Λ(b) can be approximated arbitrarily closely by finite linear combinations of the expressions Λ(b), Λ(b q ) Λ(b p ) = b p , b q E Λ(b).
All this would hold for each of the basis element Λ(b m ), m ∈ P . In this way, we prove that any π b (x), x ∈ BbB, can be approximated by a linear combination of the operators b p , b q E , p, q ∈ P .
Write V = span b p , b q E : p, q ∈ Γ ⊆ B(H b ) . Then: Proposition 6.7. For the operators b p , b q E , b p ′ , b q ′ E ∈ V , we have: As a result, we conclude that V op ∼ = B 0 (H b ), the algebra of compact operators on H b .
Proof. These are rather standard and straightforward.
(1). Let ξ ∈ H b be arbitrary. we have: (3). The results of (1), (2) show that V is a * -algebra contained in B(H b ), and the generators behave exactly like the matrix units (see, for instance, Lemma 4.5). Therefore, the operator norm closure of V coincides with B 0 (H b ).
The following results are immediate: Theorem 6.8. Let b ∈ B be a fixed self-adjoint element. Then we have: (1) BbB = π b (BbB) ⊆ B 0 (H b ).
Proof. (1). The result of Proposition 6.6 is that π b (BbB) is contained in the closure of V . Therefore, by Proposition 6.7, the result follows.
(2). Since BbB is a C * -algebra contained in the algebra of compact operators, it is liminal, in the sense of [7]. 6.3. B is a postliminal C * -algebra. By definition, a C * -algebra A is said to be postliminal, if every non-zero quotient C * -algebra of A possesses a non-zero liminal closed two-sided ideal (see [7], [1]).
In our case, suppose J is any nontrivial proper (closed, two-sided) ideal of B, and let B/J be its corresponding quotient C * -algebra. Since J = B, we can consider a self-adjoint element b ∈ B with b ∈ J. Consider the ideal BbB of B. Then BbB + J is an ideal of B such that (BbB + J)/J ∼ = BbB/(BbB ∩ J). Since the latter is a quotient of a liminal C * -algebra BbB, it is also liminal. In this way, we have shown that the quotient C * -algebra B/J contains an ideal (BbB + J)/J, which is isomorphic to a liminal C *algebra. By definition above, we can see that B is postliminal: Theorem 6.9. Let (E, B, ν) be a separability triple. Then B is a postliminal C * -algebra.
Remark. For separable C * -algebras, it is well-known that being postliminal is equivalent to being type I . The result that B is postliminal (or type I) seems compatible with the result in the purely algebraic case [24], where it was shown that the base algebra B has to be a direct sum of matrix algebras.
Finally, here is the result when B is commutative: Proposition 6.10. If B is commutative, its spectrum is totally disconnected.
Proof. If B is commutative, any of its ideals is also commutative. In particular, each building-block ideal BbB (for b self-adjoint), being a commutative subalgebra contained in the algebra of compact operators, would be isomorphic to C, spanned by a single projection. Our B would be spanned by its projections. If we write B ∼ = C(X), where X is the spectrum of B, this means that X is totally disconnected.