Representations of Semigroupoid and Springs

Chapter 1

∗-Algebras and Semigroupoid In this chapter we show that a semigroupoid is a set equipped with a partially defined associative operation. Given a semigroupoid Λ we construct a C*-algebra (Λ) from it. We then present two main examples of semigroupoids, namely the Markov semigroupoid associated to an infinite 0-1 matrix, and the semigroupoid associated to a row-finite higher- rank graph without sources [1]. Section (1.1): Representations of Semigroupoid and Springs The theory of ∗-algebras has greatly benefited from Combinatorics in the sense that some of the most interesting examples of ∗-algebras arise from combinatorial objects, such as the case of graph ∗-algebras. More recently Kumjian and Pask have introduced the notion of higher-rank graphs, inspired by Robertson and Steger's work on buildings, which turns out to be another combinatorial object with which an interesting new class of ∗-algebras may be constructed.

The crucial insight leading to the notion of higher-rank graphs lies in viewing ordinary graphs as categories (in which the morphisms are finite paths) equipped with a length function possessing a certain unique factorization property.

Kumjian and Pask's interesting idea of viewing graphs as categories suggests that one could construct ∗-algebras for more general categories.

Since Eilenberg and Mac Lane introduced the notion of categories in the early 40's, the archetypal idea of composition of functions has been mathematically formulated in terms of categories, whereby a great emphasis is put on the domain and co-domain of a function. However one may argue that, while the domain is an intrinsic part of a function, co-domains are not so vital. If one imagines a very elementary function with domain, say = {1,2}, defined by () = , one does not really need to worry about its co-domain. But if is to be seen as a morphism in the category of sets, one needs to first choose a set containing the image of , and only then becomes an element of Hom(, ). Regardless of the very innocent nature of our function , it suddenly is made to evoke an enormous

1 amount of morphisms, all of them having the same domain , but with the wildest possible collection of co-domains.

Addressing this concern one could replace the idea of categories with the following: a big set (or perhaps a class) would represent the collection of all morphisms, regardless of domains, ranges or co-domains. A set of composable pairs (, g) of morphisms would be given in advance and for each such pair one would define a composition g. Assuming the appropriate associativity axiom one arrives as the notion of a semigroupoid, precisely defined below. Should our morphisms be actual functions a sensible condition for a pair (, g) to be composable would be to require the range of g to be contained in the domain of , but we might also think of more abstract situations in which the morphisms are not necessarily functions.

For example, let = {(, )},∈, be an infinite 0-1 matrix, where is an arbitrary set, and let Λ be the set of all finite admissible words in , meaning finite sequences = … of elements ∈ , such that (, ) = 1. Given , ∈∧ write

= … and = , , … , , and let us say that and are composable if (, ) = 1, in which case we let be the concatenated word

= … , , … , .

We shall refer to ∧ as the Markov semigroupoid. This category-like structure lacks a notion of objects and in fact it cannot always be made into a category. Consider for instance the matrix 1 1 = . 1 0

If we let the index set of be = {, }, notice that the words and may be legally composed to form the words , , and , but is forbidden, precisely because (, ) = 0.

Should there exist an underlying category, the fact that, say, is a legal composition would lead one to believe that (), the domain, or source of

2 coincides with (), the co-domain of . But then for similar reasons one would have

() = () = () = (), which would imply that is a valid composition, but it is clearly not. This example was in fact already noticed by Tomforde with the purpose of showing that a 0-1 matrix is not always the edge matrix of a graph. Although the above matrix may be replaced by another one which is the edge matrix of a graph and gives the same Cuntz-Krieger algebra, the same trick does not work for infinite matrices.

This is perhaps an indication that we should learn to live with semigroupoids which are not true categories. Given the sheer simplicity of the notion of semigroupoid, one can easily fit all of the combinatorial objects so far referred to within the framework of semigroupoids.

The goal of this work is therefore to introduce a notion of representation of semigroupoids, with its accompanying universal ∗-algebra, which in turn generalizes earlier constructions such as the Cuntz-Krieger algebras for arbitrary matrices and the higher-rank graph ∗-algebras [5]: (A graph (rank graph or higher rank graph) (Λ, ) consists of a countable small category Λ (with range and source maps and respectively) together with a functor : Λ → satisfying the factorization property: for every ∈ and , ∈ with () = + , there are unique elements , ∈ Λ such that = and () = , () = . For ∈ we write Λ ∶= (). A morphism between -graphs (Λ, ) and (Λ, ) is a functor : Λ → Λ compatible with the degree maps.), and hence ordinary graph ∗-algebras as well.

We devoted to comparing semigroupoid ∗-algebras with Cuntz-Krieger and higher-rank graph ∗-algebras. A deeper study is made of the structure of semigroupoid ∗-algebras, including describing them as groupoid ∗-algebras.

The definition of a representation of a semigroupoid Λ given, and consequently of the ∗-algebra of Λ, here denoted (Λ), is strongly inuenced, and hence it is capable of smoothly dealing with the troubles usually caused by non-row- finiteness.

Speaking of another phenomenon that requires special attention in graph ∗- algebra theory, the presence of sources, once cast in the perspective of semigroupoids, becomes much easier to deal with.

To avoid confusion we use a different term and define a spring (rather than source) to be an element of a semigroupoid Λ for which g is not a legal multiplication for any g ∈ Λ. The sources of graph theory are much the same as our springs, and they cause the same sort of problems, but there are some subtle, albeit important differences. For example, in a category any element may be right- multiplied by the identity morphism on its domain, and hence categories never have any springs. On the other hand, even though higher-rank graphs are defined as categories, sources may still be present and require a special treatment.

While springs are irremediably killed when considered within the associated semigroupoid ∗-algebra, it is rather easy to get rid of them by replacing the given semigroupoid by a somewhat canonical spring-less. This is specially interesting because a slight correction performed on the ingredient semigroupoid is seen to avoid the need to redesign the whole theoretical apparatus.

As already mentioned, the ∗-algebra ∗(Λ) associated to a higher-rank graph (Λ, ) in turns out to be a special case of our construction: since Λ is defined to be a category, it is obviously a semigroupoid, so we may consider its semigroupoid ∗-algebra (Λ), which we prove to be isomorphic to ∗(Λ).

One of the most interesting aspects of this is that the construction of (Λ) does not use the dimension function ''" at all, relying exclusively on the algebraic structure of the subjacent category. In other words, this shows that the dimension function is superuous in the definition of ∗(Λ).

It should be stressed that our proof of the isomorphism between (Λ) and ∗(Λ) is done under the standing hypotheses, namely that (Λ, ) is row-finite and has no sources. We will not find here a comparison between our construction and the more recent treatment of Farthing, Muhly and Yeend for general finitely aligned higher rank graphs.

It is a consequence of Definition (1.1.9), describing our notion of a representation of a semigroupoid Λ, that if Λ contains elements , g and ℎ such that g = ℎ, then

= . Therefore, even if g and ℎ are different, that difference is blurred when these elements are seen in (Λ) via the universal representation. This should probably be interpreted as saying that our representation theory is not really well suited to deal with general semigroupoids in which non monic elements are present. An element is said to be monic if

g = ℎ ⟹ g = ℎ. Fortunately all of our examples consist of semigroupoids containing only monic elements.

No attempt has been made to consider topological semigroupoids although we believe this is a worthwhile program to be pursued. Among a few indications that this can be done is Katsura's topological graphs and Yeend's topological higher- rank graphs, not to mention Renault's pioneering work on groupoids.

After recognizing the precise obstruction for interpreting Cuntz-Krieger algebras from the point of view of categories or graphs, one can hardly help but to think of the obvious generalization of higher-rank graphs to semigroupoids based on the unique factorization property. Even though we do not do anything useful based on this concept we spell out the precise Definition. As an example, the Markov semigroupoid for the above 2 × 2 matrix is a rank 1 semigroupoid which is not a rank 1 graph.

We would also like to mention that although we have not seriously considered the ultra-graph ∗-algebras of Tomforde from a semigroupoid point of view, we believe that these may also be described in terms of naturally occurring semigroupoids.

In this section we introduce the basic algebraic ingredient of our construction.

Definition (1.1.1) [1]: A semigroupoid is a triple Λ, Λ(), . such that Λ is a set, Λ() is a subset of Λ × Λ, and

∙ ∶ Λ() → Λ is an operation which is associative in the following sense: if , g, ℎ ∈ Λ are such that either

(i) (, g) ∈ Λ() and (g, ℎ) ∈ Λ() or (ii) (, g) ∈ Λ() and (g, ℎ) ∈ Λ() or (iii) (g, ℎ) ∈ Λ() and (, gℎ) ∈ Λ(), then all of (, g), (g , ℎ), (g, ℎ) and (, gℎ) lie in Λ(), and

(g)ℎ = (gℎ). Moreover, for every ∈ Λ, we will let

Λ = g ∈ Λ: (, g) ∈ Λ(). From now on we fix a semigroupoid Λ.

Definition (1.1.2) [1]: Let f, g ∈ Λ. We shall say that divides g, or that g is a multiple of , in symbols |g, if either

(i) = g, or (ii) There exists ℎ ∈ Λ such that ℎ = g. When |g, and g|, we shall say that and g are equivalent, in symbols ≃ .

Perhaps the correct way to write up the above definition is to require that (, ℎ) ∈ Λ() before referring to the product ''ℎ''. However we will adopt the convention that, when a statement is made about a freshly introduced element which involves a multiplication, then the statement is implicitly supposed to include the requirement that the multiplication involved is allowed.

Notice that in the absence of anything resembling a unit in Λ, it is conceivable that for some element ∈ Λ there exists no ∈ Λ such that = . Had we not explicitly included (1.1.2) (i), it would not always be the case that |.

A useful artifice is to introduce a unit for Λ, that is, pick some element in the universe outside Λ, call it 1, and set Λ = Λ ∪̇ {1}. For every ∈ Λ put

1 = 1 = . Then, whenever |g, regardless of whether = g or not, there always exists ∈ Λ such that g = .

We will find it useful to extend the definition of Λ , for ∈ Λ by putting

Λ = Λ. Nevertheless, even if 1 is a meaningful product for every ∈ Λ, we will not include 1 in Λ.

We should be aware that Λ is not a semigroupoid. Otherwise, since 1 and 1g are meaningful products, axiom (1.1.1) (i) would imply that (1)g is also a meaningful product, but this is clearly not always the case.

It is interesting to understand the extent to which the associativity property fails for Λ. As already observed, (1.1.1) (i) does fail irremediably whe g = 1. Nevertheless it is easy to see that (1.1.1) generalizes to Λ in all other cases. This is quite useful, since when we are developing a computation, having arrived at an expression of the form (g)ℎ, and therefore having already checked that all products involved are meaningful, we most often want to proceed by writing

⋯ = (g)ℎ = (gℎ). The axiom to be invoked here is (1.1.1) (i) (or (1.1.1) (iii) in a similar situation), and fortunately not (1.1.1) (i)!

Proposition (1.1.3) [1]: Division is a reflexive and transitive relation.

Proof: That division is reflexive follows from the definition. In order to prove transitivity let , g ∈ Λ be such that |g and g|ℎ. We must prove that |ℎ.

The case in which = g, or g = ℎ is obvious. Otherwise there are , in Λ (rather than in Λ) such that = g, and g = ℎ. As observed above, it is implicit that (, ), (, ) ∈ Λ(), which implies that

(, ), (, ) ∈ Λ(). By (1.1.1) (ii) we deduce that (, ) ∈ Λ() and that

() = () = g = ℎ, and hence |ℎ.

Division is also invariant under multiplication on the left:

Proposition (1.1.4) [1]: If , , g ∈ Λ are such that |g, and (, ) ∈ Λ(), then (, g) ∈ Λ() and |g.

Proof: The case in which = g being obvious we assume that there is ∈ Λ such that = g. Since (, ), (, ) ∈ Λ() we conclude from (1.1.1) (i) that (, ) and (, ) = (, ) lie in Λ(), and that

() = () = g, so |g.

The next concept will be crucial to the analysis of the structure of semigroupoids.

Definition (1.1.5) [1]: Let , g ∈ Λ. We shall say that and g intersect if they admit a common multiple, that is, an element ∈ Λ such that | and g|. Otherwise we will say

8 that and g are disjoint. We shall indicate the fact that and g intersect by writing ⋒ g, and when they are disjoint we will write ⊥ .

If there exists a right-zero element, that is, an element 0 ∈ Λ such that (, 0) ∈ Λ() and 0 = 0, for all ∈ Λ, then obviously |0, and hence any two elements intersect. We shall be mostly interested in semigroupoids without a right-zero element.

Employing the unitization Λ notice that ⋒ g if and only if there are , ∈ Λ such that = g.

The last important concept, borrowed from the Theory of Categories, is as follows:

Definition (1.1.6) [1]: We shall say that an element ∈ Λ is monic if for every g, ℎ ∈ Λ we have

g = ℎ ⟹ g = ℎ. We would now like to discuss certain special properties of elements ∈ Λ for which Λ = . It would be sensible to call these elements sources, following the terminology adopted in Graph Theory, but given some subtle differences we'd rather use another term:

Definition (1.1.7) [1]: We will say that an element of a semigroupoid Λ is a spring when Λ = .

Springs are sometimes annoying, so we shall now discuss a way of getting rid of springs. Let us therefore fix a semigroupoid Λ which has springs.

Denote by Λ the subset of Λ formed by all springs and let be a set containing a distinct element , for every g ∈ Λ. Consider any equivalence relation '' ~ '' on according to which

~ , (1) for any spring g, and any such that g ∈ Λ. Observe that g is necessarily also a spring since Λ = Λ, by (1.1.1) (i)-(ii). For example, one can take the

9 equivalence relation according to which any two elements are related. Alternatively we could use the smallest equivalence relation satisfying (1).

We shall denote the quotient space ⁄~ by , and for every spring g we will denote the equivalence class of by . Unlike the , the are obviously no longer distinct elements. In particular we have

= , ∀ ∈ Λ, ∀ ∈ Λ ∩ Λ . We shall now construct a semigroupoid Γ as follows: set Γ = Λ⋃̇ , and put

() () Γ = Λ ⋃g, : g ∈ Λ⋃, : g ∈ Λ. Define the multiplication

∙∶ Γ() → Γ, to coincide with the multiplication of Λ when restricted to Λ(), and moreover set

g ⋅ = g and ⋅ = , ∀g ∈ Λ. It is rather tedious, but entirely elementary, to show that Γ is a semigroupoid without any springs containing Λ. To summarize the conclusions of this section we state the following:

Theorem (1.1.8) [1]: For any semigroupoid Λ there exists a spring-less semigroupoid Γ containing Λ.

Given a certain freedom in the choice of the equivalence relation ''~'' above, there seems not to be a canonical way to embed Λ in a spring-less semigroupoid. The user might therefore have to make a case by case choice according to his or her preference.

In this section we begin the study of the central notion bridging semigroupoids and operator algebras.

Definition (1.1.9) [1]: Let Λ be a semigroupoid and let be a unital ∗-algebra. A mapping ∶ Λ → will be called a representation of Λ in , if for every g ∈ Λ, one has that:

(i) is a partial isometry,

() if (, g) ∈ Λ (ii) = 0, otherwise. ∗ ∗ Moreover the initial projections = , and the final projections = , are required to commute amongst themselves and to satisfy

(iii) = 0, if ⊥ g, () (iv) = , if (, g) ∈ Λ . () ∗ ∗ Notice that if (, g) ∉ Λ , then = = 0; by (ii). Complementing (iv) above we could therefore add:

() (v) = 0, if (, g) ∉ Λ . We will automatically extend any representation to the unitization Λ by setting = 1. Likewise we put = = 1. Notice that in case Λ contains an element which is not monic, say g = ℎ, for a pair of distinct elements g, ℎ ∈ Λ, one necessarily has = , for every representation . In fact

∗ ∗ ∗ ∗ ∗ = = = = = = ∗ and similarly = , so it follows that = , as claimed.

This should probably be interpreted as saying that our representation theory is not really well suited to deal with general semigroupoids in which non monic elements are present. In fact, all of our examples consist of semigroupoids containing only monic elements.

From now on we will fix a representation of a given semigroupoid Λ in a ∗ unital -algebra . By (1.1.9) (iv) we have that ≤ , for all ℎ ∈ Λ , so if ℎ, ℎ ∈ Λ we deduce that

⋁ ∶= + − ≤ . More generally, if is a finite subset of Λ we will have

≤ . ∈ We now wish to discuss whether or not the above inequality becomes an identity under circumstances which we now make explicit:

Definition (1.1.10) [1]: Let be any subset of Λ. A subset ⊆ will be called a covering of if for every ∈ there exists ℎ ∈ such that ℎ ⋒ . If moreover the elements of are mutually disjoint then will be called a partition of .

The following elementary fact is noted for further reference:

Proposition (1.1.11) [1]: A subset ⊆ is a partition of if and only if is a maximal subset of consisting of pairwise disjoint elements.

Returning to our discussion above we wish to require that

? , (2) ∈ whenever is a covering of Λ . The trouble with this equation is that when is infinite there is no reasonable topology available on under which one can make sense of the supremum of infinitely many commuting projections.

Before we try to attach any sense to (2) notice that if g ∈ Λ and ℎ ∈ Λ⁄Λ, then

≤ 1 − , by (1.1.9) (v), and hence also

≤ 1 − , ∈ for every finite set ⊆ Λ⁄Λ. More generally, given finite subsets , ⊆ Λ, denote

Λ, = Λ ⋂ Λ⁄Λ, ∈ ∈

12 and let ℎ ∈ Λ,. By (1.1.9) (iv)-(v), we have that

≤ 1 − . ∈ ∈ As in the above cases we deduce that

≤ 1 − , ∈ ∈ ∈ for every finite subset ⊆ Λ,.

Definition (1.1.12) [1]: A representation S of Λ in a unital ∗-algebra is said to be tight if for every finite subsets , ⊆ Λ, and for every finite covering of Λ, one has that

= 1 − , ∈ ∈ ∈ Observe that if for every pair of finite subset , ⊆ Λ, one has that Λ, admits no finite covering then any representation is tight by default.

For many representation theories there is a ∗-algebra whose representations are in one-to-one correspondence with the representations in the given theory. Semigroupoid representations are no exception:

Definition (1.1.13) [1]: Given a semigroupoid Λ we shall let (Λ) be the universal unital ∗-algebra generated by a family of partial isometries subject to the relations that the ∈ correspondence ⟼ is a tight representation of Λ. That representation will be called the universal representation and the closed ∗ -subalgebra of (Λ) generated by its range will be denoted (Λ).

It is clear that (Λ) is either equal to (Λ) or to its unitization. Observe also that the relations we are referring to in the above definition are all expressable in

13 the form described. Moreover these relations are admissible, since any partial isometry has norm one. It therefore follows that (Λ) exists.

The universal property of (Λ) may be expressed as follows:

Proposition (1.1.14) [1]: For every tight representation of Λ in a unital ∗-algebra B there exists a unique ∗ -homomorphism ∶ (Λ) → , such that φ(S) = T, for every ∈ Λ. It might also be interesting to define a ''Toeplitz" extension of (Λ), as the universal unital ∗-algebra generated by a family of partial isometries ∈ subject to the relations that the correspondence ⟼ is a (not necessarily tight) representation of Λ. If such an algebra is denoted (Λ), it is immediate that (Λ) is a quotient of (Λ).

As already observed the usefulness of these constructions is probably limited to the case in which every element of Λ is monic.

Tight representations and springs do not go together well, as explained below:

Proposition (1.1.15) [1]: Let be a tight representation of a semigroupoid Λ and let ∈ Λ be a spring (as defined in (1.1.7)). Then = 0.

Proof: Under the assumption that Λ = , notice that the empty set is a covering of ∗ Λ and hence = 0, by (1.1.12). Since = , one has that = 0, as well.

We thus see that springs do not play any role with respect to tight representations. There are in fact some other non-spring elements on which every tight representation vanishes. Consider for instance the situation in which Λ consists of a finite number of elements, say Λ = {ℎ, … , ℎ}, each ℎ being a spring. Then Λis a finite cover of itself and hence by (1.1.12) we have

= = 0, which clearly implies that = 0.

One might feel tempted to redesign the whole concept of tight representations especially if one is bothered by the fact that springs are killed by them. However we strongly feel that the right thing to do is to redesign the semigroupoid instead, using (1.1.8) to replace Λ by a spring-less semigroupoid containing it.

In this case it might be useful to understand the following situation:

Proposition (1.1.16) [1]: Let be a tight representation of a semigroupoid Λ and suppose that ∈ Λ is such that Λ contains a single element such that = . Then is a projection and moreover = .

Proof: Since = , we have that = . But since is also a partial isometry, it must necessarily be a projection. By assumption we have that {} is a finite covering for Λ so

∗ = = = . With this in mind we will occasionally work under the assumption that our semigroupoid has no springs. Section (1.2): Markov Semigroupoid with Categories and Higher- rank Graphs

In this section we present a semigroupoid whose ∗-algebra which is isomorphic to the Cuntz-Krieger algebra introduced. For this let be any set and let = {(, )},∈ be an arbitrary matrix with entries in {0,1}. We consider the set Λ = Λ of all finite admissible words

= … ,

15 i.e., finite sequences of elements ∈ , such that (, ) = 1. Even though it is sometimes interesting to consider the empty word as valid, we shall not do so. If allowed, the empty word would duplicate the role of the extra element 1 ∈ Λ. Our words are therefore assumed to have strictly positive length ( ≥ 1).

Given another admissible word, say = … , the concatenated word

∶= … … is admissible as long as (, ) = 1. Thus, if we set

() Λ = {(, ) = ( … , … ) ∈ Λ × Λ: (, ) = 1}, we get a semigroupoid with concatenation as product.

Definition (2.1.1) [1]:

The semigroupoid Λ = Λ defined above will be called the Markov semigroupoid.

Observe that the springs in Λ are precisely the words = ( … ) for which (, ) = 0, for every ∈ , that is, for which the th row of is zero. To avoid springs we will assume that no row of is zero.

Theorem (1.2.2) [1]:

Suppose that has no zero rows. Then (Λ) is ∗ -isomorphic to the Exel-Laca algebra . Proof: Throughout this proof we will denote the standard generators of by , ∈ ∗ ∗ together with their initial and final projections = and = , respectively. Likewise the standard generators of (Λ) will be denoted by , ∈ ∗ ∗ along with their initial and final projections = and = . In addition, for every ∈ we will identify the one-letter word ''" with the element itself, so we may think of as a subset of Λ.

We begin by claiming that the set of partial isometries

⊆ (Λ) ∈ satisfies the defining relations of , namely TCK, TCK, and TCK.

Conditions TCK and TCK follow immediately from (1.1.9), and the observation that if and are distinct elements of , then ⊥ as elements of Λ.

() When (, ) = 1 we have that (, ) ∈ Λ and hence = = (, ), by (1.1.9) (iv). Otherwise, if (, ) = 0, we have that (, ) ∉ Λ() and hence

= 0 = (, ), by (1.1.9) (v). This proves TCK.

In order let , be finite subsets of such that

(, , ) ∶= (, ) 1 − (, ) (3) ∈ ∈ equals zero for all but finitely many 's. It is then easy to see that

∶= { ∈ : (, , ) ≠ 0} is a finite partition of Λ,, so

1 − = = (, , ) , ∈ ∈ ∈ ∈ ∈ because the canonical representation ∈ Λ ⟼ ∈ (Λ) is tight by definition. It then follows from the universal property of that there exists a ∗ homomorphism

∶ → (Λ), (4)

such that = , for every ∈ .

Next consider the map ∶ Λ → defined as follows: given ∈ Λ, write … , with ∈ , and put = … ,

We claim that is a tight representation of Λ in . The first two axioms of (1.1.9) are immediate, while the commutativity of the , and follow. Next suppose that , ∈ Λ are such that ⊥ . One may then prove that

= … … and = … … , with 1 ≤ ≤ , , and such that = for < , and ≠ . Denoting by

= … (possibly the empty word), we have that ∗ ∗ ∗ ≤ = , ∗ and similarly ≤ . It follows that

∗ ∗ ∗ ∗ ≤ = = = 0, Hence proving (1.1.9) (iii). In order to verify (1.1.9) (iv) let (, ) ∈ Λ(), so that

(, ) = 1, where n is the length of . As shown in ''Claim (1)", we have that = , so = = = , where we have used TCK in the second equality.

We are then left with the task of proving to be tight. For this let and be finite subsets of Λ and let be a finite covering of Λ,. We must prove that

= 1 − . (5) ∈ ∈ ∈

Using TCK it is easy to check the inequality ''≤" in (5) so it suffices to verify the opposite inequality.

Let h, h ∈ Z be such that h ⋒ h, and write hx = hx, where xx ∈ Λ. Assuming that the length of ℎ does not exceed that of ℎ, one sees that ℎ is an , initial segment of ℎ, and hence ℎ|ℎ. Any element of Λ which intersects ℎ must therefore also intersect ℎ. This said we see that ∶= ⁄{ℎ} is also a covering of Λ,. Since the left-hand-side of (5) decreases upon replacing by , it is clearly enough to prove the remaining inequality ''≥" with in place of .

Proceeding in such a way every time we find pairs of intersecting elements in we may then suppose that consists of pairwise disjoint elements, and hence that is a partition.

Given ∈ Λ, write = … , with ∈ , and observe that = , as already mentioned. Since Λ = Λ, as well, we may assume without loss of generality that and consist of words of length one, or equivalently that , ∈ . Let

= { ∈ : (, , ) ≠ 0}, where (, , ) is as in (5). Notice that ∈ if and only if (, ) = 1, and (, ) = 0, for all ∈ and ∈ , which is precisely to say that ∈ Λ,. In other words

= Λ, ∩ . It is clear that shares with the property of being maximal among the subsets of pairwise disjoint elements of Λ, (see (1.1.11)).

Suppose for the moment that is formed by words of length one, i.e, that ⊆ . Then ⊆ , and so = , by maximality. This implies that is finite and

= = = (, , ) = 1 − , ∈ ∈ ∈ ∈ ∈ ∈ Thus proving (5). Addressing the situation in which is not necessarily contained in , let

= { ∈ ∶ = }, ∀ ∈ . Since ⊆ Λ, it is evident that

= . ∈

Moreover notice that each is nonempty since otherwise ∪ {} will be a subset of Λ, formed by mutually disjoint elements, contradicting the maximality of . In particular this shows that is finite and hence, so that

1 − = (, , ) (6) ∈ ∈ ∈ ∈ We claim that for every ∈ one has that

= .

∈ Before proving the claim lets us notice that it does implies our goal, for then

() = = 1 − ,

∈ ∈ ∈ ∈ ∈ ∈ proving (5).

Noticing that each is maximal among subsets of mutually disjoint elements beginning in , the claim follows from the following:

Lemma (1.2.3) [1]:

Given ∈ , let Λ() = { ∈ Λ: = }, and let be a finite partition of Λ(). Then

= . ∈ Proof: Let be the maximum length of the elements of . We will prove the statement by induction on . If = 1 it is clear that = {} and the conclusion follows by obvious reasons. Supposing that > 1 observe that ∉ , or else any element in with length will intersect , violating the hypothesis that consists of mutually disjoint elements. Therefore every element of has length at least two.

Let = { ∈ : (, ) = 1} and set = { ∈ : = }. It is clear that

= . ∈

Moreover notice that every is nonempty, since otherwise ∪ {} consists of mutually disjoint elements and properly contains , contradicting maximality. In particular this implies that is finite and hence we have

= (, , ) = . (7) ∈ ∈

For every ∈ , let be the set obtained by deleting the first letter from all words in , so that ⊆ Λ(), and = . One moment of reffexion will convince that is maximal among the subsets of mutually disjoint elements of Λ(). Since the maximum length of elements in is no bigger than − 1, we may use induction to conclude that

= . ∈ Therefore

∗ ∗ ∗ () ∗ ∗ = = = = ∈ ∈ ∈ ∈ ∈

= = .

∈ ∈ ∈ Returning to the proof of (1.2.2), now in possession of the information that is a tight representation of Λ, we conclude by the universal property of (Λ) that there exists a ∗ -homomorphism

: (Λ) → , such that = , for all ∈ Λ. It is then clear that is the inverse of the homomorphism of (4), and hence both Φ and are isomorphisms.

In this section we fix a small category Λ. Notice that the collection of all morphisms of Λ (which we identify with Λ itself) is a semigroupoid under composition. We shall now study Λ from the point of view of the theory introduced in the previous section.

Given ∈ obj(Λ) (meaning the set of objects of Λ) we will identify with the identity morphism on , so that we will see obj(Λ) as a subset of the set of all morphisms.

Given ∈ Λ we will denote by () and () the domain and co-domain of , respectively. Thus the set of all composable pairs may be described as

Λ() = {(, g) ∈ Λ × Λ ∶ () = (g)}. Given ∈ Λ notice that Λ = {g ∈ Λ ∶ () = (g)}. In particular, if ∈ obj(Λ) then () = () = , so

Λ = {g ∈ Λ ∶ (g) = }. A category is a special sort of semigroupoid in several ways. For example, if () , , g and g ∈ Λ are such that (, g) ∈ Λ for all , , except perhaps for () (, ) = (2,2), then necessarily (, g) ∈ Λ , because

() = (g) = () = (g). Another special property of a category among semigroupoids is the fact that for every ∈ Λ there exists ∈ Λ such that = , namely one may take to be (the identity on) (). Thus | even if we had omitted (1.1.2) (i) in the definition of division. Clearly this also implies that Λ has no springs.

From now on we fix a representation of Λ in a unital ∗-algebra and denote by and , the initial and final projections of each , respectively. A few elementary facts are in order:

Proposition (1.2.4) [1]:

(i) For every ∈ obj(Λ) one has that is a projection, and hence

= = .

(ii) If and are distinct objects then ⊥ .

(iii) For every ∈ Λ one has that = ().

Proof: We leave the elementary proof of (i). Given distinct objects and it is clear that ⊥ , so ⊥ , by (1.1.9) (iii). With respect to (iii) we have

∗ ∗ ∗ = = () = () = () = (), where the last equality follows from (1.1.9) (iv).

Definition (1.2.5) [1]: Let be a Hilbert space and let : Λ → ℬ() be a representation. We will say that Λ is nondegenerated if the closed ∗ -subalgebra of ℬ() generated by the range of is nondegenerated.

Nondegenerated Hilbert space representations are partly tight in the following sense:

Proposition (1.2.6) [1]: Let : Λ → ℬ() be a representation. If either

(i) is nondegenerated, or (ii) obj(Λ) is infinite, then for every finite subsets , ⊆ Λ such that Λ, = , one has that

1 − = 0. ∈ ∈

Proof: Notice that Λ, = ϕ, implies that

Λ ⊆ Λ\ Λ\Λ = Λ . (8) ∈ ∈ ∈

Case (1):

Assuming that F ≠ , let ∈ F. Then either there is some ∈ F, with () ≠ (), in which case

= ()() = 0, proving the statement; or () = (), for all ∈ . Therefore we may suppose that () belongs to Λ for every ∈ , and hence by (8) there exists g ∈ such that () ∈ Λ . But this is only possible if () = (g) and hence

1 − = ()1 − () = 0,

23 concluding the proof in case (1).

Case (2): Assuming next that , we claim that

obj(Λ) = {(g): g ∈ }. In fact, arguing as in (8) one has that ⋃∈ Λ = Λ, so for every ∈ obj(Λ) there exists g in such that ∈ Λ, whence = (g), proving our claim.

Under the assumption that obj(Λ) is infinite we have reached a contradiction, meaning that case (2) is impossible and the proof is concluded. We thus proceed supposing nondegeneracy. Let

= 1 − , ∈ so, proving the statement is equivalent to proving that = 0. Given ∈ obj(Λ), let g ∈ be such that = (g). Then

= () = , from where we deduce that

= 1 − = 0. Given any ∈ Λ we then have that

∗ ∗ = () = 0 and = () = 0, so = 0, by nondegeneracy.

We next present a greatly simplified way to check that a representation of Λ is tight.

Proposition (1.2.7) [1]: Given a representation : Λ → ℬ(), consider the following two statements:

a. is right.

b. For every ∈ obj(Λ) and every finite covering of Λ one has that

⋁∈ = . Then

(i) (a) implies (b). (ii) If is nondegenerated,

Proof:

(i) Assume that is tight and that is a finite covering of Λ. Setting = {} and = , notice that Λ, = Λ, so is a finite covering of Λ,, and hence we have by definition that

= 1 − = = . ∈ ∈ ∈

(ii) Assuming nondegenerated, or obj(Λ) infinite, we next prove that (b) implies (a). So let and be finite subsets of Λ and let be a finite covering of Λ,. We must prove that the identity in (1.1.12) holds. If Λ, = , the conclusion follows from (1.2.6). So we assume that Λ, ≠ .

Case (1): ≠ : Pick ℎ ∈ Λ, and notice that for every ∈ one has that () = (ℎ), and for every g ∈ , it is the case that (g) ≠ (ℎ). It therefore follows that

Λ, = Λ, where = (ℎ), so is in fact a covering of Λ. By hypothesis we then have that

= . (9) ∈ On the other hand observe that for every g ∈ , we have that

(..)() () ⊥ , given that (g) ≠ . Noticing that for ∈ , we have = () = , we deduce that

() 1 − = , ∈ ∈ ∈ proving that the identity in (1.1.12) indeed holds in case ≠ .

Case (2): = : Let

= obj(Λ)\{(g): g ∈ }, so that

Λ, = Λ. ∈ Given that is a finite covering of Λ,, we have that for each ∈ there exists ℎ ∈ such that ⋒ ℎ, which in turn implies that (ℎ) = . Therefore is finite and hence so is obj(Λ).

Thus, case (2) is impossible under the hypothesis that obj(Λ) is infinite, and hence the proof is finished under that hypothesis. We therefore proceed supposing nondegeneracy. It is then easy to show that

= 1, ∈() and hence

(..)() 1 − () = . (10) ∈ ∈ ∈ By assumption is contained in Λ,, and hence the range of each ℎ ∈ belongs to . Thus

= , ∈ where = {ℎ ∈ : (ℎ) = }. Observe that is a covering for Λ , since if ∈ Λ, there exists some ℎ ∈ with ℎ ⋒ , but this implies that (ℎ) = () =

, and hence ℎ ∈ . Thus

() = = = 1 − .

∈ ∈ ∈ ∈ ∈ ∈ We shall now apply the conclusions above to show that higher-rank graph ∗- algebras may be seen as special cases of our construction. See the definitions and a detailed treatment of higher-rank graph ∗-algebras.

Before we embark on the study of -graphs from the point of view of semigroupoids let us propose a generalization of the notion of higher-rank graphs to semigroupoids which are not necessarily categories. We will not draw any conclusions based on this notion, limiting ourselves to note that it is a natural extension of Kumjian and Pask's interesting idea.

Definition (1.2.9) [1]: Let be a natural number. A rank semigroupoid, or a -semigroupoid, is a pair (Λ, ), where Λ is a semigroupoid and

: Λ → , is a function such that

(i) For every (, g) ∈ Λ(), one has that (g) = () + (g), (ii) If ∈ Λ, and , ∈ are such that () = + , there exists a unique pair (g, ℎ) ∈ Λ() such that (g) = , (ℎ) = and gℎ = . For example, the Markov semigroupoid is a 1-semigroupoid, if equipped with the word length function.

Let (Λ, ) be a -graph. In particular Λ is a category and hence a semigroupoid. Under suitable hypothesis we shall now prove that the ∗-algebra of the subjacent

27 semigroupoid is isomorphic to the ∗-algebra of Λ, as defined by Kumjian and Pask. In particular it will follow that the dimension function is superuous for the definition of the corresponding ∗-algebra.

As before, if ∈ obj(Λ) we will denote by Λ the set of elements ∈ Λ for which () = . For every ∈ we will moreover let

Λ = { ∈ Λ : () = }. We should observe that Λ is denoted Λ ().

According, Λ is said to have no sources if Λ is never empty. In case Λ is finite for every and one says that Λ is row-finite.

Notice that the absence of sources is a much more stringent condition than to require that Λ has no springs, according to Definition (1.1.7). In fact, since Λ is a category, and hence () ∈ Λ, for every ∈ Λ, we see that Λ ≠ , and hence higher-rank graphs automatically have no springs!

Below we will work under the standing hypotheses, but we note that our construction is meaningful regardless of these requirements, so it would be interesting to compare our construction where these hypotheses are not required. This said, we suppose throughout that Λ is a -graph for which

0 < |Λ| < ∞, ∀ ∈ obj(Λ), ∀ ∈ . (11) Lemma (1.2.10) [1]: For every object of Λ and every ∈ one has that Λ is partition of Λ . Proof: Suppose that , g ∈ Λ are such that ⋒ g. So there are , ∈ Λ such that = g. Since () = = (g) we have that = , by the uniqueness of the factorization. This shows that the elements of Λ are pairwise disjoint.

In order to show that Λ is a covering of Λ , let g ∈ Λ . By (11) pick any () ℎ ∈ Λ . Since (gℎ) = (g) + (ℎ) = (g) + = + (g),

we may write gℎ = , with () = , and () = (g). It follows that ∈ Λ and g ⋒ . We now reproduce the definition of the ∗-algebra of a -graph.

Definition (1.2.11) [1]: Given a -graph Λ satisfying (11), the ∗-algebra of Λ, denoted by ∗(Λ), is ∗ defined to be the universal -algebra generated by a family : ∈ Λ of partial isometries satisfying:

(i) { ∶ ∈ obj(Λ)} is a family of mutually orthogonal projections,

(ii) = for all , g ∈ Λ such that () = (g), ∗ (iii) = () for all ∈ Λ, ∗ (iv) For every object and every one has ∑ . ∈ = ∈ The following is certainly well known to specialists in higher-rank graph ∗- algebras:

Proposition (1.2.12) [1]: For every and g in Λ one has that

∗ (i) If ⊥ g then , ∗ ∗ (ii) commutes with .

Proof: Recall that whenever ∈ is such that (), () ≤ , we have

∗ ∗ = , where the sum extends over all pair (, ) of elements in Λ such that = g, and () = . So

∗ ∗ ∗ ∗ ∗ = = . , ,

Since the last expression is symmetric with respect to and g, we see that (ii) is proved. Moreover, when ⊥ g, it is clear that there exist no pairs (, ) for which = g, and hence (i) is proved as well.

We shall now prove that the crucial axiom (1.2.11) (iv) generalizes to coverings:

Lemma (1.2.13) [1]: Let be an object of Λ. If is a finite covering of Λ then

∗ = . ∈ Proof: Let ∈ with ≥ (ℎ), for every ℎ ∈ . For all ∈ Λ we know that there is some ℎ ∈ such that ⋒ ℎ, so we may write = ℎ, for suitable and .

Since () = ≥ (ℎ), we may write = , with () = (ℎ). Noticing that

= ℎ, we deduce from the unique factorization property that = ℎ, which amounts to ∗ ∗ saying that ℎ|. We claim that this implies that ≤ . In fact

∗ ∗ = ∗ ∗ = ∗ = ∗. Summarizing, we have proved that for every ∈ Λ, there exists ℎ ∈ , such that ∗ ∗ ≤ . Therefore

(..)() ∗ ∗ ∗ = ≤ ≤ , ∈ ∈ ∈ from where the conclusion follows.

Theorem (1.2.14) [1]: ∗ If Λ is a -graph satisfying (11) then (Λ) is ∗ -isomorphic to (Λ). Proof: Throughout this proof we denote the standard generators of ∗(Λ) by , ∈ together with their initial and final projections and , respectively. Meanwhile the standard generators of (Λ) will be denoted by , along with their initial ∈ and final projections and . In particular the are known to satisfy (1.2.11)

(i)-(iv), while the are known to give a tight representation of the semigroupoid Λ.

Working within the semigroupoid ∗-algebra (Λ), we begin by arguing that the also satisfy (1.2.11) (i)-(iv). In fact (1.2.11) (i) follows from (1.2.4) (i)-(ii), while (1.2.11) (ii) is a consequence of (1.1.9) (ii). With respect to (1.2.11) (iii) it was proved in (1.2.4) (iii). Finally (1.2.11) (iv) results from the combination of (1.2.10) and (1.2.7) (i).

Therefore, by the universal property of ∗(Λ), there exists a ∗-homomorphism

: ∗(Λ) → (Λ), such that = , for every ∈ Λ. Evidently the range of Λ is contained in the closed ∗ -subalgebra of (Λ) generated by the , also known as (Λ).

We next move our focus to the higher-rank graph algebra ∗(Λ), and prove that the correspondence

∗ ∈ Λ ⟼ ∈ (Λ) is a tight representation. Skipping the obvious (1.1.9) (i) we notice that (1.1.9) (ii) follows from (1.2.11) (ii) when () = (g). On the other hand, if () ≠ (g) we have

= ()() = 0, by (1.2.11) (i).

We next claim that the initial and final projections of the commute among themselves. That two initial projections commute follows from (1.2.11) (i) and

(iii). Speaking of the commutativity between an initial projection and a final projection , we have that = (), by (1.2.11) (iii) and ≤ () by (1.2.11)

(iv). So either ≤ , if ()(g), or ⊥ , if () ≠ (g), by (1.2.11) (i). In any case it is clear that and commute. That two final projections commute is precisely the content of (1.2.12) (ii).

Clearly (1.1.9) (iii) is granted by (1.2.12) (i). In order to prove (1.1.9) (iv) let , g ∈ Λ with () = (g). We then have that

∗ ∗ ∗ = = () = () = = . This shows that is a representation of Λ in ∗(Λ), which we will now prove to be tight. For this let

∶ ∗(Λ) ⟶ ℬ() be a faithful nondegenerated representation of ∗(Λ). Through we will view ∗(Λ) as a subalgebra of ℬ(), and hence we may consider as a representation of Λ on . It is clear that is nondegenerated, according to definition (1.2.5). By (1.2.7) (ii) it is then enough to show that for every ∈ obj(Λ) and every finite partition of one has that

= , ∈ but this is precisely what was proved in (1.2.13).

By the universal property of (Λ) there is a ∗-homomorphism

Ψ ∶ (Λ) → () ∗ such that Ψ = , for every ∈ Λ. Clearly Ψ(Λ) ⊆ (Λ), so we may then view Ψ and as maps

Ψ ∶ (Λ) → ∗(Λ) and Φ ∶ ∗(Λ) → (Λ), which are obviously each others inverses.

Chapter 2 ∗-Algebras and Tracially -Absorbing In this chapter we show that if is a ∗-algebra for which the a tracial notion of Ƶ-absorption for simple holds then has almost unperforated Cuntz semigroup, and if in addition is nuclear and separable we show this property is equivalent to having ≅ ⨂Ƶ. We furthermore show that this property is preserved under forming certain crossed products by actions satisfying a tracial Rokhlin type property [2]. Section (2.1): Tracially -Absorbing ∗-Algebras and Absorption in The Unclear Case The purpose of this section is to introduce and study a property of ∗-algebras which can be thought of as a tracial version of -absorption, in a way reminiscent of the definition of tracially algebras and ∗-algebras of higher tracial rank. There are several motivations for looking at this property. A property closely related to ours (the two coincide in certain cases) appeared as a technical step in Winter’s work concerning -absorption for ∗-algebras of finite nuclear dimension and the recent work of Matui-Sato on strict comparison and -absorption, and it may thus be profitable to isolate this property for further study. Additionally, this property may be easier to establish in certain instances, in particular when considering crossed products by an action which satisfies a tracial version of the Rokhlin property. Indeed, we study a generalization of the tracial Rokhlin property which does not require projections and show that under certain conditions, crossed products of tracially -absorbing ∗-algebras by such actions are again tracially Z-absorbing. The Rokhlin property has been instrumental in the study of group actions on ∗-algebras. However, the Rokhlin property might be harder to establish in certain cases, and might not exist in others. In the finite group case, the Rokhlin property is uncommon and its existence requires restrictive theoretic constraints on the algebra and the action. While for the single automorphism case the Rokhlin property is much more common (and even generic in certain cases), it still requires the existence of many projections. The Rokhlin property has been generalized to a tracial version, although this generalization still requires the existence of projections. Further tracial-type generalizations in which

33 the projections are replaced by positive elements were considered, and we will study a closely related variant. We note that a different generalization which does not require projections has been to view the Rokhlin property as a zero- dimensional level of what can be called Rokhlin dimension.

A unital ∗-algebra is -absorbing if and only if for any ∈ ℕ, any finite subset ⊆ and for any > 0 there exist c.p.c. order zero maps ∶ → and ∶ ∶ → such that the image of any normalized matrix under or commutes up to with the elements of , and such that (,)(,) = (,) and (,) = 1 − (1). In this characterization, 1 − (1) must be “small” in a tracial sense. We introduce a property called tracial -absorption which basically amounts to dropping the requirement for an order zero map from into and instead asking that 1 − (1) is arbitrarily small in the sense of Cuntz comparison. We show that tracially -absorbing ∗-algebras have almost unperforated Cuntz semigroups. This generalizes results of Rørdam concerning -absorption. We then use ideas to show that tracial -absorption implies absorption for simple, unital, separable, nuclear ∗-algebras.

Recall that a c.p.c. map ∶ → ℬ is said to have order zero if ()() = 0 whenever , ∈ satisfy = 0. For positive elements a, ∈ we say that a is Cuntz-subequivalent to (written ≾ ) if there is a sequence ∈ such ∗ that lim→‖ − ‖ = 0. Definition (2.1.1) [2]: We say that a unital ∗-algebra is tracially -absorbing if ≇ ℂ and for any finite set ⊂ , > 0 and non-zero positive element ∈ and ∈ ℕ there is an order zero contraction ∶ → such that the following hold: (I) 1 − (1) ≾ .

(II) For any normalized element ∈ and any ∈ we have ‖(), ‖ < .

Proposition (2.1.2) [2]: Let be a simple unital ∗-algebra. If is -absorbing then is tracially - absorbing.

Proof: First assume is stably finite. We know that has strict comparison. Let

, , , be given. Since is simple there exist ℓ ∈ and , … , ℓ ∈ such that

ℓ ∗ = 1.

This clearly implies that () ≥ 1⁄ℓ for all ∈ (). Let be some number such that > and n divides . Since is -absorbing there is a unital homomorphism ∶ , → such that for any normalized element ∈ , and for any ∈ we have

‖(), ‖ < .

We have a c.p.c. order zero map ∶ → ∈ , such that

1 − (1) ≾ ,.

Let = ∘ ∶ → . For any ∈ () we have that

1 − (1) ≤ , ≤ 1⁄ ≤ () which entails that 1 − (1) ≾ . Since embeds unitally into we restrict to obtain the map we are looking for.

If is not stably finite then, is purely infinite. Set = and continue as before. The condition 1 − (1) ≾ is automatically satisfied.

In what follows we identify () with ⊗ in the usual way. Lemma (2.1.3) [2]: Let be a simple, unital, non type ∗-algebra and let ∈ ℕ. For every non- zero positive element ∈ () there exists a non-zero positive element ∈ such that ≿ 1 ⊗ .

Proof:

Since a is positive and non-zero, there exists such that ( ⊗ 1)( ⊗ 1) = 0. Assume without loss of generality that = 1. Furthermore, by replacing a with the element , ⊗ 1, ⊗ 1 we may assume that a is of the form , ⊗ for some non-zero positive ∈ .

Note that both and have a strictly positive element and hence, by Brown’s Theorem, we have that is stably isomorphic to , this implies that the former is a simple, infinite dimensional, ∗-algebra not isomorphic to the compact operators. In particular is non-type . Therefore, there is a non-zero homomorphism ∶ → . Let ∈ (0,1] denote the identity function.

Using the picture = (0,1] ⊗ we denote ℎ = ( ⊗ 1) and =

( ⊗ ,). Observe that , ⊗ ℎ ∼ 1 ⊗ in (). Consequently we have that ≿ 1 ⊗ as needed.

Lemma (2.1.4) [2]: Let be a simple unital ∗-algebra. If is tracially -absorbing then so is () for any .

Proof:

It is clear that () ≅ ⊗ satisfies the conditions of Definition (2.1.1) provided the positive element a is of the form 1 ⊗ , ∈ . Thus, it suffices to show that every non-zero positive element ∈ () Cuntz-dominates a non- zero positive element of the form 1 ⊗ . But this follows from the previous lemma.

Notation (2.1.5) [2]: Let be a separable ∗-algebra. We denote

= ℕ ℕ

We view as embedded into as equivalence classes of constant sequences and we denote by

∩ the relative commutant of in . The following result can help simplify proofs.

Lemma (2.1.6) [2]: Let be a separable, unital ∗-algebra. If is tracially -absorbing then for any ∈ ℕ and any non-zero positive contraction ∈ there exists a c.p.c. order zero map ∶ → ∩ such that 1 − (1) ≾ in . Proof:

Let ()∈ℕ ⊆ be a dense sequence and denote = {, . . . , }. For each ∈ ℕ find a c.p.c. order zero map ∶ → such that 1 − (1) ≾ and ‖[(), ]‖ < for all normalized ∈ and for all ∈ . Let be the composition of the map

()∈ℕ ∶ → ℕ with the quotient map into . It is easy to see that has the desired properties. The following is a restatement along with some additional notation:

Notation (2.1.7) [2]: Let ∶ → be an order zero c.p.c. map, where is finite dimensional. Denote ℎ = (1). Recall that there is a homomorphism ∶ → ∗∗ ∩ {ℎ} such that () = ()ℎ where ℎ = (1).

If ∈ (0,1] , we denote by [] the c.p. order zero map given by

[]() = ()(ℎ). If is the square root function, we may use the notation [] for [].

For a c.p.c. map : → from a finite dimensional algebra , and an element ∈ , we write

‖[, ]‖ < to mean ‖(), ‖ < for all normalized elements ∈ . If ⊆ is some subset, we shall use the notation

‖[, ]‖ < to mean ‖, ‖ < for all ∈ .

We have

Lemma (2.1.8) [2]: For any ∈ (0,1] and any > 0 there is an > 0 such that whenever : → is a c.p.c. order zero map and b is a contraction satisfying ‖, ‖ < , we have that ‖[], ‖ < .

Proof:

Let and as above be given. We can find ∈ (0,1] such that ‖ − ‖ < where is the identity function on (0,1]. Find > 0 such that ‖[, ]‖ < implies ‖[(), ]‖ < for any normalized elements , ∈ with a positive (this is easily done by approximating uniformly by polynomials). We may of course assume that < . Let ∶ → be a c.p.c. order zero map and ∈ a contraction such that ‖, ‖ < . Let , ℎ be as in the notation above. For any normalized ∈ we have the following: []() = ()(ℎ)

≈ ⁄3 ()ℎ(ℎ)

= ()(ℎ)

≈ ⁄3 ()(ℎ)

≈ ⁄3 []().

We will also be needing a slight variation of this previous lemma. We omit the proof since it is essentially the same as that of the preceding lemma.

Lemma (2.1.9) [2]: For any ∈ (0,1] and any > 0 there is an > 0 such that whenever ∶ → is a c.p.c. order zero map and ∈ Aut() is an automorphism satisfying () − () < for all normalized elements ∈ , we have that []() − []() < for all normalized ∈ .

In this section we will prove that tracially -absorbing ∗-algebras have almost unperforated Cuntz semigroup and therefore strict comparison. The proof mixes ideas with ideas originating from the study of tracially algebras.

Recall that a positive element ∈ is called purely positive if a is not Cuntz- equivalent to a projection. This is equivalent to saying that 0 is an accumulation point of ().

Lemma (2.1.10) [2]: Let be a simple non type ∗-algebra. For any projection ∈ and > 0 there is a purely positive element ≤ such that ( − 1)〈〉 ≤ 〈〉.

Proof: Since is simple and non type , the algebra ℬ = is also non type (as in the proof of Lemma (2.1.1)). Hence, there is an injective homomorphism

∶ → ℬ. Let denote the identity function on (0,1] and let = ( ⊗ ). Let = − . It is clear by functional calculus (and because is injective) that a is purely positive.

Additionally, we have that 〈〉 is represented by 1 ⊗ ∈ ⊗ ℬ, and notice that

1⨂ ≥ (1 − )⨂ + ⨂ .

The Cuntz class of the right hand side is ( − 1)〈〉 + ( − 1)〈〉, which dominates ( − 1) 〈 + 〉 = ( − 1)〈〉.

Lemma (2.1.11) [2]: Let be a unital tracially -absorbing ∗-algebra. If , ∈ such that 〈〉 ≤ 〈〉 in () for some ∈ ℕ and is purely positive then ≾ .

Proof: Fix > 0. Without loss of generality we may assume that ‖‖ = ‖‖ = 1. We ∗ choose = () ∈ () and > 0 such that [( − )⨂1] =

( − )⨂1. Let ∈ (0,1] be a non-negative function such that = 0 on (⁄2 , 1], > 0 on (0, ⁄2) and ‖‖ = 1 and denote = (). Note that ≠ 0 because is purely positive. We now fix > 0. We shall find ∈ such that ∗ ‖[( − ) + ] − ( − )‖ < . Since is arbitrary this will show that ( − ) ≾ ( − ) + ≾ . By replacing each with () for some function ∈ (0,1] that vanishes on (0, ⁄2] and = 1 on [, 1] we may ∗ assume that = 0. Note that after doing this we still have [( − )⨂1] =

( − )⨂1 which yields

( − ) , = ( − ) ∗ = (1) ℓ ℓ 0, ≠ ℓ

Let , ℎ ∈ (0,1] be defined by

⎧ 7 ⎪ , ≤ () = 7, ℎ() = 1 − √1 − ⎨ ⎪ 1, ≥ ⎩ 7 We observe that those functions satisfy the following. |() − | < , 1 − ℎ() = √1 − (2) 6 in the algebra (0,1].

Find a c.p.c order zero map ∶ → such that 1 − (1) ≾ and

‖[, ]‖ < where = {( − ), ( − )} ∪ , and where > 0 is chosen using Lemma (2.1.8) such that

‖[[], ]‖ < , [], < , ‖[ℎ[], ]‖ < . 36 36 3 ⁄ ⁄ Let = (1)( − ). Denote = 1 − (1) and set = ( − ) . We have ‖( − ) − ( + )‖ < 3

⁄ since = 1 − ℎ[](1). We denote = ([]). We now define ̂ = [](1) and ̂ = ∑ ̂ .

Our goal is to first show that ‖̂( − ) ̂∗ − ‖ < :

∗ ∗ ̂( − )̂ = ̂ ( − )̂ ℓ ,,,ℓ

∗ = [](1) ( − )ℓℓ [](1) ,,,ℓ

∗ ≈ ⁄6 ℓ( − )ℓ ,,,ℓ

∗ = (1)[](1) ( − )ℓ ,,

∗ = (1)[](1) ( − )ℓ ,

= (1)[](1) ( − )

= (1) [](1)( − )

≈ ⁄6 (1)( − )

= where the first approximation is due to our choice of and the second follows from (2).

We now deal with . Since ≾ ≾ , we may find ∈ such that ∗ ‖ − ‖ < ⁄3. Furthermore, by replacing s with () where ∈ (0,1] is some function that is 1 on [0, ⁄2] and vanishes on [, 1], we may assume that

( − ) = 0. We take = ̂ + and calculate:

∗ ∗ ∗ ‖[( − ) + ] − ( − )‖ = ‖̂( − )̂ + − ( − )‖

∗ ∗ ≤ ‖̂( − )̂ − ‖ + ‖ − ‖ + ‖ + − ( − )‖ < .

Since this holds for every > 0 we have that ( − ) ≾ . Finally, since is arbitrary, we have ≾ .

Theorem (2.1.12) [2]: Let be a simple unital ∗-algebra. If is tracially -absorbing then () is almost unperforated, and therefore has strict comparison.

Proof:

Let , ∈ () such that 〈〉 ≤ ( − 1)〈〉 for some . We want to show that 〈〉 ≤ 〈〉. Since () is also tracially -absorbing, we may assume without loss of generality that , ∈ . If is purely positive then we are through because in particular 〈〉 ≤ 〈〉 and now we can apply Lemma (2.1.11).

If is not purely positive then is Cuntz equivalent to a projection ∈ and then by Lemma (2.1.10) we may find a purely positive element ∈ such that ( − 1)〈〉 ≤ 〈〉 and ≤ . We may now apply Lemma (2.1.11) to get ≾ ≤ ∼

The main result of this section is the following theorem.

Theorem (2.1.13) [2]: Let be a simple, separable, unital, nuclear ∗-algebra. If is tracially - absorbing then ≅ ⊗ .

Proof: By Theorem (2.1.13) we know that has strict comparison. Thus, if is traceless then is purely infinite. By Kirchberg’s absorption Theorem ≅ ⨂ so we are done. Let us then assume that () ≠ ∅. This implies that is stably finite so the conclusion of Lemma (2.1.16) holds. We now apply Theorem (2.1.14) to complete the proof.

We begin this section with a general lemma that we will need in this section and the following. We will only use it for actions of finite groups or of the integers.

We have the following theorem.

Theorem (2.1.14) [2]: Let be a unital, separable, simple, nuclear ∗-algebra such that has strict comparison, () ≠ ∅, and the following condition holds:

For any ∈ ℕ there exists a c.p.c. order zero map ∶ → ∩ and a representative

()∈ℕ ∈ ∈ℕ of ψ(e,) ∈ such that c ∈ is a positive contraction for all n and

lim max |( ) − 1⁄| = 0. → ∈() Then ≅ ⊗ .

Lemma (2.1.15) [2]: Let be a simple, separable, unital, infinite dimensional C∗-algebra. For any there exists a positive contraction ∈ such that ≠ 0 and () ≤ 1⁄ for all ∈ ().

Lemma (2.1.16) [2]: Let be a simple, separable, stably finite, unital ∗-algebra. If is tracially -absorbing then for any ∈ ℕ, we can find a sequence of c.p.c. order zero maps

∶ → such that (i). lim max, − 1⁄ = 0 for all ∈ ℕ. → ∈

(ii). lim ‖[, ()]‖ = 0 for all ∈ and for all ∈ . → Proof: Using Lemma (2.1.15) we can find a sequence of non-zero positive contractions ∈ such that

lim max () = 0. → ∈()

Let ()∈ℕ ⊆ be a dense sequence. We will denote = {, … , }. For each we can now find a c.p.c. order zero map ∶ → such that

‖[, ]‖ ≤ 1⁄ and

1 − (1) ≾ .

Clearly property (ii) holds, so it remains to see that the maps satisfy (i). It is clear that that , = , for any trace , so it suffices to show that

1 − max , = min , ⎯⎯ 0. ∈() ∈() → We claim that

1 − , ~1 − (1) for all . To see this, fix , and note that by functional calculus we have a homomorphism ∶ ([0,1]) → given by

() = (1). Letting denote the identity function on [0,1], a simple calculation yields:

(1 − ) = 1 − , , (1 − ) = 1 − (1) Since homomorphisms of ∗-algebras induce maps on the level of the Cuntz semigroup, our claim now follows from the simple fact that 1 − ∼ 1 − in ([0,1]).

We thus have

0 ≤ 1 − , ≤ 1 − ,

= 1 − (1)

≤ () and since lim→ max∈() () = 0, we have that (i) holds.

Lemma (2.1.18) [2]: Let ∶ → Aut() be an action of a discrete group on a simple, unital ∗- algebra . Suppose that is outer for all ∈ \{1}. Then for every non-zero positive element ∈ ⋊, in the reduced crossed product there exists a non- zero positive element ∈ such that ≾ .

Proof:

Let ∶ ⋊, → be the canonical faithful conditional expectation. By replacing a with ‖()‖ we may assume that ‖()‖ = 1. We can find a finite sum = ∑ ∈ ⋊ such that ‖ − ‖ < where is the , canonical unitary implementing for ∈ . We can assume that is positive and non-zero, and we label the indices such that = 1. Note that is a non- zero positive element in because = ( ). Since is a contractive map, we have

1 2 ‖ ‖ ≥ 1 − = . 3 3 Let 0 < < . By Lemma (2.1.11) we can find a positive element ∈ () with ‖‖ = 1 such that 2 ‖ ‖ > ‖ ‖ − ≥ − , () < , ∀ ≠ 1. 3 This implies that for ≠ 1 we have

( ) ( ) = = < . We deduce that ‖ − ‖ < and thus 1 ‖ − ‖ < + . 3 Let = − + , note that is a non-zero positive element by choice of and we have

≾ ≾ where the first Cuntz subequivalence follows.

Definition (2.1.19) [2]: If ∶ → Aut() is an action of a finite group on a simple, unital ∗- algebra , then is said to have the generalized tracial Rokhlin property if for any > 0, any finite subset ⊆ , and any non-zero positive element ∈ there exist normalized positive contractions ⊆ such that: ∈

(i) ⊥ when ≠ ℎ.

(ii) 1 − ∑∈ ≾ .

(iii) , < for all ∈ , ∈

(iv) () − < for all , ℎ ∈ We first collect a few basic properties of such actions.

Proposition (2.1.20) [2]: Let α ∶ G → Aut() be an action of a finite group G on a simple, unital C∗- algebra . If has the generalized tracial Rokhlin property then is outer for all ∈ \{1}.

Proof: Let ∈ and assume that = Ad() for some ∈ . Let be ∈ elements as in Definition (2.1.19) taking = 1⁄2 , = 1, and = {}. We have

1 = − 1 < ∗ − + 2 1 = ( ) − + 2 1 1 < + 2 2 = 1 which is a contradiction.

We state that if ∶ → Aut() is an action of a discrete group on a simple ∗ -algebra such that is outer for all ≠ 1 then ⋊, is simple.

Corollary (2.1.21) [2]: Let ∶ → Aut() be an action of a finite group on a simple, unital ∗- algebra . If has the generalized tracial Rokhlin property then ⋊ is simple.

Lemma (2.1.22) [2]: Let be a simple, separable, unital tracially Z-absorbing C∗-algebra and let ∶ → Aut() be an action of a finite group on . Assume that has the generalized tracial Rokhlin property. Then for any finite set ⊂ , > 0 and

47 non-zero positive element ∈ and ∈ ℕ there is a c.p.c. order zero map

∶ → such that: (i) 1 − (1) ≾ .

(ii) For any normalized element ∈ and any ∈ we have ‖(), ‖ < .

(iii) For any normalized element ∈ and any ∈ we have

() − () < .

Proof: Let , , , be given as in the statement of the lemma. Since is simple and unital, we can find > 0 such that () ≥ for all ∈ (). Use Lemma (2.1.15) to find a positive element ∈ such that () < for all ∈ || (). Since ∘ is also a trace, we have that () < (3) || + 2 for all ∈ and ∈ ().

Let η > 0 be as in Lemma (2.1.8) such that ‖[h(ψ), y]‖ < ε⁄2 whenever

∶ → is a c.p.c order zero map and ∈ is a contraction such that ‖[, ]‖ < where ℎ ∈ ((0, 1]) is given by 2, < 1⁄2 ℎ() = . 1, ≥ 1⁄2 Let be positive contractions such that ∈

(i) , < for all ∈ and for all ∈ . (ii) ( ) − < . ||

(iii) 1 − ∑∈ ≾ as guaranteed by the generalized tracial Rokhlin property.

By Lemma (2.1.6) we can find a c.p.c. order zero map ∶ → ∩ such that 1 − () ≾ . We denote by the automorphism of induced by

, and define ∶ → by

() = (). ∈ is clearly an order zero c.p.c. map.

First we remark that ∩ is invariant under . Thus we have

() = () ≈ () = () ∈ ∈ for all ∈ . This is the first property we will be needing.

We now consider () for normalized ∈ :

() = () ∈

≈ ⁄2 () ∈

= () = () ∈ The third and last property we will be needing of is:

1 − (1) = 1 − + − (1) ∈ ∈

⁄ ⁄ = 1 − + 1 − (1) ∈ ∈

≾ ⨁ () ≾ ∈ where the last Cuntz-subequivalence follows from (iii) and Theorem (2.1.12). By Proposition (2.1.4), we can find ∈ such that ∗ = 1 − (1) − 1⁄4 . We now use liftability of order zero maps with finite dimensional domains to lift to a sequence of order zero maps ℓ ∶ → . Let (ℓ)ℓ∈ℕ ∈ ∏ℕ be a representative of . We have

ℓ→ ∗ ‖ℓℓ − (1 − ℓ(1) − 1⁄4)‖ ⎯ 0.

The properties of that we have established imply that for ℓ ∈ ℕ large enough, the following properties hold:

(i) ℓ , < for all ∈ , ( ) ( ) (ii) ℓ − ℓ < for all normalized elements ∈ ,

(iii) ∗ − 1 − (1) − 1⁄4 < 1⁄4, ℓ ℓ ℓ By Proposition (2.1.2), the third property above implies that

∗ 1 − ℓ (1) − 1⁄2 = 1 − ℓ (1) − 1⁄4 − 1⁄4 ≾ ℓ ℓ ≾ .

Define = ℎ[ℓ ]. By functional calculus, one checks that

1 − (1) = 21 − (1) − 1⁄2 ℓ which gives

1 − (1) ≾ .

Property (ii) above together with (2.1.9) ensures that () − () <

⁄2 < for all normalized ∈ and ∈ . This shows that meets our requirements.

We are now ready to prove the following theorem.

Theorem (2.1.23) [2]: Let be a simple unital ∗-algebra and let ∶ → Aut() be an action of a finite group on . Assume that has the generalized tracial Rokhlin property. If

is tracially -absorbing then ⋊ is also tracially -absorbing. Proof: Lemma (2.1.22) implies that we can always find c.p.c. order zero maps as in Definition (2.1.2) provided that the positive element a is taken from . By Lemma (2.1.18) we may always assume without loss of generality that ∈ (otherwise replace it by a non-zero positive element from which it dominates).

Putting together Theorems (2.1.13) and (2.1.23) we obtain the following partial generalization.

Corollary (2.1.24) [2]: Let be a simple, separable, unital, nuclear ∗-algebra and let ∶ → Aut() be an action of a finite group with the generalized tracial Rokhlin property. If is absorbing then so is ⋊ . As a non-trivial example of an action satisfying the generalized tracial Rokhlin property, we consider the symmetric group acting by permutation on the -fold tensor power of . This example was studied, Corollary (2.1.24) gives a different proof of those results.

We recall the following special case.

Theorem (2.1.25) [2]: Let τ denote the unique tracial state on Z. There exists a unital embedding ∶ ([0,1]) → such that

() = ()

Corollary (2.1.26) [2]: Let denote the unique tracial state on , let ⊆ be a finite subset, and let > 0. There exists a unital embedding ∶ ([0,1]) → such that

() = () and ‖[, ()]‖ < for all ∈ and normalized ∈ ([0,1]).

Proof: We may assume that all elements of are of norm at most 1. Now, we may decompose as ≅ ⊗ such that each element ∈ is close to within ⁄2 to an element in the unit ball of ⊗ 1. Now choose a unital embedding ∶ ([0,1]) → as in Theorem (2.1.25). Define ∶ ([0,1]) → ⊗ ≅ by

() = 1 ⊗ (). clearly satisfies the commutation requirement, and since the restriction of the unique trace on ⊗ to the second component 1 ⊗ is the unique trace on that ∗ -algebra, we see that (()) is indeed given by integrating f against Lebesgue measure, as required.

Example (2.1.27) [2]:

Let = be the symmetric group on the set {1, … , }, and let = ⨂ ≅ . Let ∶ → Aut() be the action defined on elementary tensors by the formula

(⨂ … ⨂) = ()⨂ … ⨂()

Let , , be as in Definition (2.1.19). We denote = () where is the unique trace on . Let

∈ ([0,1] ) be a continuous function of norm one, such that:

(i) is supported on the set ∶= {(, … , ) ∈ [0,1] | < ⋯ < }, (ii) ({ ∈ | () ≠ 1}) < ! where is Lebesgue measure on [0,1] . For ∈ we now define ∈ ([0,1] ) by

(, … , ) = (), … , ()

The are pairwise orthogonal and sum up to an element of norm 1 that equals 1 on a subset of [0,1] of measure larger than 1 − .

Our next step will be to define an embedding of ([0,1]) into . We may assume that the finite set consists of elementary tensors. Denote = () () ⨂ … ⨂ where is some finite index set. For = 1, … , we take unital ∈ embeddings ∶ ([0,1]) → such that

() = () () and z , ψ(g) < ε for all i ∈ I and normalized g ∈ C([0,1]). Using the identification ([0,1]) ≅ ([0,1])⨂ we define ∶ ([0,1]) → by

(⨂ … ⨂) = ()⨂ … ⨂()

We define = (). Clearly we have that ‖[, ]‖ < for all ∈ . It is also not hard to see that

() = [,] for any ∈ ([0,1] ) which implies that for the elements = () we have (1 − ∑∈ ) < . Since has strict comparison and is the only trace on this entails 1 − ∑∈ ≾ . section (2.2): Actions of ℤ

Definition (2.2.1) [2]: Let be an automorphism of a simple, unital ∗-algebra . We say that has the generalized tracial Rokhlin property if for any finite set ⊆ , any > 0, and ∈ ℕ, and any non-zero positive element ∈ there exist normalized orthogonal positive contractions , . . . , ∈ such that the following holds:

(i) 1 − ∑ ≾ .

(ii) ‖[, ]‖ < for all and for all ∈ .

(iii) ‖() − ‖ < for all ≤ ≤ − 1.

Note that in the definition we do not require that () be close to . Notation (2.2.2) [2]:

Elements , . . . , as in Definition (2.2.1) are said to satisfy the relations ℛ(, , , ).

We begin as we did in the previous section with some basic properties.

Proposition (2.2.3) [2]: Let be a simple, unital ∗-algebra and let ∈ Aut() have the generalized tracial Rokhlin property. Then is outer for all ∈ ℤ\{0}. Proof: Suppose is implemented by some unitary ∈ . Let , … , ∈ be elements satisfying ℛ + 1, , 1 , {}. We have

1 = ‖ − ‖ 1 < ‖ ∗ − ‖ + 2 + 2 1 = ‖( ) − ‖ + 2 + 2 1 1 < + + + 2 2 + 2 < 1

Corollary (2.2.4) [2]: Let ∈ Aut() be an automorphism on a simple, unital ∗-algebra . If has the generalized tracial Rokhlin property then ⋊ is simple. Lemma (2.2.5) [2]: Let be a simple, separable, unital, ∗-algebra and let ∈ Aut() be an automorphism with the generalized tracial Rokhlin property. Recall that acts naturally on () via ⟼ ∘ . Assume that there exists ∈ ℕ such that the

action of on () is trivial. Then for any > 0 there exist ∈ ℕ, such that whenever ≥ , 0 < < , ∈ satisfies 〈〉 ≤ 〈1〉, ⊆ is some finite subset and , … , ∈ are elements satisfying the relations ℛ(, , , ) then () < for all ∈ () and for all . Proof: Choose such that < . Let , > 0, ∈ be as in the statement of the lemma and let , … , be elements satisfying ℛ(, , , ). We partition the set = {1, … , } into m sets

= where = ∩ ( + ℤ). We have that

‖ () − ‖ < which implies that

( () − ) = (( − )) ≾ whenever 1 ≤ ≤ − . Since acts trivially on (), this entails that

(( − mε)) ≤ () for all 1 ≤ ≤ − . Similarly we have that

(( − )) ≤ () whenever + 1 ≤ ≤ . By iterating this argument we get that

(( − )) ≤ ()

for all 1 ≤ ≤ and for all , ∈ . Note that ≤ for all . This implies that, if ∈ we have 1 (( − ) ) ≤ ( ) ≤ (1) < . 2

Denote = and let , ∈ ((0,1]) be defined by 0, ≤ , ≤ () = 1 , () = − () . ( − ), > (1 − ), > 1 − 1 −

Notice that () ∼ ( − ) for all ∈ . Together with the previous observations this implies that ( ) < 2 for all and for all ∈ (). Now define another function ℎ ∈ ((0,1]) by , ≤ 1 − ℎ() = . , > 1 − We have that

ℎ()(1 − )ℎ() = ℎ()(1 − ) = (), thus

ℎ()(1 − )ℎ() = () for all . Since is orthogonal to for ≠ , this implies that

ℎ() 1 − ℎ() = () entailing that

() ≾ 1 − ≾ for all . Finally we have that

= () + ()

≾ () ⊕ ()

≾ ⊕ () In particular we have

() ≤ () + () <

Lemma (2.2.6) [2]: Let be a simple, separable, unital, tracially -absorbing ∗-algebra and let ∈ Aut() be an automorphism with the generalized tracial Rokhlin property. If acts trivially on () for some m, then for any finite set ⊂ , > 0 and non-zero positive element ∈ and ∈ ℕ there is a c.p.c. order zero map

∶ → such that: (i) 1 − (1) ≾ .

(ii) For any normalized element ∈ and any ∈ we have ‖(), ‖ < .

(iii) For any normalized element ∈ we have () − () < .

Proof: Let , , , be given as in the statement of the lemma. We assume throughout that < 1. As in the proof of Lemma (2.1.22) we can find > 0 such that () > for all ∈ (). Let ∈ ℕ be such that < . Denote = . By Lemma (2.2.5) we can find such that whenever ≥ , 0 < < , ∈ with 〈〉 ≤ 〈1〉, ⊆ is some finite subset, and , … , ∈ are elements satisfying ℛ(, , , ) then we have () < for all and for all ∈ (). We may furthermore assume > 2. Use Lemma (2.1.15) to find a positive element ∈ such that () < min , for all ∈ (). Choose ≥ and choose > 0 such that 2 + + < . In particular < . Let = ()| ∈ , 1 ≤ ≤

Let , … , ∈ be elements satisfying ℛ(, , , ). We now modify the elements to obtain new elements as follows: , ≤ ≤ − ⎧ ⎪ , 1 ≤ < = ⎨ − ⎪ , − < ≤ ⎩ By Lemma (2.1.6) we can find a c.p.c. order zero map φ ∶ M → ∩ such that 1 − (1) ≾ . We define ∶ → by

() = (). is clearly a c.p.c. order zero map. Our first goal is now to find a nice bound on () − () for normalized ∈ . First, note that ‖( ) − ‖ < for all < and also ‖ ‖, ‖ ‖ < . Now assume ≠ , we have

(() − ) − = −() −

≤ () +

≤ + + 2

= 2.

We use this estimate to obtain that for normalized ∈

() − ()

= (() − ) ()

≤ (() − ) − () () ,

+ (() − ) ()

4 ≤ 2 + + .

Now we are ready to consider the expression we are interested in. Let ∈ be a normalized element. We have

() − ()

≤ () − () + (1)

+ (1)

4 ⁄ 2 ≤ 2 + + +

< . Next, we claim that 1 − (1) ≾ . To see this, first note that

1 − ≾ 1 −

≾ ⨁ Using this, we may write

1 − (1) = 1 − (1) + (1) 1 −

≾ 1 − (1) ⊕ 1 −

≾ ⊕ Since has strict comparison and () < for all ∈ (), we have that ⨁ ≾ which proves our claim.

The final condition we need on is that (), < for all normalized

∈ and for all ∈ . This follows immediately from our construction. We now continue as in the proof of Lemma (2.1.22) and use projectivity of order zero maps to lift to a sequence of c.p.c. order zero maps from into . Going far enough along this sequence we obtain the map which we need.

Combining this with Theorem (2.1.13), we obtain the following.

Theorem (2.2.7) [2]: Let be a simple, separable, unital, tracially -absorbing ∗-algebra and let ∈ Aut() be an automorphism with the generalized tracial Rokhlin property. If acts trivially on () for some m then ⋊ is also tracially -absorbing. Example (2.2.8) [2]: It is shown that the bilateral tensor shift automorphism on ⨂ ≅ satisfies what is there defined as the weak Rokhlin property. Since has strict comparison, this implies that has the generalized tracial Rokhlin property. Since has unique trace this shows that α satisfies the conditions of Theorem (2.2.7).

Chapter 3

∗-Algebras and Non-Stable -Theory

In this chapter we consider the properties weak cancellation, −surjectivity, ∗ good index theory, and −injectivity, for the class of extremally rich −algebras, and for the smaller class of isometrically rich ∗−algebras. We establish all four properties for isometrically rich ∗−algebras and for extremally rich ∗−algebras that are either purely infinite or of real rank zero, −injectivity in the real rank zero case following from a prior result of H. Lin [3]. Section (3.1): Weak cancellation We defined the concept of extremal richness. One of several equivalent criteria for the ∗-algebra to be extremally rich is that the closed unit ball of , the unitization, is the convex hull of ℰ(), the set of its extreme points. Further review of the concept is given in the next section. simple ∗-algebra is extremally rich if and only if it is either of stable rank one or purely infinite, and a theme of our work has been that extremal richness is a generalization of the stable rank one property which is suitable for infinite algebras. Since much of the success in the classification of simple ∗-algebras has been for algebras that are either purely infinite or of stable rank one, it seems worthwhile to study non-simple extremally rich ∗-algebras.

J. Cuntz defined purely infinite simple ∗-algebras and showed that they have many good non-stable K-theoretic properties. And M. Rieffel, motivated by algebraic results of H. Bass, defined topological stable rank and showed that ∗- algebras of (topological) stable rank one have similarly good properties. We therefore investigated whether extremally rich ∗-algebras also have the good properties. Although we haven’t proved that all extremally rich algebras have good non-stable -theoretic properties, we have found large subclasses that do. In particular, the summary includes all four properties listed in the abstract for isometrically rich ∗-algebras and for extremally rich ∗-algebras which are either purely infinite (in the sense of E. Kirchberg and M. Rørdam) or of real rank zero. All three cases of cover purely infinite simple ∗-algebras.

Murray-von Neumann [6]: Two subspaces belong to are called Murray von Neumann if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informly, if '' Knows '' that the subspace are isomorphic).

A ∗-algebra has weak cancellation if whenever and are projections in which generate the same (closed, two-sided) ideal and have the same class in

(), then p is Murray-von Neumann equivalent to ( ∼ ). Of course, Typeset ∼ implies that they generate the same ideal and that [] = [] in (). Nevertheless, it was observed by Rieffel, that ∗-algebras of stable rank one

(tsr() = 1) satisfy a stronger property: If [] = [] in (), then ∼ . Of course this stronger property can’t hold in infinite algebras. Cuntz showed that if is purely infinite simple, if and are both non-zero, and if [] = [] in (), then ∼ ; and the concept of weak cancellation was designed to specialize to this property in the simple case. More about weak cancellation and its history is given in the next section.

We say that has -surjectivity if the map from to () is surjective, -injectivity if this map is injective, and -bijectivity if it is bijective.

Here is the unitary group and the connected component of the identity (so that is the set of homotopy classes). Rieffel showed that tsr() = 1 implies that has -bijectivity and Cuntz showed the same for purely infinite simple. A result of P. Ara, states that every quotient ∗-algebra of a ∗ ∗ Rickart -algebra has -surjectivity. Below states that every extremally rich - algebra with weak cancellation has -surjectivity. This should be compared, which states that every ∗-algebra with real rank zero and stable weak cancellation has -surjectivity.

We defined the extremal -set, (), which is roughly analogous to () with extremal partial isometries used in place of unitaries. The equivalence relation for is more complicated than that for ; we show that if is extremally rich with weak cancellation, then () =⎯ ℰ homotopy), in exact analogy with the -case. We say that has -surjectivity, -injectivity, or -bijectivity if the map from

ℰ⁄homoopy to () is respectively surjective, injective, or bijective. These properties actually imply the corresponding -properties. We show that every ∗ extremally rich -algebra with weak cancellation has -surjectivity. We say that a (non-unital) ∗-algebra has good index theory if whenever is embedded as an ideal in a unital ∗-algebra and is a unitary in / such that ([]) = 0 in (), there is a unitary in which lifts . Of course, the boundary map, : (/) → (), from the -theory long exact sequence is often regarded as an abstract index map. Using that long exact sequence, we can reformulate the good index theory property: If ∈ (/), ∈ (), and lifts [] , then u lifts to (). This suggests a stronger property–require that the lift of lie in the class . However, there is no need to name this stronger property (still demanded for all choices of ), since it is equivalent to requiring that have both -surjectivity and good index theory. Good index theory has been considered, so far as we know, no one previously proposed a name for it. M. Pimsner, S. Popa, and D. Voiculescu proved that () ⊗ has good index theory when is compact, where denotes the algebra of compact operators on a separable, infinite dimensional Hilbert space. J. Mingo showed that () can be replaced by an arbitrary unital ∗-algebra and asked whether every stable ∗-algebra has good index theory. (In both of these results the property is considered only for = (), the multiplier algebra, but this is sufficient to imply that it holds for all .) Shortly after, . Nagy proved, that csr() ≤ 2 implies that has good index theory. Here csr denotes connected stable rank (Rieffel), and csr() ≤ 2 also implies -surjectivity for . It had already been proved by . Sheu, and V. Nistor, that csr() ≤ 2 for all stable . So Nagy completed the affirmative answer to Mingo’s question (but was apparently unaware of Mingo’s work). Since Rieffel proved that csr() ≤ tsr() + 1, Nagy also established good index theory for ∗-algebras of stable rank one. The fact that purely infinite simple ∗-algebras have good index theory should also be considered previously known.

A unital ∗-algebra has stable rank one if , the set of invertible elements, is dense in , is isometrically rich if ⋃ , the set of one-sided invertible elements, is dense, and is extremally rich if , the set of quasi-invertible

63 elements is dense. If is non-unital, we say that has one of these properties if has the property. All three properties pass to (closed, two-sided) ideals and hereditary ∗-subalgebras, quotient algebras, and matrix algebras and stabilizations.

Before reviewing the definition of quasi-invertibility, we recall R. Kadison’s criterion for extreme points of the unit ball of a ∗-algebra . Extreme points exist if and only if is unital, and is extremal if and only if

(1 − ∗)(1 − ∗) = 0.

Equivalently, is a partial isometry and ∩ = 0, where = id(1 − ∗) and = id(1 − ∗). Here id(⋅) denotes the ideal generated by ⋅, 1 − ∗ and 1 − ∗ are called the left and right defect projections of , and and are the left and right defect ideals of .

We showed that seven conditions on an element t of a unital ∗-algebra are equivalent, and these conditions are the definition of quasi-invertible. (Non-unital algebras have no quasi-invertibles.) One of these conditions amounts to saying that t has closed range and that if = || is its canonical polar decomposition (the closed range condition implies ∈ ), then ∈ ℰ(). Another is that there are ideals and with ∩ = 0 such that + is left invertible in / and + is right invertible in /. Clearly the minimal choices for and are the defect ideals of , and we also call these the defect ideals of . Of course, if is prime, one of and must be 0; thus every quasi-invertible element is one-sided invertible and every extremal partial isometry is an isometry or co-isometry.

In general, there is an analogy in the two-step progressions from stable rank one (through isometrically rich) to extremally rich, from invertibility to quasi- invertibility, and from unitary to extremal partial isometry. It is not always necessary to pursue all three concepts in parallel because A (unital) has stable rank one if and only if it is extremally rich and every extremal is unitary, and is isometrically rich if and only if it is extremally rich and every extremal is an isometry or co-isometry. However, the most general results are usually not proved first. Rørdam proved Robertson’s conjecture: For A unital, tsr() = 1 if and only if the closed unit ball is the convex hull of ().

For technical reasons it is sometimes necessary to consider extremals, quasi- invertibility, and extremal richness for objects other than ∗-algebras, namely bimodules of the form , where and are projections in . S. Sakai’s criterion for u to be an extreme point of the unit ball of is:

( − ∗)( − ∗) = 0 or equivalently, (1 − ∗)(1 − ∗) = 0.

Quasi-invertibility and extremal richness for bimodules are treated analogously to the treatments for ∗-algebras with no difficulty. It is not necessary to be explicitly aware of the fact that is a bimodule or even to know what “bimodule” means. Nevertheless, the abstract setting was discussed. One warning: There are no concepts of unitality or unitization for bimodules. It is required that ℰ() ≠ ∅ in order for to have a chance to be extremally rich. (When = 0, it is automatically extremally rich.) The extremal richness of does not imply that of , but it is important to our main results that sometimes is extremally rich . Since tsr() ≤ if and only if “left invertibles” are dense in the bimodule (= ()), we once looked at extremal richness for ; but it turned out that extremally rich does not imply extremally rich.

An example that does invoke the abstract setting may be interesting. The right module called ℋ by . Kasparov is also an ⊗ – -imprimitivity bimodule. Regardless of whether or not is extremally rich, ℋ is an extremally rich bimodule if and only if is unital.

Definitions (3.1.1) [3]: If and are projections in a ∗-algebra, we write − to mean ∼ ≤ for some projection . The relations ∼ and ≾ can be extended to projections in

⋃ () either by replacing and by ⊕ 0 and ⊕ 0 for suitable and or by allowing the partial isometries to be non-square matrices. The projection is infinite if it is equivalent to a proper subprojection of itself, otherwise finite, and a ∗ unital -algebra is finite or infinite according as is, and stably finite if all the matrix algebras () are finite. If and are projections in () and (), 0 respectively, then ⊕ denotes the projection in (), and ⊕ 0

… ⊕ is defined similarly. In this context is used for the -fold sum, ⊕ … ⊕ . A projection is called properly infinite if 2 ≾ .

It is possible to reformulate weak cancellation in a way that does not mention -theory. Note that if , , and are projections in an ideal and if is full in [ ] (i.e., = id()), then () = []() if and only if ⊕ ∼ ⊕ for sufficiently large . Thus the hypotheses, id() = id() = and []() =

[](), can be replaced by, ⊕ ∼ ( + 1) and ⊕ ∼ ( + 1) for sufficiently large . It was pointed out to us by . Goodearl that the concept can be simplified further if we demand weak cancellation for the stabilization of : ⊗ has weak cancellation if and only if 2 ∼ ⊕ ∼ 2 implies ∼ for all projections and in ⊗ . Moreover, this is equivalent to “separativity,” a term which was introduced into semigroup theory by A. Clifford and G. Preston. The set of Murray-von Neumann equivalence classes of projections in is not in general a semigroup, and it is only the stable version of weak cancellation that is literally equivalent to separativity. However, can be used to show that weak cancellation is a stable property in the real rank zero case.

Defect ideals were treated. The defect ideal of , denoted (), is the ideal generated by all defect projections of elements of ℰ(). (All of these defect projections are in .) If A is extremally rich, then () is the smallest ideal such that tsr(/) = 1. It follows that defect ideals are compatible with Rieffel-Morita equivalence among extremally rich ∗-algebras. In particular, if is a full hereditary ∗-subalgebra, then () = ∩ (). The symbol () denotes the -fold iteration of .

The primitive ideal space of will be denoted by ⋁. If is an ideal of , then ⋁ is identified with an open subset of ⋁ (namely, the complement of hull()), and if is a hereditary ∗-subalgebra, then ⋁ is identified with id()⋁. For hereditary, denotes the two-sided annihilator of , which is again hereditary, and is an ideal if is an ideal.

Also, = denotes norm closure and denotes the extended Toeplitz algebra, which was discussed.

Definitions (3.1.2) [3]: Recall that a ∗-algebra has weak cancellation if any pair of projections , in that generate the same closed ideal of and have the same image in ()

66 must be Murray–von Neumann equivalent in (hence in ). If () has weak cancellation for every , equivalently, if ⊗ has weak cancellation, we say that has stable weak cancellation. We shall show below that weak cancellation implies stable weak cancellation if is extremally rich, but for now we need the distinction.

Proposition (3.1.3) [3]: If and are projections in an extremally rich ∗-algebra such that [] = [] in () then is extremally rich. Proof:

Since (A) ⊂ (A) and pAq = we may assume that is unital. Then also [ − ] = [ − ] in (), so

( − ) ⊕ ~( − ) ⊕ (in ()) for sufficiently large. Since () is extremally rich conclude that

= ( ⊕ 0)()( ⊕ 0) is extremally rich.

Lemma (3.1.4) [3]: Let and be projections in a ∗-algebra and for each element in let id() denote the closed ideal generated by . If now ∈ ℰ() then

() = () ∩ ().

Proof: Since ∗ ≤ we have ∗ ∈ id(), whence ∈ id(). Similarly ∗ ≤ , so ∈ id().

Conversely, let ∶ → /() denote the quotient map. Then the extremality equation gives

()()() = 0, whence (() ∩ ()) = 0; so () ∩ () ⊂ (), as desired.

Although we will prove later that every extremally rich ∗-algebra with weak cancellation also has -surjectivity, it will facilitate some of the following arguments to impose it as a condition on the algebras.

Lemma (3.1.5) [3]: Let and be full projections in an extremally rich ∗-algebra such that

[] = [] in (), and assume that we have found an extreme partial isometry in ℰ() such that

∗ ∗ = − and = − are projections in an ideal of . Assume further that / has -surjectivity for every full projection in . Then

~ ⊕ and = +

for some full projection in and projections and in , with [] = [] in (). Proof:

Note first that the element [] − [] of () belongs to the kernel of the natural map from () into (). By the six–term exact sequence in -theory ∗ there is therefore an in (⁄) such that = [] − []. Since = is a full projection in by Lemma (3.1.4), we may identify (⁄) with (⁄), and by -surjectivity we can therefore find in such that + is unitary with [ + ] = . Since is extremally rich, extreme points lift from quotients, so we may assume that ∈ ℰ(). Computing in () we find that

∗ ∗ [] − [] = = index = [ − ] − [ − ]. Let

∗ ∗ ∗ ∗ = + ( − ) , = + ( − ), = .

Since ∈ ℰ() we see from Lemma (3.1.4) that is full in , and since is full in , we have also that is a full projection in . By construction ∗ ∗ − and − belong to , so and belong to ; and evidently [] = [] in (). Finally, 68

∗ ∗ ∗ = + = + ~ ⊕ ,

∗ = + = + . Theorem (3.1.6) [3]: For an extremally rich ∗-algebra the following conditions are equivalent:

(i) has weak cancellation;

(ii) If = for some projection p in and ∈ ℰ(()), there is a projection in such that

∗ ⊕ 0~( ⊕ ) − ();

(iii) If = for some projection in , and {, … , } is a finite subset of ℰ() there is a projection in such that ∗ ⊕ 0~ ( − ) (). = 1 Proof: (i) ⇒ (ii) Since homotopic projections are equivalent we may assume that

∗ ∗ ∗ ∗ ∗ ( ⊕ ) − = ( − − ) ⊕ ( − − ), ∗ where ∈ ℰ() and ∈ ℰ(), and where we set = − and ∗ ∗ = − for , = 1,2. Evidently the two projections − and ∗ − have the same image in () for any ideal containing . To show that they also generate the same ideal in it suffices to show that they both generate as an ideal (inside ). However,

∗ ∗ ∗ ∗ − ≥ and − ≥ , and both and generate as an ideal by Lemma (3.1.4). By assumption there is therefore a partial isometry in such that

∗ ∗ ∗ ∗ = − and = .

Thus

∗ ∗ = ( − )

∗ ∗ ∗ is a projection equivalent to − , and since ≤ = − and ∗ is centrally orthogonal to − (because ∈ ℰ()) it follows that actually

∗ ≤ − = . ∗ ∗ But then is a projection in equivalent to − and orthogonal to ∗ − = . We may therefore take

∗ ∗ = − + , which is a projection in equivalent to ( ⊕ ) − ∗.

(ii) ⇒ (iii) We use induction on , the case n = 1 being trivial. By assumption, used on {, … , }, we can therefore find a projection in such that − 1 ∗ ∗ ~ − ~2 ( − ). = 1 = 1

This means that we can write = + ( − ), where − 1 ∗ ~ − ( − ). = 1

In particular, ≾ − = .

Set = . Since

− + ≾ 2

∗ ∗ there is a partial isometry in () such that = ⊕ 0 and = ≤ ⊕ . It follows that if we define

∗ = ⊕ − + ( ⊕ 0) then ∈ ℰ(()). Moreover,

∗ ∗ ∗ ∗ ⊕ − = (( − ) ⊕ 0) ~ − .

Applying (ii) to and we find a projection in such that ∼ − ∗ . Thus we may take = + to complete the induction step. (iii) ⇒ (i) Let and be projections in that generate the same closed ideal and for which [] = [] in (). Replacing by does not effect the conditions in (iii) so we may assume that = ; i.e., and are full projections in . Since

[] = [] in () it follows from Proposition (3.1.3) that is extremally rich (and non–zero). Take therefore in ℰ() and define

∗ ∗ = − , = − .

Then [] = [] in (). If ∶ → ⁄() denotes the quotient map, we have [()] = [()] in

(()). Since tsr(()) = 1, this implies that () ∼ (). Thus () is isometrically isomorphic to (), which has stable rank one. It follows that

() is “unitary” so that () = () = 0. Thus both and belong to (). Since tsr(⁄()) = 1 for every projection in we may apply Lemma

(3.1.5) with = () to obtain a full projection in and projections and in () such that [] = [] in (()) and

~ ⊕ , = + .

Let = . Then is a full corner of , so () = () is a full hereditary ∗-subalgebra of (). Consequently the set of projections

∗ = { − | ∈ ℰ()} generates () as an ideal. For some finite subset {} of we therefore have

∗ ∗ ≾ ( − ), ≾ ( − ) ; and since [] = [] in (()) we can assume, possibly after enlarging the subset, that

∗ ∗ ⊕ ( − ) ~ ⊕ ( − ).

Applying condition (iii) we find a projection in such that ∼⊕ ∗ ( − ). Thus ⊕ ∼ + with ≤ , whence

~ ⊕ ~ + = .

Corollary (3.1.7) [3]: Every isometrically rich ∗-algebra has stable weak cancellation.

The term “purely infinite” will be used in the Kirchberg- Rørdam sense. Every projection in a purely infinite ∗-algebra is properly infinite. We made the definition that is purely properly infinite if every non-zero hereditary ∗- subalgebra is generated as an ideal by its properly infinite projections. This is equivalent to purely infinite when is simple but stronger in general. However, the two concepts are equivalent if has enough projections. We showed that an extremally rich ∗-algebra is purely properly infinite if and only if () = for every ideal of . We also showed that for extremally rich every non-zero projection in is properly infinite if and only if () = for every ideal which is generated (as an ideal) by a projection. The next result has a still weaker hypothesis.

Corollary (3.1.8) [3]: If is an extremally rich ∗-algebra such that () = whenever is the left defect ideal of an element of ℰ(), then has weak cancellation. In particular, this applies if every defect projection is properly infinite or if is purely infinite.

Proof:

To show that satisfies condition (iii) let , … , be in ℰ(), where = ∗ for a projection in , and let = − . Since is also a defect projection for , the hypothesis implies that (id()) = id(); and since is Rieffel- Morita equivalent to id(), this implies () = . Then it implies that is properly infinite for some . But is equivalent to a projection in , namely the left defect projection of . Now the proper infiniteness of implies that there is an isometry in such that ≾ − ∗ ∗ . So if = − + , then is an isometry in and ≾ − . ∗ Finally, if = ∏ , then is an isometry in and ⊕ ≾ ( − ).

Lemma (3.1.9) [3]: ∗ Let , and be projections in a unital -algebra such that

~ ≤ and − ~ − ~.

Then also − ∼ . Proof:

We compute (in and in ())

− = − + − ∼ ( − ) ⊕ ( − )

∼ ⊕ ( − ) = ( − + ) ⊕ ( − )

∼ ( − + ) ⊕ − ∼ − + + − = .

Lemma (3.1.10) [3]: ∗ Let and be projections in a unital -algebra of real rank zero such that ∼ and − ∼ − ∼ . If I denotes the closed ideal of generated by (and ) then − ∼ 1 − in = + ℂ. Proof: By assumption there are and in such that

∗ ∗ ∗ ∗ = = , = − , = − . Let ∶ → / be the quotient map and note that () and () are unitaries. If ∗ ∗ is the partial isometry in for which = and = (so that ∈ ) then ∗ + is unitary in .

∗ ∗ Since () has -injectivity by Lin, and since [( )] = 0 in (()), we must have

∗ ∗ ( ) = () for some unitary () in the identity component of (()). However, (()) = (()) so we may assume that ∈ (). Consider now the ∗ unitary = ( + ) and note that

∗ ∗ ∗ () = ( ) = ( ).

∗ Therefore = is a partial isometry in , and by computation

∗ ∗ ∗ ∗ = = − and = = − . Finally,

∗ ∗ () = (( ) ) = () , so that ∈ , as desired. Theorem (3.1.11) [3]: Every extremally rich ∗-algebra of real rank zero has stable weak cancellation.

Proof: The given data are stable so it suffices to show that has weak cancellation.

We do this by verifying condition (ii) in Theorem (3.1.6), and since the given data are also hereditary it suffices to verify the condition for alone, assuming that is unital. Finally, using we may assume that the defect projection in () has the form

∗ ∗ ∗ ∗ ( − − ) ⊕ ( − − ) , ∗ ∗ where ∈ ℰ() and ∈ (), for = − and = − , , = 1,2.

Let be the closed ideal of generated by the two interesting projections ∗ ∗ − and − . We have

∗ ∗ ( − )( − ) = 0 since ∈ ℰ(). Moreover,

∗ ∗ ( − )( − ) = 0

∗ since already = 0. It follows that − ∈ . Since is isomorphic to its image in ⁄ we may replace by ⁄ without changing notation. In

∗ other words we may assume that = 0. In that case − = 0, so if we ∗ put = we have ∼ ≤ .

Let be the closed ideal of generated by (and ). Since = 0 we see that ∈ . But since ≾ and = 0 also ∈ . With ∶ → ⁄ the quotient map this means that the three projections

() , () , () satisfy the assumptions of Lemma (3.1.9). Consequently ( − ) ∼ () ∼ ∗ ( − ). But now Lemma (3.1.10) applies to show that ( − ) = and ∗ ( − ) = for some partial isometry u in (). Since () is isomorphic to this means that − ∼ − in . Since we already had

∗ ∗ − = ~ = − , and since − ≤ 1 − , we conclude that − ∼ for some projection ≤ − . But then

∗ = − + is a projection in and

∗ ∗ ~( − ) ⊕ ( − ) . Since we will prove later that all extremally rich ∗-algebras with weak cancellation also have -surjectivity, the use of -surjectivity in the hypothesis of the next lemma is just a temporary expedient.

Lemma (3.1.12) [3]: Let be an extremally rich ∗-algebra and a closed ideal of . Assume that both and ⁄ have weak cancellation and that ⁄ has -surjectivity for every projection in . Then has weak cancellation.

Proof: Let and be projections in which generate the same closed ideal in and have the same image in (). Since weak cancellation passes to ideals we may

75 replace by , i.e. we may assume that and are full projections. If ∶ → ⁄ denotes the quotient map then the conditions above are also satisfied for () and

() relative to () and (()). By hypothesis there is therefore an element u in such that

(∗) = () and (∗) = () .

Since is extremally rich by Proposition (3.1.3), and () ∈ ℰ(()), we can choose in ℰ(). Let

∗ ∗ ∗ = − , = − , = .

Then and belong to and we can apply Lemma (3.1.5) to obtain a full projection in , projections and in such that [] = [] in (), and such that

~ ⊕ , = + . Next, an argument similar to part of the proof of (iii) ⇒ (i) in Theorem (3.1.6) yields

∼ ⊕ , = + , where is a full projection in and , are projections in () such that [] = [] in (()), as follows:

Let ∶ → ⁄() denote the quotient map. From [()] = [()] and tsr(()) = 1 we conclude that () ∼ (). Then there is in ℰ() such that () implements this equivalence. Thus

~ ⊕ , = + ,

∗ ∗ ∗ where = , = − , = − , and , ∈ (). Since [] − [] is in the kernel of ∗ ∗∶ (()) → (), there is in (/()) ∗ with = [] − []. Since is full in , is a full hereditary - subalgebra of and () is full in /(). Thus = [()], where () is unitary in (), and may be taken in ℰ(). Hence

⊕ ~ ⊕ , + = + ,

∗ ∗ ∗ where = , ~ ⊕ ( − ), = + − , and is full in . Now let = + .

Finally we note that () = (⋃ ) , where {} is an upward directed family of ideals each of which is generated by finitely many defect projections − ∗ , ∈ ℰ(). Here we are identifying with + ℂ. Since -theory is compatible with direct limits, and since every projection in () is contained in some , there are and a finite collection, , … , in ℰ() such that ∗ ∗ − , … , − generate , , ∈ , and [] = [] in ( ). Now an argument similar to part of the proof shows that there is a projection in

() which generates (as an ideal). Since has weak cancellation, it follows that ⊕ ∼ + , whence ∼ ⊕ ∼ + = . (Since ⊕ ≾ , we are not here assuming stable weak cancellation for ). Theorem (3.1.6) (iii) implies that the extremally rich ∗-algebra has weak cancellation if the set of equivalence classes of projections in is “convex” in a suitable sense for all of the form . Any extra hypotheses that guarantee this convexity imply additional positive results.

Corollary (3.1.13) [3]: If is an extremally rich ∗-algebra with weak cancellation, then has weak cancellation.

Proposition (3.1.14) [3]: If is a closed ideal in an extremally rich ∗-algebra and if has weak cancellation, then ⁄ has weak cancellation.

Proof: Let be a projection in ⁄ and put = (⁄). Then let be the inverse image of in . Since is a hereditary ∗-subalgebra of it has weak cancellation and is extremally rich. (But if the projection does not lift we may never find a unital substitute for ). It follows that extreme points lift from to , so if {, … , } is a finite subset of ℰ() it can be lifted to a finite subset {, … , } in ℰ(). Since has weak cancellation by Corollary (3.1.14) we can

∗ find a projection in (actually in () = ()) such that ∼⊕ (1 − ). It follows that

∗ ()~ ( − ), where ∶ → is the quotient map, whence / has weak cancellation by Theorem (3.1.6).

Proposition (3.1.15) [3]: If is an extremally rich ∗-algebra with weak cancellation, then:

(i) There is for every projection in ( ⊗ ) (= () ⊗ ) a projection in () such that ∼ .

(ii) If is a projection in () ⊗ , then there is an infinite set {} of mutually orthogonal projections in () such that ∼ , ∀. (iii) If () is -unital, then () has a full, hereditary, stable, -unital ∗-subalgebra .

Proof: (i) By Corollary (3.1.14) we may assume that is unital. Since () is generated as an ideal by the set = { − ∗| ∈ ℰ()} it follows that ( ⊗ ) is generated by the set ⊗ . There is therefore a finite subset {} in ℰ() such that

≾ ( − ∗).

Applying condition (iii) of Theorem (3.1.6) with = we find a projection in ∗ with ∼⊕ ( − ). Thus ∼ ≤ and evidently ∈ ( ⊗ ) ∩ = (). (ii) We do a recursive construction. At step we construct + 1

mutually orthogonal and equivalent projections, , … , , . The first step is done by applying part (i) with 2 in place of . At step

+ 1 we apply part (i) to = her( − ), where = + ⋯ + . Since is full in , () = ∩ (), and thus ∈ ().

Hence, we can obtain + 1 and + 1 by applying part (i) with 2 in place of .

(iii) The hypothesis implies that there is a countable set, {}, of projections in () such that () = id({}). Then the same technique as in part (ii) produces a countable set, {}, of mutually orthogonal projections which consists of infinitely many equivalent

copies of each . Then take = her({}). Corollary (3.1.16) [3]: If is an extremally rich ∗-algebra with weak cancellation, then also has stable weak cancellation.

Proof: We apply Theorem (3.1.6) to prove that () ⊗ has weak cancellation. This is immediate since each subalgebra of the form (() ⊗ ) is isomorphic to an algebra (). Then Lemma (3.1.12) implies ⊗ has weak cancellation, since tsr(( ⊗ )/(() ⊗ )) = 1.

We have

Corollary (3.1.17) [3]: If is an extremally rich ∗-algebra with weak cancellation and if is a projection in such that () = (equivalently () = where = id()), then is properly infinite. In particular, the hypotheses of Corollary (3.1.8) imply that every defect projection is properly infinite.

Proposition (3.1.18) [3]: If is a unital ∗-algebra which is extremally rich and has weak cancellation there is for each and every in ℰ(()) a in ℰ() such that

∗ ∗ ∗ ∗ − ~ − and − ~ − . Proof:

By Proposition (3.1.16) we can find projections and in () such that

∗ ∗ 2( )~ and 2( − )~.

Thus for subprojections ≤ and ≤ we have

∗ ∗ − ~ ≾ − and − ~ ≾ − . But then both 1 − and 1 − generate as an ideal, and evidently [1 − ] = ∗ [1 − ] in () (since [] = []). By weak cancellation − = and ∗ ∗ ∗ − = for some partial isometry in . But since − and − are centrally orthogonal in () we see that and are centrally orthogonal in , whence ∈ ℰ().

Corollary (3.1.19) [3]:

With as above, if () contains a proper isometry so does . (Finiteness implies stable finiteness).

Section (3.2): -Surjectivity and Good Index Theory - Injectivity and -Surjectivity In the next three lemmas we shall be concerned with a closed ideal in a unital ∗-algebra and will denote + ℂ1. (The possibility = is not excluded). In ∗ () we consider the unital -subalgebra consisting of matrices of the form

, ∈ ℂ , ∈ , ∈ . + Note that the subset of determined by = 0 is an ideal of which is Rieffel- Morita equivalent to . (If = this ideal is the whole of ). Thus every ideal of

gives rise to an ideal of (which is determined by ∈ , ∈ ∩ , and = 0). We shall commit a slight abuse of notation and denote both ideals by the same symbol. We shall denote by the connected component containing 1 of the group of invertible elements in () ∩ .

Lemma (3.2.1) [3]: Assume has weak cancellation and , ∈ . If ∈ ℰ() and ∗ + ∗ = , and if furthermore is the second column of a left invertible element of , then − ∗ ∼ ∗ in .

Proof: Since ∈ ℰ() we know that w is a partial isometry (∗ = − ∗) and

[ − ∗] = [] − [∗] = [∗] = [∗] in (). Thus the lemma follows by weak cancellation if we can show that both projections generate as an ideal.

In fact it is equivalent to show that they generate as an ideal. The condition for a projection in to generate as an ideal is twofold:

(i) Either ∉ or = . (ii) The algebra generates as an ideal.

But if generates as an ideal, then (i) is obviously true, and also is a full hereditary ∗-subalgebra of . And from this we easily deduce that (= ∩ ) is full in .

Now the fullness of ∗ follows from Lemma (3.1.4). Let denote the closed ideal of generated by − ∗. If ≠ we pass to / without changing notation. The conditions on and w are unchanged, but now ∗ = . Since ∈ ℰ() we have − ∗ centrally orthogonal to ∗ (= − ∗), and this now forces − ∗ = 0. Thus both and are co–isometries. But this contradicts the last part of the hypothesis.

Lemma (3.2.2) [3]: Suppose that is a left invertible element of and that is extremally rich with 0 weak cancellation. There is then an element in such that = for 0 some left invertible element in .

Proof: ⋆ Write = . By assumption ∗ is invertible so ∗ + ∗ ∈ ().If ⋆ ∗ ∗ ∗ = | | is the polar decomposition of and is a function in (0, ‖‖] ∗ ∗ ∗ such that |() − | is small for all in [0, ‖‖] then with = (| |)| | we ∗ ∗ ∗ still have + ∈ () . There is an element in ℰ() such that + ∗ is quasi–invertible. Thus + = for some in ℰ() and in () . Multiplying from the left by the element

0 0 ∗ ∗ 0 | | ⋆ −∗ in gives a matrix . Left multiplication by , also in , ⋆ 0 ⋆ gives the matrix , where = ( − ∗). Since we still have ∗ + ∗ ⋆ invertible and ∗ is a projection this implies that ∗ ≥ ( − ∗) for some > 0. If therefore = || is the polar decomposition of then ∈ (actually

∈ ) and we consider ℎ(||) in for some ℎ in (0, ‖‖] such that ℎ(||)|| = − ∗. Left multiplication with ∗ (| ∗|) − + ℎ 0 , 0 ⋆ ⋆ which belongs to , transforms the matrix into , where now ⋆ ⋆ ∗ = − ∗. ⋆ The elements , now satisfy the conditions in Lemma (3.2.1) since is ⋆ ∗ ∗ ∗ ∗ left invertible. Thus = and − = for some partial isometry in . Since ∈ ℰ() it has the form = + for some c in and in with

0 || = 1. After left multiplication with we can assume that = 1 without 0 ̅ ∗ ∗ ∗ ∗ any other changes. Since + = and + = the element = + is unitary in . For 0 ≤ ≤ 1 define − 0 ∗ = . 0 ( − )∗ 0 Since − ∈ it follows by routine calculations that ∈ for all . However, ⋆ 0 = = ⋆ for some elements in and in . Since the original element was left invertible it now follows that a must be left invertible (in ); so if 1 = we perform the final left multiplication by the element

0 in − 0 to obtain the desired solution . 0 Lemma (3.2.3) [3]: Suppose that is a quasi–invertible element in and that is extremally rich with weak cancellation. We can then find and in such that = 0 for some quasi–invertible element in . 0 Proof: Without changing notation, we replace with ⊕ ℂ, with ⊕ 0, and with ⊕ . This avoids an annoying but essentially trivial complication that would occur if ≠ but ( + )/ = /.

There are orthogonal closed ideals and of with corresponding quotient morphisms and such that () is left invertible and () is right ∗ invertible. Using Lemma (3.2.2) with (), () and ( ) in place of , and , which is legitimate by Proposition (3.1.15), we find an element () in such that

( ) 0 ( ) () = . 0 Since invertible elements in the connected component of the identity are always liftable we may assume that ∈ , and we can write

= , with in = ∩ . + Define the ∗-algebra

= ⊂ and note that the restriction of the morphism to is an isomorphism except possibly at the (1,1)–corner since ∩ = {0}. As () (= ) has weak cancellation, we can apply Lemma (3.2.2) with (), () and () in place of , and ; and find an element () in () such that ( ) 0 ( ) ( ) = . 0 Again we may assume that ∈ () , but since | is an isomorphism except at the (1,1)–corner this implies that 0 = 0 for some in . Necessarily then ∈ , as desired.

Theorem (3.2.4) [3]: ∗ Every extremally rich -algebra with weak cancellation has -surjectivity. Theorem (3.2.5) [3]: In the category of extremally rich ∗-algebras the subcategory of algebras that also have weak cancellation is stable under the formation of quotients, hereditary ∗-subalgebras (in particular ideals), matrix tensoring, Rieffel-Morita equivalence, arbitrary extensions, and inductive limits. Also if the extremally rich ∗-algebra has a composition series of ideals, {|0 ≤ ≤ }, such that = 0, = , and ⁄ has weak cancellation for each < , then has weak cancellation.

Proof: The wording of the result reflects the fact that the category of extremally rich ∗-algebras is not itself stable under (arbitrary) extensions, and stable only under extreme point preserving inductive limits.

We defined two analogues of which use extremal partial isometries in place of unitaries, or equivalently, quasi-invertibles in place of invertibles. One of these, denoted ℰ(), takes two extremals, each in some matrix algebra over , to be equivalent if ⊕ is homotopic to ⊕ in ℰ() for some (large) and suitable , . The equivalence relation for () is coarser. For example, the ∗ ∗ defect ideals, = id(1 − ) and = id(1 − ), are invariants of the −class ∗ ∗ of , as are also the classes of − and − in () and (), respectively. But even the Murray-von Neumann equivalence classes of − ∗ ∗ and − are invariants of the ℰ− class of . If this were the only difference between and ℰ, then obviously weak cancellation would imply = ℰ. Although the difference is more extensive, we shall prove in the next section that

() = ℰ() when is extremally rich with weak cancellation. Neither nor ℰ is a group, but both contain and the group acts on both. The next result is that extremal richness with weak cancellation implies -surjectivity. The same proof shows “ℰ-surjectivity”, a property which is formally stronger, but equivalent in this situation.

Theorem (3.2.6) [3]: ∗ Every extremally rich -algebra with weak cancellation has -surjectivity. Theorem (3.2.7) [3]: Every extremally rich ∗-algebra with weak cancellation has good index theory.

Proof: We are given a unital ∗-algebra , an ideal which is extremally rich with weak cancellation, the quotient map ∶ → ⁄, a unitary in ⁄, and in ∗ () such that () = [] in (⁄). We wish to find a unitary in such that () = . Let denote the natural map from () to (⁄). We may choose , power of 2, so that there is a unitary in () which belongs to the class such that () is homotopic to ⊕ − in ((⁄)). Then

( ⊕ )(()) can be lifted to in (()). We replace with , without changing notation, and thus achieve that () = ⊕ . Thus belongs to the algebra called in connection with Lemma (3.2.2), with ⁄() in place of , and that Lemma provides in such that has the form ⊕ ⁄. The (1,1)− corner of is congruent to a scalar modulo , and clearly we may assume this scalar is 1. Thus satisfies the same properties as , relative to /2, except for the inconsequential fact that it is only invertible instead of unitary. We may remedy this, if desired, with a polar decomposition. Continuing in this way, we attain our goal.

Proposition (3.2.8) [3]: Assume and are extremal partial isometries in matrix algebras over the ∗ unital -algebra which lie in the same -class. If the defect ideals are extremally rich with weak cancellation, then and lie in the same ℰ-class. Proof: Let and be the left and right defect ideals, which are the same for and . (Here we regard the defect ideals as ideals of , using the identification of ideals of

with ideals of ()). By replacing with () for suitable , with ⊕ , with ⊕ , and changing notation, we may assume and are in . Let ∶ → /( + ), ∶ → /, and ∶ → / be the quotient maps.

Since () = ()() is a unitary whose class in (/( + )) is 0, it follows from Theorems (3.2.4) and (3.2.6) that we may take to be a unitary whose class in () is 0. Thus () = ( ), where = . We now construct unitaries ∈ + and ∈ + such that the classes of and in () and (), respectively, are trivial, () = ( ), and () = ( ). Once this is done, we have that = = , since ∩ = 0. Since all of the w’s are trivial in (), it follows that and are equivalent in ℰ().

To construct , we may replace with /, since is an isomorphism on . Then and are isometries which agree modulo . The defect projections = − ∗ and = − ∗ each generate the ideal and have the same class ∗ ∗ ∗ in (). Thus there is x such that = and = . Then = + is a unitary in + , and = . Now since () = () and has -

surjectivity, there is in () which induces the same class as in (). So ∗ we can take = ( − + ). The construction of is similar. Corollary (3.2.9) [3]:

If is extremally rich with weak cancellation, then () = ℰ(). Extremal Analogues of Good Index Theory (3.2.10) [3]: Four different properties are listed below. In all cases is an ideal in a unital ∗ -algebra and ∶ () → (/) are the quotient maps. ( ) [ ] (i) If ∈ ℰ(/), ∈ () , and [()] = , then there is

∈ ℰ() such that () = and [] = [] . ( ) [ ] (ii) If ∈ ℰ(/), ∈ ℰ () , and [()]ℰ = ℰ , then there is

∈ ℰ() such that () = and []ℰ = []ℰ. ( ) [ ] (iii) If ∈ ℰ(/), ∈ ℰ () , and [()] = , then there is ∈ ℰ() such that () = . ( ) [ ] (iv) If ∈ ℰ(/), ∈ () , and [()]ℰ = ℰ , then there is ∈ ℰ() such that () = .

Theorem (3.2.11) [3]: Let be a closed ideal in a unital ∗-algebra and assume that is extremally ( ⁄ ) ( ) rich with weak cancellation. If ∈ ℰ , ∈ , and [()]ℰ = [ ] ( ⁄ ) ℰ , where ∶ () → is the quotient map, then there is ∈

ℰ() such that () = and []ℰ = []ℰ. Proof: The proof is almost identical to that of Theorem (3.2.7), except that we use Lemma (3.2.3) instead of (3.2.2) We may assume that n is a power of 2 and that

() is homotopic to ⊕ in ℰ((⁄)). There are () and () in ((⁄)) such that ⊕ = ()()(). We may assume , ∈ (()). Then without changing notation, we replace with to achieve () = ⊕ . Then coninue as in (3.2.7).

Corollary (3.2.12) [3]: Let be a closed ideal in a unital ∗-algebra and assume that is extremally ( ⁄ ) ( ) rich with weak cancellation. If ∈ ℰ , ∈ ℰ () , and [()] = ⁄ [], where ∶ () → ( ) is the quotient map, and if the defect ideals of are extremally rich with weak cancellation, then there is in ℰ() such that

() = and []ℰ = []ℰ. Corollary (3.2.13) [3]: Let be a closed ideal in a unital ∗-algebra and assume that is extremally rich with weak cancellation. If ∈ (⁄), ∈ ℰ(), the defect ideals of [ ] ⁄ are in , and [()] = , where ∶ () → ( ) is the ( ) quotient map, then there is ∈ ℰ() such that = and [] = [] . Corollary (3.2.14) [3]: Let be a closed ideal in a unital ∗-algebra and assume that is extremally ( ⁄ ) ( ) rich with weak cancellation. If ∈ , and if [] is in the image of , then can be lifted to ℰ().

Proposition (3.2.15) [3]: Let be a closed ideal in a unital ∗-algebra and assume that is extremally ( ⁄ ) ( ) rich with weak cancellation. If ∈ ℰ , ∈ ℰ () , and [()] = ( ⁄ ) [] , where ∶ () → is the quotient map, and if ( + ) ∩ = 0, where and are the defect ideals of , then there is ∈ ℰ() such that

() = and [] = []. Proof: Consider the pullback diagram

⎯⎯ /( + ) ↓ ↓ / ⎯⎯ /( + + ) where the maps are the obvious ones and = . Then () is a unitary whose -class is lifted by [()] ∈ (⁄( + )). By Theorems (3.2.7) and (3.2.4)

88 there is in (⁄( + )) such that () = () and [] = [()] in (⁄( + )). Thus there is in such that () = and () = , whence ∈ ℰ(). Now the defect ideals of are contained in + and map under to the defect ideals of , the defect ideals of are the same as those of (), and | is an isomorphism. Thus u also has defect ideals and . Then it follows that

[] = [] . Remarks (3.2.16) [3]: (i) The relations of the results in this section to classical index theory become clearer if reformulated in an equivalent way. Thus for good index theory we would start with in such that () is invertible in ⁄ (such an is called a Fredholm element relative to ) and seek a -perturbation of which is invertible. Similarly, for the extremal analogues of good index theory we would start with such that () ∈ (⁄) (such an was called quasi-Fredholm) and seek a -perturbation in . Quasi-Fredholm elements are meant to be analogous to classical semi-Fredholm operators (but we have used the name quasi-invertible, not semi-invertible). Also, instead of

hypothesizing in ℰ(()), we would hypothesize a class in (). Furthermore, we also defined an index space, Inde(), which is the orbit space of (⁄) under the image of (). The existence of could be reformulated in terms of the index in this

sense of [] (in (3.2.11) (i) or (iii)). Now every element of Ind() has built into it a pair (, ) of defect ideals and a class in (), where = ( + ), and is obtained from the boundary map just as in the Fredholm case. Moreover, for given (, ), is determined by . (However, it is awkward to describe which classes

arise in this way.) Since lives in () instead of (), it seems reasonable that we should use hypotheses on the defect ideals as well as on to prove (3.2.11) (i) or (iii). (ii) Corollaries (3.2.14) and (3.2.15) should be compared to a result of . Nagy. This implies a fortiori.

If has general stable rank (gsr) at most 2, and if is an element of ( ) ( ⁄ ) [ ] such that ( ) ≤ 0, then can be lifted to an isometry in . Now the hypothesis gsr() ≤ 2 is not comparable with our hypothesis on , but it is implied by tsr() = 1 or even csr() ≤ 2. Aside from this difference, () is intermediate in strength between (3.2.14) and (3.2.15). It is fairly routine to deduce [ ] ( ) from ( ) ≤ 0 the existence of an isometry in some such that [] [ ] ( ) lifts . Thus (3.2.14) gives the conclusion of and also allows us to control the -class of the lift if [] is given, whereas (3.2.15) states only that can be lifted to ℰ() and doesn’t require that the lift be an isometry. It is also interesting that even though our hypothesis on doesn’t imply Nagy’s, nevertheless Nagy’s proof will work with our hypothesis.

(iii) If ⁄ is extremally rich with weak cancellation, then the hypothesis on defect ideals in Corollary (3.2.13) is automatically satisfied. Also, by the last remark in (3.2.11), if the Corona algebra () = ()⁄ is extremally rich with weak cancellation, then (3.2.11) (iii) is true for all with no hypotheses other than those on . If the Corona algebra hypothesis is satisfied also for all ideals of , then we even get (3.2.11) (1). To see this we first use Proposition (3.2.16), applied to ⁄ ∩ ( + ), to reduce to the case ⊂ + . Then the argument based applies not just to give a lift but to show that has the same

defect ideals as . But implies that [] = [] . Example (3.2.17) [3]: For ease of notation set = () for some infinite dimensional separable Hilbert space and choose a projection in such that both spaces () and (1 − )() are infinite-dimensional. Let be the ∗-subalgebra of ⊗ consisting of convergent sequences = () such that

lim( − ) = lim ( − ) = 0 . This algebra was considered to give an example of a ∗-algebra which is the inductive limit of extremally rich ∗-algebras (actually von Neumann algebras) without itself being extremally rich.

Let = ⊗ , which is clearly a closed ideal in , and consider the (split) extension

0 → → → ⊕ → 0 .

In this piquant situation all -groups vanish; but the extremal -sets do not, and they control the quasi-Fredholm elements in since and ⁄ are extremally rich with weak cancellation. The Fredholm theory is trivial in this example: Every invertible element in ⁄ lifts to a invertible element in – as it must by Theorem (3.2.7).

Writing ℤ = ℤ ∪ {±∞} we find that () is the set of sequences in ℤ that are eventually zero, whereas (⁄) = (ℤ ) . An element = () belongs to if and only if every is either left or right invertible and there is an > 0 such ∗ that for all , || (and ||) has a gap (0, ) in its spectra. Since () converges to a block diagonal operator in we can describe () as eventually constant sequences () in ℤ together with an element (, ) in the first or third quadrant of (ℤ ) such that + = lim α. Elements () and () in () are composable if and only if and have the same sign for 1 ≤ ≤ ∞, in which case () + () = ( + ).

Thus the image of () in (⁄) is (ℤ) ∪ (ℤ) , and a quasi-Fredholm element can be perturbed to an element of if and only if its index is in the union of the first and third quadrants. Equivalently, an element of ℰ(⁄) can be lifted to ℰ() if and only if both components of its -class have the same sign.

It is also interesting to consider = ⊗ and = { ∈ | ∈ , ∀}.

Then = (), and ⁄ = (⁄). Here the ordinary -groups do not all vanish, and it can be seen that (⁄) = (⁄) = (), , : , , ∈ ℤ , (, ) ≠ (∞, −∞), (−∞, ∞) = + , eventually.

Proposition (3.2.18) [3]: Suppose that is an extremally rich ∗-algebra with weak cancellation such that the natural map

()() → () is injective. Then also the following map is injective:

()() → ().

Proof: Consider the commutative diagram

() ⎯⎯⎯⎯ ()

↓ ↓

() ⎯⎯⎯⎯ ()

If is in (()) such that [] = 0 then by Theorems (3.2.7) and (3.2.4) we have = () for some v in (()) with [] = 0. By assumption this means that

∈ (()), whence ∈ (()). The next result does not mention extremal richness, but it illustrates an application of good index theory.

Corollary (3.2.19) [3]: If is -unital and stable, then we have a short exact sequence of groups

0 → () → () → () → 0 .

The main goal of this section is to prove that extremal richness plus weak cancellation implies -injectivity. This is accomplished in Theorem (3.2.28), the main step being Lemma (3.2.26), which already includes all the extremally rich ∗-algebras which are purely properly infinite. Our proof is partly modeled on

Cuntz’s -injectivity, but we need some additional ideas, in particular the

92 introduction of -surjectivity. We also use a technique similar to one used by Zhang.

The map (3.2.20) [3]:

: (⁄) → (). Since Bott periodicity identifies (⁄) with ((⁄)), where S denotes suspension, we may consider to be defined on this latter group. We need to know the form of in the special case = [], ∈ ((⁄)). Thus, is given by a continuous function ∶ [0, 1] → (⁄) such that (0) = (1) = 1. Then can be lifted to ∶ [0, 1] → () such that

(0) = 1 and (1) ∈ (1 + ) ∩ (), and = [(1)]. It is important to note that (1) is null-homotopic in ().

Definitions (3.2.21) [3]:

We say that has (strong) -surjectivity if the group () is generated by {[]| is a projection in }. Thus Zhang’s result shows that ∗-algebras of real ∗ rank zero have strong -surjectivity. Cuntz showed that purely infinite simple - algebras satisfy a still stronger property, which, however, is too strong for our purposes below. We say that has weak -surjectivity if has -surjectivity. Then strong -surjectivity implies weak -surjectivity because the function defined by () = exp(2) is a unitary in ()~ which corresponds to under Bott periodicity, for each projection in . Since Rieffel showed that csr() ≤ 2 implies -surjectivity for and also that tsr() ≤ 1 implies csr() ≤ 2, we see ∗ that all -algebras of stable rank one have weak -surjectivity. Proposition (3.2.22) [3]:

If is extremally rich with weak cancellation, then () has strong - surjectivity.

Proof:

Since () is generated as an ideal by projections, (()) is generated by {[]| is a projection in () ⊗ }. But by Proposition (3.1.16), each such is equivalent to a projection in ().

Lemma (3.2.23) [3]: If is a ∗-algebra and is a -unital hereditary ∗-subalgebra such that contains an infinite set {} of mutually orthogonal and equivalent projections which are full (in ), then there is a full hereditary ∗-subalgebra ⊃ such that ≅ ⊗ , where is unital. In particular, has an approximate identity,

(), consisting of full projections, such that for each and , men ≾ . Lemma (3.2.24) [3]: ∗ If is an extremally rich -algebra with weak cancellation, then () has - injectivity.

Proof: Let = () and let be the forced unitization. Assume there is in

(1 + ) ∩ () whose -class is trivial but is not null-homotopic in (). We claim then that there is an ideal of which is maximal with respect to the property that + fails to be null-homotopic in ⁄. To prove this by Zorn’s

Lemma, we may assume a totally ordered collection {} of ideals such that, with = (⋃), + is null-homotopic and prove that for some , + is null- homotopic in ⁄. Since ⁄ is the image of , is homotopic in to some ∈ ( + ) ∩ (). Writing = + , ∈ , we see that for some

, ‖ + ‖ < 1, whence + is null-homotopic. Now, extremal partial isometries lift from ⁄ to , and thus (⁄) = ⁄. Therefore we may replace with ⁄ and with ⁄, without changing notation, and seek to obtain a contradiction.

Next choose a continuous function ∶ → [0, ∞), where is the unit circle, such that {|() ≠ 0} = { | 2⁄3 < < 4⁄3}. Let = her(()), and note that ⊂ , since (1) = 0. If = 0, then the spectrum of omits −1, a contradiction.

Case (i): If tsr() = 1, let = id(). By construction, + is null- homotopic. Hence we see as above that is homotopic to some ∈ (). If is the class of in (), then maps to 0 in (). It follows that = for some in (⁄). Now Proposition (3.2.24) implies that ⁄ has -surjectivity (since (⁄) = ⁄). Thus (3.2.22) applies, and we see that is represented by a

94 unitary w in which is null-homotopic in (). But now ∗ is a unitary in whose -class vanishes and tsr() = 1. Thus Rieffel’s -injectivity result, now applies to give our contradiction.

Case (ii): If tsr() > 1, then let p be a non-zero defect projection for and note that 1 − is full (by Lemma (3.1.4), for example). Let = id(), and let ∶ → ⁄ be the quotient map. Since Re() ≤ − , is invertible in and is homotopic to within (). It follows that is homotopic to

+ , where is a unitary element of ( − )( − ) such that () = (). To see this, first homotop within to

( − ( − )()) − ()( − ), which has the form + , invertible in ( − )( − ).

By construction () is null-homotopic. Since () = (( − )( − )), every element of ⁄ lifts to (( − )( − )). It follows that is homotopic (within (( − )( − ))) to a unitary in ( − )( − ).

If is the class of in (), then maps to 0 in (). It follows that = for some in (⁄). As above we use the -surjectivity of ⁄ and (3.2.22) to get a special form for , but now we use ( − )( − ) in place of . So we find a unitary in ( − )( − ) such that is null-homotopic in ∗ (( − )( − )) and [] = in (). Then if = , we see that + is homotopic to in () and [ + ] = 0 in (). From now on all the action takes place in , and we will make no further use of the assumption that has a null-homotopic image in any non-trivial quotient.

Next let ∶ → ⁄() be the quotient map. By which we apply within

(( − )( − )), () is null-homotopic. (Here we are using the fullness of − , which implies that [] = 0 in (( − )( − ))). Since (( − )( − )) has weak -surjectivity, we may use the above argument to homotop to an element in ( − )()( − ) such that [] = 0 in (()).

Then we write () = ⋃ , where each is the ideal generated by finitely many defect projections, and {} is directed upward. Then is homotopic to in some (( − )( − )), and because of the compatibility of with direct limits, we may assume [] = 0 in (). Clearly we may also assume ∈ −

+ ( − )( − ). Since is full in , () = ∩ (), a full hereditary ∗-subalgebra of (). Hence every projection in () is equivalent to a projection in () ⊗ . So Proposition (3.1.16) (ii) can be applied to to find an infinite set, {}, of mutually orthogonal and mutually equivalent projections in , which are full in . (So that actually ∈ ).

Finally we apply Lemma (3.2.25) with in place of and = her( − + ). If and {} are as in the Lemma, let = + ( − + ). Since → + , for large there is a unitary in such that − + is homotopic to + in (). Since is full in , () is naturally isomorphic to (), and hence [] = 0 in (). It follows that for some

, ⊕ is null-homotopic in (()). But the conclusion of

Lemma (3.2.25) states that men ≾ − . Thus − + is null- homotopic in (), and is null-homotopic in ().

Proposition (3.2.25) [3]: Let be an ideal of a ∗-algebra . Then:

(i) If and ⁄ have -injectivity and ⁄ has weak -surjectivity, then has -injectivity. (ii) If and ⁄ have weak -surjectivity and has -injectivity, then has weak -surjectivity. Proof: We may assume unital. Let ∶ → ⁄ be the quotient map.

The argument for part (i) already occurred several times in the proof of Lemma

(3.2.26). If [] = 0, ∈ (), then () is null-homotopic, whence there is in () which is homotopic to . If = [](), then = , and (3.2.22)

96 applies to give in () which is null-homotopic in () and which represents ∗ the class . Then the -injectivity of is applied to .

For part (ii) we are given in (), which is identified with (). Then ()() is represented by in (⁄), and u is represented by a function ∶ [0, 1] → (⁄) such that (0) = (1) = . Lift f to ∶ [0, 1] → () such that (0) = . Then, since (()()) = 0, we see from (3.2.22) that [(1)]() = 0 and so (1) is null-homotopic in (). Thus there is ℎ ∶ [0, 1] → () such that ℎ(0) = and ℎ(1) = (1), so that ℎ∗ gives an element of () which represents a class in (). Finally, − is in the image of () (which is the kernel of ()) and is therefore represented by a unitary. Theorem (3.2.26) [3]: ∗ If is an extremally rich -algebra with weak cancellation, then has - injectivity and weak -surjectivity. Theorem (3.2.27) [3]: If is an extremally rich ∗-algebra with weak cancellation, then also has

-injectivity. Proof:

Let and be elements of ℰ() which lie in the same -class. Exactly as in the proof of Proposition (3.2.9), we show that = , where all the ’s are unitaries in whose -classes vanish. Thus, by Theorem (3.2.28), all the ’s are in (), and hence is homotopic to in ℰ(). The next result should be regarded as an example.

Corollary (3.2.28) [3]: If is a purely infinite simple ∗-algebra, then any two proper isometries in are homotopic within the set of isometries.

The next result is a summary theorem which includes the results we have proved about three classes of extremally rich ∗-algebras, except that it omits the results related to the extremal analogues of good index theory.

Theorem (3.2.29) [3]: If A is an extremally rich C∗-algebra, then, under any one of the hypotheses listed below, A has weak cancellation, -bijectivity (i.e., (A) = (A)A), good index theory, -bijectivity (i.e., K(A) = ℰ(A)⁄homotopy), and weak K- surjectivity. Moreover, (A) has strong K-surjectivity. (i) has real rank zero. (ii) If is the left defect ideal of an element of ℰ(), then () = . In particular, this applies if is purely infinite or if every defect projection is properly infinite. (iii) is isometrically rich.

Lemma (3.2.30) [3]: If is an extremally rich ∗-algebra whose primitive ideal space is Hausdorff, then has weak cancellation.

Proof: We verify condition (iii) in Theorem (3.1.5). Thus let = for a projection in , and consider ∈ ℰ() with left and right defect ideals and . Since ⋁ is a compact-open subset of ⋁, which is Hausdorff, we see that ⋁ is closed. Thus = ⊕ . Let = ⊕ . From ⊂ it follows that is an isometry and a co-isometry. Hence if = ⊕ , then is an isometry with the same left defect projection as . Now if , , … , are in ℰ(), then = ∏ is an isometry in and ∗ ∗ − ~ ( − ) = 1

Proposition (3.2.31) [3]: If is an extremally rich ∗-algebra whose primitive ideal space is almost Hausdorff, then has weak cancellation.

Proof:

The hypothesis implies that has a composition series of ideals, {|0 ≤ ≤ ⋁ }, such that = 0, = , and (⁄) is Hausdorff for each < . Thus the result follows from the Lemma and the last sentence of Theorem (3.2.5).

Corollary (3.2.32) [3]: If is an extremally rich ∗-algebra which is of type , then has weak cancellation.

We state with at most minimal indications of proof some results which are relevant mainly to non-extremally rich ∗-algebras.

Theorem (3.2.33) [3]: Assume is an ideal of a ∗-algebra , and ⁄ have weak cancellation, has real rank zero, and ⁄ has -surjectivity for each projection in . Then has weak cancellation.

Theorem (3.2.34) [3]: Assume is an ideal of a ∗-algebra , ⁄ has weak cancellation, has stable ∗ weak cancellation for every hereditary -subalgebra of , and ⁄ has - surjectivity for each projection in . Then has weak cancellation.

Theorem (3.2.35) which has a similar conclusion when has real rank zero. The proof of (3.2.35) is somewhat similar to that of Lemma (3.1.11), one difference being that a different method is used for lifting partial isometries. The proof of (3.2.36) is also somewhat similar. Here a key difference is that the boundary map, ∶ (⁄) → (), is dealt with by the method applicable to general ∗-algebras –i.e., the unitary in ⁄ need not be liftable to a partial isometry in .

The proof of the next theorem is somewhat similar to parts of the proof of Lemma (3.2.26).

Theorem (3.2.35) [3]: ∗ Every purely properly infinite -algebra has -injectivity. Just as we found it necessary to link weak cancellation with -surjectivity and -injectivity with - surjectivity to facilitate several proofs, so the following more obvious linkage can be useful.

Proposition (3.2.36) [3]: Let be an ideal of a ∗-algebra . Then:

(i) If and ⁄ have -surjectivity and has good index theory, then has -surjectivity. (ii) If and ⁄ have good index theory and ⁄ has -surjectivity, then has good index theory.

Propositions (3.2.27) and (3.2.38), and Theorem (3.2.36), combined with easy direct limit arguments, can be used to derive results for ∗-algebras that have composition series with well-behaved quotients. Here is one example. Note that has generalized stable rank one if has a composition series of ideals{| 0 ≤ ≤ } , such that = 0, = , and tsr(⁄) = 1 for each < . Since implies that every type extremally rich ∗-algebra has generalized stable rank one, the following result includes Corollary (3.2.34).

Proposition (3.2.37) [3]: Every ∗-algebra of generalized stable rank one has stable weak cancellation,

-bijectivity, good index theory, and weak -surjectivity. Of the three parts of Theorem (3.2.31), case (iii) is the widest in scope, and it is this case which most justifies the belief that extremal richness is a useful hypothesis for proving weak cancellation. Note that it is also unknown whether real rank zero implies weak cancellation.

100

Chapter 4 Nuclear ∗-algebras A noncommutative Amir- Cambern Theorem for von Neumann algebras In this chapter we show that an intermediate result, we compare the Banach- Mazur cb-distance and the Kadison-Kastler distance. Finally, we show that if two ∗-algebras are close enough for the cb-distance, then they have at most the same length [4]. Section (4.1): Almost Completely Isometric Maps We concern perturbations of operator algebras as operator spaces, more precisely perturbations relative to the Banach-Mazur cb-distance. G. Pisier introduced the Banach-Mazur cb-distance (or cb-distance in short) between two operator spaces , :

(, ) = inf{‖‖‖ ‖}, where the infimum runs over all possible linear completely bounded isomorphisms : → . This extends naturally the classical Banach-Mazur distance for Banach spaces when these are endowed

With their minimal operator space structure (in particular, the Banach-Mazur distance and the cb-distance between two ()-spaces coincide).

For ()-spaces, being isomorphic (or equivalently cb-isomorphic) is a very flexible relation; Milutin's theorem states that () is isomorphic to ([0,1]) for any uncountable compact metric space . this theorem has some noncommutative generalization for separably acting injective von Neumann algebras [7]: (Avon Neumann algebra is a ∗-algebra of bounded operators on a Hillbert space that is closed in the weak operator topology and contains the identity operator) and for separable non-type unclear ∗-algebras. But here, we are interested in perturbation results when the cb-distance is small. Let us recall the generalization of Banach-Stone theorem obtained independently by D. Amir and M. Cambern: if the Banach-Mazur distance between two ()-spaces is strictly smaller than 2, ∗ then they are ∗ -isomorphic (as -algebras). Actually, this is also true for spaces of continuous functions vanishing at infinity on locally compact Hausdorff spaces.

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One is tempted to extend the Amir-Cambern Theorem to noncommutative ∗- algebras. R. Kadison described isometries between ∗-algebras, in particular the isometric structure of a ∗-algebra only determines its Jordan structure, hence to recover the ∗-structure we need a priori assumption on the cb-distance (not only on the classical Banach-Mazur distance). Here, we show:

Theorem (4.1.1) [4]: Let be a separable nuclear ∗-algebra or a von Neumann algebra, then there ∗ exists an > 0 such that for any -algebra ℬ, the inequality (, ℬ) < 1 + implies that and ℬ are ∗ -isomorphic.

∗ When is a separable nuclear -algebra, one can take = 3. 10 . When is a von Neumann algebra, = 4. 10 is sufficient. Such a result can not be extended to all unital ∗-algebras, see Corollary (4.2.9) below for a counter-example involving nonseparable ∗-algebras.

The proof of Theorem (4.1.1) is totally different from the commutative case, the cb-distance concerns only the operator space structure, hence the basic idea is to gain the algebraic structure. It is crucial to work with the completely bounded cohomology here, because we can exploit the deep result that every completely bounded cohomology group of a von Neumann algebra over itself vanishes, this is unknown for the bounded cohomology. This allows us to conclude for von Neumann algebras.

For separable nuclear ∗-algebras, the strategy is different, because vanishing of completely bounded cohomology groups is not available. First, we compare the cb-distance and the completely bounded Kadison- Kastler distance , (see the definition below):

Proof: The proof of Theorem (4.1.1) is a variant of the proof of Theorem (4.1.2). For clarity we postpone the quantitative estimate to the next Remark.

When is a separable nuclear ∗-algebra, this follows directly from Theorem (4.1.2).

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When is a von Neumann algebra, one does not need to go to the bidual ∗∗ in the preceding proof (to apply Proposition (4.2.2)), so that we directly conclude that

∶ → ℬ is a ∗ -is omorphism. But we should mention another way (which improves theoretically our bound in the von Neumann algebras case): the second step in the proof is not necessary, just define directly a new multiplication using the cb-isomorphism (instead of ). Then is just an algebra isomorphism (not necessarily selfadjoint).

Theorem (4.1.2) [4]: There exists a constant > 0, such that for any ∗-algebras and ℬ, there exist faithful ∗ -representations and ℬ of and ℬ on the same Hillerbt space such that

,(), ℬ(ℬ) ≤ ln (, ℬ).

One can choose = 3620, when (, ℬ) < 1 + 10.

In order to prove this theorem, we need to control explicitly the defect of selfadjointness of a unital almost completely isometric map. Then we will use stability of separable nuclear ∗-algebras for the Kadison- Kastler distance, this is a major result in perturbation theory. Let us recall this result more precisely. As usual H denotes a Hilbert space and () its bounded linear endomorphisms. Let , ℬ be subalgebras of (), the Kadison-Kastler distance between and ℬ inside () is

(, ℬ) = (), (ℬ), where denotes the Hausdorff distance and () (respectively (ℬ)) denotes the unit ball of (ℬ respectively). We also recall its completely bounded version defined by:

( ) ( ) ( ) , , ℬ = supℓ⨂ , ℬ (using the well-known identification () = (ℓ⨂)). Then we have the main result: for any < 42. 10, if ⊂ () is a separable nuclear sub-∗- ∗ agebra and ℬ ⊂ () is another sub- -agebra if (, ℬ) ≤ then and ℬ are

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∗ ∗ -isomorphic. Therefore, it is clear that -case of Theorem (4.1.1) is a corollary of this last result and our Theorem (4.1.2).

We already mentioned that an Amir- Cambern type theorem is false for any ∗ -algebras, however we can try to prove that some ∗-algebraic invariants are preserved under perturbation relative to the cb-distance. The notion of length of an operator algebra has been defined by G. Pisier in order to attack the Kadison similarity problem (he proved that a ∗-algebra has finite length if and only if it has the Kadison similarity property). It has been proved that having finite length is a property which is stable under perturbation for the Kadison- Kastler distance. Here, we prove that if two ∗-algebras are close enough for the cb-distance, then they have at most the same length.

Proof:

As , is bounded, the statement is only interesting when (, ℬ) is close to 1. The proof uses ideas.

Let ∶ → ℬ be a cb-isomorphism with ‖‖ ≤ 1 and ‖ ‖ sufficiently closed to 1.

Consider the bidual extension still denoted by ∶ ∗∗ → ℬ∗∗, it remains a - isomorphism and satisfies the same norm estimates.

We suppose unital, we will treat the non-unital case afterwards. The first step is to unitize . By Proposition (4.1.7), (1) is invertible in ℬ. If ‖ ‖ < √2. Let = (1), then is unital and

‖ ‖ ‖‖ ≤ 2 − ‖ ‖ and ‖ ‖ ≤ ‖ ‖. Note also that () = ℬ. The second step is to make our cb-isomorphism selfadjoint. By Theorem ⋆ (4.1.10), we have ‖ − ‖ ≤ 2(‖ ‖) for some continuous function with (1) = 0. Let = ( + ⋆). Then : ∗∗ → ℬ∗∗ is unital ∗ -preserving, ‖ − ‖ ≤ (‖ ‖) and () ⊂ ℬ. So if ‖ ‖ is close enough to 1, is also a cb-isomorphism such that () = ℬ with norm estimates ‖‖ ≤

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(‖ ‖) and ‖ ‖ ≤ (‖ ‖), for some continuous function at 1 with (1) = 1. Define on ∗∗ a new multiplication by, for , ∈ ∗∗

m(, ) = ()().

The multiplication m is associative and ∗ -preserving. It is obviously completely bounded and clearly

∨ ‖m − m‖ ≤ ‖ ‖‖ ‖.

Thus, the estimate in Theorem (4.1.10) gives that ‖m − m∗∗‖ ≤ (‖ ‖) for some continuous function with (1) = 0. If ‖ ‖ is close enough to 1, we get from Proposition (4.2.2), that there is a completely bounded ∗ -preserving linear isomorphism ∶ → with

‖ − id∗∗‖ ≤ (‖ ‖)

∗∗ for some continuous function with (1) = 0 and for , ∈

()() = m(, ) = ()().

Note that necessarily (1) = 1, and is ∗ -preserving. Let = ∶ ∗∗ → ℬ∗∗, it is a ∗-preserving cb-isomorphism. Moreover, for ∗∗ , ∈ , () = ()(), hence is actually a ∗ -isomorphism. Now represent faithfully ℬ∗∗ inside () for some Hilbert space and let us check that the ∗-algebras () and ℬ are close for the Kadison-Kastler distance inside ().

Actually we just need to check this for the Kadison- Kastler distance, as above is close to id∗∗ in cb-norm. For any ∈ (), we have:

‖() − ()‖ ≤ (‖ ‖), (1) and as () = ℬ, for ∈ (ℬ), we have

− () ≤ ‖ ‖(‖ ‖) = (‖ ‖).

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From which one easily deduces ,((), ) ≤ (‖ ‖), for some continuous function with (1) = 0. Now if is non-unital, in the preceding proof, (1) is now invertible in ℬ∗∗ (here 1 denotes the unit of ∗∗), so () = ℬ is not valid anymore. But the inequality (1) above still holds and we deduce that for ∈ ()

‖() − ()‖ ≤ ( + )(‖ ‖). Now from Lemma (4.1.8), there is a unitary in ℬ∗∗ such that ‖ − (1)‖ ≤ (‖ ‖) for some continuous function with (1) = 0. Therefore

‖() − ()‖ ≤ ( + + )(‖ ‖). Taking the adjoints we obtain

∗ ∗ ∗ ‖() − ( ) ‖ ≤ ( + + )(‖ ‖). Write = , for some and in the (), then

‖() − ()‖ ≤ 2( + + )(‖ ‖). (2) As ()(∗)∗ belongs to (ℬ), we conclude that the ∗-algebra () is nearly included in the ∗-algebra ℬ∗. Let us prove the converse near inclusion. Let ∈ (ℬ), we can factorize = ()(∗)∗ with , ∈ such that ‖‖ ≤ ‖‖ and ‖‖ ≤ ‖‖. From inequality (2), we get

∗ ∗ ∗ ‖() − ()( ) ‖ ≤ 2‖ ‖ ( + + )(‖ ‖).

∗ Finally, ,((), ℬ ) ≤ ‖ ‖ ( + + )(‖ ‖). Theorem (4.1.3) [4]: Let ≥ 1 and ℓ ∈ ℕ\{0} fixed but arbitrary constants. If and ℬ are unital ∗ ℓ -algebras with (, ℬ) < 1 + 10 4 and has length at most ℓ and length constant at most , then ℬ has length at most ℓ.

This section starts with few technical lemmas relating algebraic properties to norm estimates in operator algebras. We next use them to study almost completely isometric isomorphisms.

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It is well known that unital completely isometric isomorphisms between operator algebras are necessarily multiplicative. Hence one can hope that unital almost completely isometric bijections are almost multiplicative and almost selfadjoint. This can be checked easily by an ultraproduct argument, but the important point is to control explicitly the defect of multiplicativity and the defect of selfadjointness.

When : → ℬ is a map between two operator algebras, the defect of multiplicativity of is denoted by ∨. It consists of the bilinear map ∨: → ℬ given by ∨(, ) = () − ()(). As usual when dealing with bilinear maps, the completely bounded norm of ∨ is the cb-norm of the induced linear ∨ map : ⨂ → ℬ on the Haagerup tensor product. Given a cb map ∶ → between two operator systems, we use the notation ⋆ defined on by ⋆() = (∗)∗. The linear map − ⋆ measures the defectness of selfadjointness of . We recall the following proof.

Proof: As in the proof of Theorem (4.1.2), let ∶ → ℬ a linear cb-isomorphism with ‖‖‖ ‖ < 1 + . We consider = (1) and the multiplication m on defined by

m(, ) = ()().

Hence

mℓ − mℓ ≤ ‖‖ ∨ℓ .

Since ‖ ‖ ≤ 1 + () and ‖‖ ≤ 1 + () with () tending to 0 when tends to 0, by Theorem (4.1.10), ‖‖ ∨ℓ < 1/, if is small enough. ∨ℓ ∨ℓ (Naturally here, denotes the ℓ-linear map defined on Al by (, … , ℓ) = ( … ℓ) − () … (ℓ).) Therefore by Proposition (4.2.7), equipped with m has also length at most ℓ. But S is a cb-isomorphic algebra isomorphism from equipped with m onto ℬ, so ℬ has length at most ℓ as well.

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For the quantitative estimates, clearly ∨ℓ ≤ ‖∨‖ ∑ℓ‖‖ , hence to ℓ apply Proposition (4.2.7) we need ‖ ‖ ≤ 88√2 < 1/, which gives the explicit bound.

Proposition (4.1.4) [4]: For any > 0, there exists ∈ (0,1) such that for any unital operator algebras , ℬ, for any unital cb-isomorphism : → ℬ, ‖‖ ≤ 1 + and ‖ ‖ ≤ ∨ 1 + imply ‖ ‖ < . Proof:

Suppose the assertion is false. Then there exists > 0 such that for every positive integer ∈ ℕ\{0}, there is a unital cb-isomorphism : → ℬ between some unital operator algebras satisfying 1 1 ‖ ‖ ≤ , ‖‖ ≤ 1 + and ⋁ ≥ .

Let be a nontrivial ultrafilter on ℕ, let us denote (resp. ℬ) the ultraproduct ∏ ( ⨂)⁄ (resp. ∏ ( ⨂ℬ)⁄), here denotes the unitization ∗ of the -algebra of all compact operators on ℓ. Then (resp. ℬ) is a unital operator algebra. Now consider : → ℬ the ultraproduct map obtained from the id ⊗ ’s. Hence is a unital surjective linear complete isometry between ∨ operator algebras, so is multiplicative hence = 0.

∨ ∨ This contradicts the hypothesis for all , ‖‖ ≥ . Indeed ‖‖ = ∨ ‖(id ⨂) ‖, so there are , in the closed unit ball of ⨂ such that

‖(id ⨂)(, ) − (id ⨂)()(id ⨂)()‖ ≥ , which implies that

‖(̇ ̇) − (̇ )(̇)‖

(where ̇ denotes the equivalence class of () in ). A similar proof gives.

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Proposition (4.1.5) [4]: For any η > 0, there exists ρ ∈ (0,1)] such that for any operator systems S, , for any unital cb-isomorphism T ∶ S → , ‖T‖ ≤ 1 + ρ and ‖T ‖ ≤ 1 + ρ ⋆ imply ‖T − T ‖ < . We turn to quantitative versions of the previous Propositions for ∗-algebras.

The next Lemma is interesting because it gives an operator space characterization (it involves computations on 2 × 2 matrices) of invertibility inside a von Neumann algebra.

Lemma (4.1.6) [4]: Let be a von Neumann algebra and ∈ ℳ, ‖‖ ≤ 1. Then, is invertible if and only if there exists > 0 such that for any projection ∈ ℳ, ≥ + ‖‖ and ‖[ ]‖ ≥ + ‖‖ (3) If this holds, the maximum of the ’s satisfying (3) equals ‖‖ and (3) holds for any ∈ ℳ.

Proof: If x ∈ ℳ is invertible and y ∈ ℳ is arbitrary, by functional calculus, ∗ ≥ ‖‖, thus

∗ ∗ ∗ = ‖ + ‖ ≥ ‖‖ ‖ + ‖ = ‖ ‖ + ‖‖ . Thanks to a similar argument for the row estimate, we get that (3) holds with = ‖‖.

∗ Assume that (3) is satisfied. Fix ≥ 0 and let = [,]( ) ∈ ℳ be the spectral projection of || corresponding to [0, ]. By the functional calculus ∗ 1 + ≥ + as ‖‖ ≤ 1. Taking = in the first part of (3) gives,

1 + ≥ ≥ + ‖‖ . ∗ ∗ Hence = 0 for < . Thus is invertible and ‖( ) ‖ ≤ . Similarly ∗ must have the same property and is left and right invertible hence invertible.

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In the polar decomposition of = ||, must be a unitary so that we finally get ≤ ‖‖ and the proof is complete.

The next Proposition generalizes the well-known fact that a complete isometry between ∗-algebras sends unitaries to unitaries.

Proposition (4.1.7) [4]: Let , ℬ be two ∗-algebras. Let ∶ → ℬ be a cb-isomorphism such that ‖‖‖ ‖ < √2. Then, for any unitary ∈ , () is invertible and ‖‖ ‖()‖ ≤ 2 − ‖‖‖ ‖ Proof: Passing to biduals, we can assume that and ℬ are von Neumann algebras as ∈ ℬ is invertible in ℬ if and only if it is invertible in ℬ∗∗. Replacing by ⁄‖‖, we may assume that ‖‖ = 1 and ‖ ‖ < √2. Let ∈ ℬ with ‖‖ = 1, as ‖()‖ ≤ 1:

() 1 1 + ‖()‖ 2 ≥ ≥ ≥ . () ‖‖ ‖ ‖ ‖ ‖ Hence () satisfies (1) with = − 1 > 0. Finally applying Lemma ‖ ‖ (4.1.3), we obtain

1 ‖‖ ( ) ‖ ‖ ≤ ≤ . 2 − ‖ ‖ We have the Lemma.

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Lemma (4.1.8) [4]: Let be a unital ∗-algebra and ∈ invertible. Then there exists a unitary ∈ such that ‖ − ‖ = max ‖‖ − 1,1 − . ‖‖

Proof: Write the polar decomposition of = ||. As x is invertible, || is strictly positive element of , so is a unitary of . Obviously, ‖ − ‖ = ‖|| − 1‖. Seeing || as a strictly positive function thanks to the functional calculus, it is not difficult to conclude.

The next Lemma is the key result to compute explicitly the defect of multiplicativity. As in Lemma (4.1.6), operator space structure is needed.

Lemma (4.1.9) [4]: Let , be two unitaries in (). Let ∈ () and ≥ 1 such that ≤ √2, −1 then ‖ − ‖ ≤ 2√ − 1.

Proof: Note first that ∗ 0 1 0 1 ∗∗ = , −1 1 −1 0 ∗ −1 1 hence without loss of generality we can assume that = = 1. Take ℎ ∈ , then 1 −ℎ −ℎ ≤ √2 . −1 1 ℎ ℎ

Therefore ‖(ℎ) − ℎ‖ + 4‖ℎ‖ ≤ 4‖ℎ‖, which implies ‖ − 1‖ ≤ 2√ − 1.

We are now ready to prove the main result.

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Theorem (4.1.10) [4]: Let , ℬ be two unital ∗-algebras. Let ∶ → ℬ be a cb-isomorphism with (1) = 1 and ‖‖‖ ‖ < √2, then

∨ ‖ ‖ ≤ 2(‖‖) − 1 + ()(1 + ‖‖).

( ) ⋆ ‖ − ‖ ≤ 2‖‖ + − 1 + 2(), √2

where () = max ‖‖ − 1,1 − − ‖‖. ‖‖

Proof: We start with the defect of multiplicativity. From the definition of the Haagerup tensor norm and the Russo-Dye Theorem, it suffices to show that for any unitaries

, ∈ ℳ() we have

() ‖() − ()()‖ (ℬ) ≤ ‖‖ + − 1 + ()(1 + ‖‖) (4) √2 where = ⊗ . Without loss of generality, we can assume = 1. Let , ∈ unitaries, as = 2, we get −1 √ () () ≤ ‖‖ √2. −1 () From Lemma (4.1.8) and Proposition (4.1.7) we deduce that there are unitaries , ∈ ℬ with ‖() − ‖ ≤ () and ‖() − ‖ ≤ (). The triangular inequality gives

() ≤ ‖‖ √2 + (). −1

() Lemma (4.1.9) implies that ‖() − ‖ ≤ 2‖‖ + − 1, so that √ we get the estimate using the triangular inequality once more.

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⋆ ⋆ For the second estimate, as = ( ) we may also assume = 1. Thanks to the Russo-Dye Theorem, we just need to check that for any ∈ unitary

( ) ∗ ∗ ‖() − ( ) ‖ ≤ 2‖‖ + − 1 + 2(). √2 Taking = ∗ in the above arguments leads to ‖(∗) − ‖ ≤ () ∗ () 2‖‖ + − 1. Hence ‖ − ‖ ≤ 2‖‖ + − 1 and we √ √ conclude using the triangular inequality (4) above using the triangle inequality.

Section (4.2): A noncommutative Amir-Cambern Theorem

Definition (4.2.1) [4]: Let be an operator space. A bilinear map ∶ × → is called a multiplication on if it is associative and extends to the Haagerup tensor product

⊗ .

We denote by m the original multiplication on an operator algebra .

In the following, (, ) denotes the completely bounded cohomology group of over itself.

The next proposition is the operator space version. It gives a precise form of small perturbations of the product on anoperator algebra under cohomological conditions.

As before, the quantity ‖m − m‖ is the cb-norm of m − m as a linear map from ⊗ into . Proposition (4.2.2) [4]: Let be an operator algebra satisfying

(, ) = (, ) = 0. (5) Then there exist , > 0 such that for every multiplication m on satisfying

‖m − m‖ ≤ , there is a completely bounded linear isomorphism ∶ → such that

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‖ − id‖ ≤ ‖m − m‖ and m(, ) = ()(). If is a von Neumann algebra, then (5) is automatically satisfied with values = 1⁄11 and = 10. Moreover if m satisfies m(∗, ∗) = m(, )∗ for all , ∈ , then can be taken to satisfy (∗) = ()∗ for all ∈ .

In the bounded situation, one has to apply an implicit function theorem to the right spaces of multilinear maps. This is done in details. With the notation there (taking ℳ = ) one simply need to replace ℒ(ℳ, ℳ)by their cb-version ℒ(, ) which are obviously Banach spaces. The statement about selfadjointness of is justified.

If is a von Neumann algebra, all completely bounded cohomology groups of over itself vanish and we can choose = = 1, which gives = 11. Now let us compute , the rational function satisfies () ≤ 9.75, for small enough. Hence the sequence () defined by = () (and = ‖m − m‖) verifies ≤ 9.75 . As = = 1, we have ‖‖ ≤ + 2. Then, using the previous estimates of the ’s we get ‖ − id‖ ≤ exp(∑‖‖) ≤ 10. Remark (4.2.3) [4]: We give a rough estimate for the constants in Theorems (4.1.1) and (4.1.2).

With notation from the proof of Theorem (4.1.2), we start with ‖‖ ≤ 1 and = ‖ ‖ − 1, then 1 + ‖‖ ≤ (2 − (1 + ))⁄ and ‖‖ ≤ 1 + . From now, we assume that ≤ . One easily checks, computing derivative that

2 ( ) = max ‖‖ − 1,1 − − ‖‖ ≤ 2, ‖ ‖

‖‖ < 1 + 3. Then

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∗ ‖ − ‖ ≤ 21 + 3 + √2 − 1 + 4 ≤ 10√ = 2().

We get that is invertible as soon as 5√ < + . Let us now assume that < so that

‖‖ ≤ ‖‖ + ‖ + ‖ ≤ 1 + 6√, ‖ ‖ 1 + ‖ ‖ ≤ ≤ ≤ 1 + 8√. ‖ ‖‖ − ‖ 1 − 5(1 + )√ Now we get () ≤ 40√. Thus basic estimates lead to ‖‖ ⋁ ≤

180√, so we can choose (1 + ) = 180√ for < 1. We need (1 + ) < 1⁄11 to apply Proposition (4.2.2), so we assume < 2.10. Hence for the von Neumann algebras case of Theorem (4.1.1), we could choose = 2.10 , but we will improve this bound later.

As C = 10 in Proposition (4.2.2), f = 10f. Moreover f(1 + δ) = ‖(1)‖ − ∗ 1 = 3. Finally we obtain ,((), ℬ ) ≤ 3620√.

∗ We need ,((), ℬ ) < 1/420000 to conclude for separable nuclear ∗-algebras. Finally, Theorem (4.1.1) for separable nuclear ∗-algebras is true with = 3.10 . For von Neumann algebras, as explained in the proof of Theorem (4.1.1), we only need to deal with . Hence = 4.10 is enough to ensure ∨ ‖ ‖‖ ‖ ≤ 88√ < 1/11 and to get the conclusion. We now give a counter-example to Amir-Cambern theorem for general (nuclear) ∗ -algebras. We have the following:

Theorem (4.2.4) [4]: ∗ There exist a family ()∈[,[ of non separable -subalgebras of (ℓ) containing (ℓ) such that (, ) ≤ | − |, ℓ(, ) ≤ | − | for some > 0 but is not isomorphic to with ≠ 0.

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Actually the isomorphisms → are completely positive. This shows somehow that Theorem (4.1.1) is optimal in full generality. We conclude with the following application. Corollary (4.2.5) [4]: There exists two (non separable) non isomorphic ∗ -algebras and such that

(, ) = 1. Proof: Just take =⊕∈ℚ∩[,[ and =⊕∈ℚ∩[,[ . For any > 0, there is a bijection ∶ ℚ ∩ [0, 1[→ ℚ ∩]0,1[ with |() − | < , thus (, ) < . We now explain why and are not isomorphic. First notice that as any contains the compact operators of the space it acts on, we deduce that the minimal centeral projections in and are exactly the projection onto the 's. Hence there is a minimal central projection ∈ with = whereas ≠ for all minimal central projections ∈ .

For details on the notion of length. To prove that the length function is locally constant, the first step is to notice that the length is stable for cb-close multiplications, then an application of Theorem (4.1.10) allows to conclude. We have the next lemma. Lemma (4.2.6) [4]: Let , ∶ → be two completely bounded linear maps between operator spaces such that ∶ ⁄ker → is a cb-isomorphism with ≤ . If ‖ − ‖ < 1⁄ then ∶ ⁄ker → is also a cb-isomorphism and ≤ . 1 − ‖ − ‖ Proof:

Let be in the unit ball of (). Then there exists in (), ‖‖ < such that () = . Hence

‖ − ()‖ < ,

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where = ‖ − ‖ < 1. Applying the same procedure to ( − ( )), we obtain in (), ‖‖ < such that

‖ − ( + )‖ < proceeding by induction, we obtain the result.

We know that an operator space endowed with a multiplication m (in the sense of Definition (4.2.1) is cb-isomorphic via an algebra homomorphism to an actual operator algebra. As the length is invariant under algebraic cb- isomorphisms, it makes sense to talk about the length of equipped with m. As max is functorial, remain true in the case of a completely bounded multiplication (not necessarily completely contractive). We denote by mℓ the ℓ-linear map defined (by associativity) on ℓ.

Proposition (4.2.7) [4]: Let be a unital operator algebra of length at most ℓ and length constant at most . Let m be another multiplication on such that mℓ − mℓ < 1⁄. Then equipped with m has also length at most ℓ. Proof: ⨂ℓ ⨂ℓ Let us denote by ℓ : max() → (resp. ℓ ∶ max() → ) the completely bounded linear map induced by the original multiplication m (resp. by the new multiplication m) on the ℓ-fold Haagerup tensor product of max() (i.e. A endowed with its maximal operator space structure). The hypothesis that has length at most ℓ and length constant at most exactly means that ℓ ∶ max()⨂ℓ⁄ker → is a cb-isomorphism with ≤ . But mℓ − ℓ ℓ mℓ < 1⁄ implies that ‖ − ‖ < 1/. By Lemma (4.2.6), ∶ ℓ ℓ ℓ ⨂ℓ max() ⁄ker ℓ → is also a cb-isomorphism, so the result follows. We have just proved that the length is stable under cb-isomorphisms with small bounds. More generally, we have the following general principle: any property which is stable under perturbation by cb-close multiplications is also stable under perturbation by cb-isomorphisms with small bounds.

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List of symbols

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