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UNITARY CUNTZ SEMIGROUPS OF IDEALS AND QUOTIENTS

LAURENT CANTIER

Abstract. We deﬁne a notion of ideal for objects in the category of abstract unitary Cuntz semigroups

introduced in [3] and termed Cu∼. We show that the set of ideals of a Cu∼-semigroup has a complete lattice

structure. In fact, we prove that for any C∗-algebra of stable rank one A, the assignment I!−→ Cu1(I) deﬁnes

a complete lattice isomorphism between the set of ideals of Aand the set of ideals of its unitary Cuntz

semigroup Cu1(A). Further, we introduce a notion of quotients and exactness for the (non abelian) category

Cu∼. We show that Cu1(A)/Cu1(I)≃Cu1(A/I) for any ideal Iin Aand that the functor Cu1is exact.

Finally, we link a Cu∼-semigroup with the Cu-semigroup of its positive elements and the abelian group of

its maximal elements in a split-exact sequence. This result allows us to extract additional information that

lies within the unitary Cuntz semigroup of a C∗-algebra of stable rank one.

1. Introduction

In the last decade, the Cuntz semigroup has emerged as a suitable invariant in the classiﬁcation of

non-simple C∗-algebras. It is now well-established that this positively ordered monoid is a continuous

functor from the category of C∗-algebras to the category of abstract Cuntz semigroups, written Cu (see

[6] and [1]). Moreover, an abstract notion of ideals and quotients in the category Cu has been considered

in [5] and it has been proved that the Cuntz semigroup nicely captures the lattice of ideals of a C∗-algebra

A, that we write Lat(A). In fact, for any C∗-algebra, the assignment I!−→ Cu(I) deﬁnes a complete lattice

isomorphism between Lat(A) and the set of ideals of Cu(A), that we write Lat(Cu(A)) (see [1, §5.1.6]).

These results make the Cuntz semigroup a valuable asset whenever considering non-simple C∗-algebras.

While the Cuntz semigroup has already provided notable results for classiﬁcation (see e.g. [9], [10]), one

often has to restrict itself to the case of trivial K1since the Cuntz semigroup fails to capture the K1-group

information of a C∗-algebra. To address this issue, the author has introduced a unitary version of the

Cuntz semigroup for C∗-algebras of stable rank one, written Cu1(see [3]). This invariant, built from

pairs of positive and unitary elements, resembles the construction of the Cuntz semigroup and deﬁnes

a continuous functor from the category of C∗-algebra of stable rank one to the category Cu∼of (not

necessarily positively) ordered monoids satisfying the order-theoretic axioms (O1)-(O4) introduced in

[6].

In this paper, we investigate further this new construction and we aﬃrmatively answer the question

whether this unitary version of the Cuntz semigroup also captures the lattice of ideals of a C∗-algebra of

stable rank one. We specify that the category Cu∼does not require the underlying monoids to be posi-

tively ordered, which hinders the task to generalize notions introduced in the category Cu. For instance,

Key words and phrases. Unitary Cuntz semigroup, C∗-algebras, Cu∼-ideals, Exact sequences.

The author was supported by MINECO through the grant BES-2016-077192 and partially supported by the grants MDM-2014-

0445 and MTM-2017-83487 at the Centre de Recerca Matem`

atica in Barcelona.

1

2 LAURENT CANTIER

we cannot characterize a Cu∼-ideal of a countably-based Cu∼-semigroup by its largest element, as is done

for countably-based Cu-semigroups, since such an element might not exist in general. As a result, two

axioms, respectively named (PD), for positively directed and (PC), for positively convex appear as far as

the deﬁnition of a Cu∼-ideal is concerned. The axiom (PD) has already been introduced in [3], where

the author has established that any positively directed Cu∼-semigroup Seither has maximal elements

forming an absorbing abelian group, termed Smax , or else has no maximal elements. We ﬁnally point out

that any Cu-semigroup Ssatisﬁes these axioms and that the generalization of a Cu∼-ideal matches with

the usual deﬁnition of a Cu-ideal for any Cu-semigroup S. In the course of this investigation, we also

show that the functor Cu1satisﬁes expected properties regarding ideals, quotients and exact sequences.

These results help us to dig in depth the functorial relations between Cu,K1and Cu1found in [3, §5].

More concretely, this paper shows that the set of Cu∼-ideals of such a Cu∼-semigroup Sis a complete

lattice naturally isomorphic to the complete lattice of Cu-ideals of its positive cone S+. Furthermore, we

prove that:

Theorem 1.1. For any C∗-algebra A of stable rank one, the unitary Cuntz semigroup Cu1(A)is positively

directed and positively convex.

Moreover, the assignment I !−→ Cu1(I)deﬁnes a complete lattice isomorphism between Lat(A)and

Lat(Cu1(A)) that maps the sublattice Latf(A)of ideals in A that contain a full, positive element onto the

sublattice Latf(Cu1(A)) of ideals in Cu1(A)that are singly-generated by a positive element. In particular,

I is simple if and only if Cu1(I)is simple.

Theorem 1.2. Let A be a C∗-algebra of stable rank one and let I ∈Lat(A). Consider the canonical short

exact sequence: 0−→ Ii

−→ Aπ

−→ A/I−→ 0. Then:

(i) Cu1(π)induces a Cu∼-isomorphism Cu1(A)/Cu1(I)≃Cu1(A/I).

(ii) The following sequence is short exact in Cu∼:

0!!Cu1(I)i∗!!Cu1(A)π∗!!Cu1(A/I)!!0

Theorem 1.3. Let S be a positively directed Cu∼-semigroup that has maximal elements. Then the fol-

lowing sequence in Cu∼is split-exact:

0!!S+

i!!Sj!!Smax

q

""!!0

where i is the canonical injection, j(s) :=s+eSmax and q(s) :=s.

The paper is organized as follows: In the ﬁrst part, we deﬁne an abstract notion of a Cu∼-ideal for

any positively directed Cu∼-semigroup. We then see that the smallest ideal containing an element might

not always exist since the intersection of two Cu∼-ideals is not necessarily a Cu∼-ideal. However, the

smallest ideal containing an element sof a positively directed and positively convex Cu∼-semigroup S,

where the notion of positively convex is to be speciﬁed, always exists and is explicitly computed. We

ﬁnally build a complete lattice structure on the set of Cu∼-ideals of a positively directed and positively

convex Cu∼-semigroup S, relying on the natural set bijection between Lat(S)≃Lat(S+), where S+∈Cu

is the positive cone of S.

UNITARY CUNTZ SEMIGROUPS OF IDEALS AND QUOTIENTS 3

We also study the notion of quotients and exactness in the category Cu∼. Among others, we show that

a quotient of a positively directed and positively ordered Cu∼-semigroup by an ideal is again a positively

directed and positively ordered Cu∼-semigroup. Moreover, the functor Cu1preserves quotients and short

exact sequence of ideals. We ﬁnally use the split-exact sequence 0 −→ S+−→ S−→ Smax −→ 0

described above to unravel commutative diagrams with exact rows linking Cu,K1and Cu1of a separable

C∗-algebra with stable rank one -and its ideals-.

Note that this paper is the second part of a twofold work (following up [3]) and completes the proper-

ties of the unitary Cuntz semigroup established during the author’s PhD thesis. We also mention that the

unitary Cuntz semigroup -through these results- will be used in a forthcoming paper to distinguish two

non-simple unital separable C∗-algebras with stable rank one, which originally agree on K-Theory and

the Cuntz semigroup; see [4].

Acknowledgments The author would like to thank Ramon Antoine for suggesting a more adequate

version of the ‘positively convex’ property, and both Ramon Antoine and Francesc Perera for insightful

comments about the paper. The author also thanks the referee for his/her pertinent comments that have

helped to reformulate some part of the manuscript in a better way.

2. Preliminaries

We use Mon≤to denote the category of ordered monoids, in contrast to the category of positively

ordered monoids, that we write PoM. We also use C∗

sr 1 to denote the full subcategory of C∗-algebras of

stable rank one.

2.1. The Cuntz semigroup. We recall some deﬁnitions and properties on the Cuntz semigroup of a

C∗-algebra. More details can be found in [1], [2], [6], [11].

2.1. (The Cuntz semigroup of a C∗-algebra.) Let Abe a C∗-algebra. We denote by A+the set of

positive elements. Let aand bbe in A+. We say that ais Cuntz subequivalent to b, and we write a≲Cu b,

if there exists a sequence (xn)n∈Nin Asuch that a=lim

n∈Nxnbx∗

n. After antisymmetrizing this relation, we

get an equivalence relation over A+, called Cuntz equivalence, denoted by ∼Cu.

Let us write Cu(A) :=(A⊗K)+/∼Cu, that is, the set of Cuntz equivalence classes of positive elements

of A⊗K. Given a∈(A⊗K)+, we write [a] for the Cuntz class of a. The set Cu(A) is equipped with

an addition as follows: let v1and v2be two isometries in the multiplier algebra of A⊗K, such that

v1v∗

1+v2v∗

2=1M(A⊗K). Consider the ∗-isomorphism ψ:M2(A⊗K)−→ A⊗Kgiven by ψ(a0

0b)=

v1av∗

1+v2bv∗

2, and we write a⊕b:=ψ(a0

0b). For any [a],[b] in Cu(A), we deﬁne [a]+[b] :=[a⊕b]

and [a]≤[b] whenever a≲Cu b. In this way Cu(A) is a partially ordered semigroup called the Cuntz

semigroup of A.

For any ∗-homomorphism φ:A−→ B, one can deﬁne Cu(φ) : Cu(A)−→ Cu(B), a semigroup map,

by [a]!−→ [(φ⊗idK)(a)]. Hence, we get a functor from the category of C∗-algebras into a certain

subcategory of PoM, called the category Cu, that we describe next.

2.2. (The category Cu). Let (S,≤) be a positively ordered semigroup and let x,yin S. We say that xis

way-below y and we write x≪yif, for all increasing sequences (zn)n∈Nin Sthat have a supremum, if

4 LAURENT CANTIER

sup

n∈N

zn≥y, then there exists ksuch that zk≥x. This is an auxiliary relation on Scalled the way-below

relation or the compact-containment relation. In particular x≪yimplies x≤yand we say that xis a

compact element whenever x≪x.

We say that Sis an abstract Cuntz semigroup, or a Cu-semigroup, if it satisﬁes the following order-

theoretic axioms:

(O1): Every increasing sequence of elements in Shas a supremum.

(O2): For any x∈S, there exists a ≪-increasing sequence (xn)n∈Nin Ssuch that sup

n∈N

xn=x.

(O3): Addition and the compact containment relation are compatible.

(O4): Addition and suprema of increasing sequences are compatible.

A Cu-morphism between two Cu-semigroups S,Tis a positively ordered monoid morphism that pre-

serves the compact containment relation and suprema of increasing sequences.

The category of abstract Cuntz semigroups, written Cu, is the subcategory of PoM whose objects are

Cu-semigroups and morphisms are Cu-morphisms.

2.3. (Countably-based Cu-semigroups.) Let Sbe a Cu-semigroup. We say that Sis countably-based if

there exists a countable subset B⊆Ssuch that for any a,a′∈Ssuch that a′≪a, then there exists b∈B

such that a′≤b≪a. The set Bis often referred to as a basis. An element u∈Sis called an order-unit

of Sif for any x∈S, there exists n∈Nsuch that x≤nu, where N:=N⊔{∞}.

Let Sbe a countably-based Cu-semigroup. Then Shas a maximal element, or equivalently, it is

singly-generated. Let us also mention that if Ais a separable C∗-algebra, then Cu(A) is countably-based.

In fact, its largest element, that we write ∞A, can be explicitly constructed as follows: Let sAbe any

strictly positive element (or full positive) in A. Then ∞A=sup

n∈N

n[sA]. A fortiori, [sA] is an order-unit of

Cu(A).

2.4. (Lattice of ideals in Cu.) Let Sbe a Cu-semigroup. An ideal of Sis a submonoid Ithat is closed

under suprema of increasing sequences and such that for any x,ysuch that x≤yand y∈I, then x∈I.

It is shown in [1, §5.1.6], that for any I,Jideals of S,I∩Jis again an ideal. Therefore for any x∈S,

the ideal generated by x, deﬁned as the smallest ideal of Scontaining x, and written Ix, is exactly the

intersection of all ideals of Scontaining x. An explicit computation gives us Ix:={y∈S|y≤ ∞x}.

Moreover it is shown that I+J:={z∈S|z≤x+y,x∈I,y∈J}is also an ideal. Thus we

write Lat(S) :={ideals of S}, which is a complete lattice under the following operations: for any two

I,J∈Lat(S), we deﬁne I∧J:=I∩Jand I∨J:=I+J.

Furthermore, for any C∗-algebra A, we have that Cu(I) is an ideal of Cu(A) for any I∈Lat(A). In fact,

we have a lattice isomorphism as follows:

Lat(A)≃

−→ Lat(Cu(A))

I!−→ Cu(I)

Finally, whenever Sis countably-based, any ideal Iof Sis singly-generated, for instance by its largest

element, that we also write ∞I. In particular, for any C∗-algebra A, any a,b∈(A⊗K)+, if [a]≤[b]

in Cu(A), then Ia⊆Ib, or equivalently I[a]⊆I[b]. (The converse is a priori not true: Ix=Ik x for any

x∈Cu(A), any k∈Nbut in general x!kx.)

UNITARY CUNTZ SEMIGROUPS OF IDEALS AND QUOTIENTS 5

2.5. (Quotients in Cu.) Let Sbe a Cu-semigroup and I∈Lat(S). Let x,y∈S. We write x≤Iyif:

there exists z∈Isuch that x≤z+y. By antisymmetrizing ≤I, we obtain an equivalence relation ∼I

on S. Deﬁne S/I:=S/∼I. For x∈S, write x:=[x]∼Iand equip S/Iwith the following addition and

order: Let x,y∈S. Then x+y:=x+yand x≤y, if x≤Iy. These are well-deﬁned and (S/I,+,≤)

is a Cu-semigroup, often referred to as the quotient of S by I. Moreover, the canonical quotient map

S−→ S/Iis a surjective Cu-morphism. Finally, for any C∗-algebra Aand any I∈Lat(A), we have

Cu(A/I)≃Cu(A)/Cu(I); see [5, Corollary 2].

2.2. The unitary Cuntz semigroup. We recall some deﬁnitions and properties on the Cu1-semigroup

of a C∗-algebra with stable rank one. More details can be found in [3].

2.6. (The unitary Cuntz semigroup of a C∗-algebra - The category Cu∼.) Let Abe a C∗-algebra

of stable rank one, let a,b∈A+such that a≲Cu b. Using the stable rank one hypothesis, there exist

standard morphisms θab : her(a)∼↩−→ her(b)∼such that [θab (u)]K1does not depend on the standard

morphism chosen, for any unitary element u∈her(a)∼. That is, there is a canonical way (up to homotopy

equivalence) to extend unitary elements of her(a)∼into unitary elements of her(b)∼. Now, let u,vbe

unitary elements of her(a)∼,her(b)∼respectively. We say that (a,u) is unitarily Cuntz subequivalent to

(b,v), and we write (a,u)≲1(b,v), if a≲Cu band θab(u)∼hv. After antisymmetrizing this relation, we

get an equivalence relation on H(A) :={(a,u)|a∈(A⊗K)+,u∈U(her(a)∼)}, called the unitary Cuntz

equivalence, denoted by ∼1.

Let us write Cu1(A) :=H(A)/∼1. The set Cu1(A) can be equipped with a natural order given by

[(a,u)] ≤[(b,v)] whenever (a,u)≲1(b,v), and we set [(a,u)] +[(b,v)] :=[(a⊕b,u⊕v)]. In this way

Cu1(A) is a semigroup called the unitary Cuntz semigroup of A.

Any ∗-homomorphism φ:A−→ Bnaturally induces a semigroup morphism Cu1(φ) : Cu1(A)−→

Cu1(B), by sending [(a,u)] !−→ [(φ⊗idK)(a),(φ⊗idK)∼(u)]. Hence, we get a functor from the category

of C∗-algebras of stable rank one into a certain subcategory of ordered monoids, denoted by Mon≤, called

the category Cu∼, that we describe in the sequel.

Let (S,≤) be an ordered monoid. Recall the compact-containment relation deﬁned in Paragraph 2.2.

We say that Sis a Cu∼-semigroup if Ssatisﬁes axioms (O1)-(O4) and 0 ≪0. We emphasize that we do

not require the monoid to be positively ordered. A Cu∼-morphism between two Cu∼-semigroups S,Tis

an ordered monoid morphism that preserves the compact-containment relation and suprema of increasing

sequences.

The category of abstract unitary Cuntz semigroups, written Cu∼, is the subcategory of Mon≤whose

objects are Cu∼-semigroups and morphisms are Cu∼-morphisms. Actually, as shown in [3, Corollary

3.21], the functor Cu1from the category C∗

sr 1 to the category Cu∼is arbitrarily continuous.

2.7. (Alternative picture of the Cu1-semigroup.) We will sometimes use an alternative picture de-

scribed in [3, §4.1]. First, recall that for a C∗-algebra A, Latf(A) is the sublattice of Lat(A) consisting

of ideals that contain a full, positive element. Also recall that {σ-unital ideals of A}⊆Latf(A) and if

moreover Ais separable, then the converse inclusion holds. Finally, for any I∈Lat f(A), we deﬁne

Cuf(I) :={x∈Cu(A)|Ix=Cu(I)}to be the set of full elements in Cu(I).

6 LAURENT CANTIER

Let Abe a C∗-algebra of stable rank one such that Lat f(A)={σ-unital ideals of A}. Then Cu1(A) can

be pictured as !

I∈Latf(A)

Cuf(I)×K1(I)

that we also write Cu1(A). The addition and order are deﬁned as follows: For any (x,k),(y,l)∈Cu1(A)

"

#

#

$

#

#

%

(x,k)≤(y,l) if: x≤yand δIxIy(k)=l.

(x,k)+(y,l)=(x+y,δIxIx+y(k)+δIyIx+y(l)).

where δIJ :=K1(Ii

↩−→ J), for any I,J∈Lat f(A) such that I⊆J.

Let A,Bbe C∗-algebras of stable rank one and let φ:A−→ Bbe a ∗-homomorphism. For any

I∈Latf(A), we write J:=Bφ(I)B, the smallest ideal of Bthat contains φ(I). Then J∈Latf(B) and

Cu1(φ) can be rewritten as (Cu(φ),{K1(φ|I)}I∈Latf(A)), where φ|I:I−→ J. Observe that we might write

α,α0,αIto denote Cu1(φ),Cu(φ),K1(φ|I) respectively.

3. Ideal structure in the category Cu∼

In this section we deﬁne and study the notion of ideals in the category Cu∼. Since the underlying

monoid of a Cu∼-semigroup might not be positively ordered, deﬁnitions and results of the category Cu

cannot be applied and some extra work is needed. When it comes to a concrete Cu∼-semigroup, -that

is, coming from a C∗-algebra of stable rank one A- we wish that a Cu∼-ideal satisﬁes natural properties,

e.g. Cu1(I) is an ideal of Cu1(A) or Lat(A) is entirely captured by the set of Cu∼-ideals of Cu1(A). For

that matter, we ﬁrst have to study the set of maximal elements of a Cu∼-semigroup. We show that under

additional axioms -satisﬁed by any Cu1(A)-, namely the axioms (PD) and (PC), the set maximal elements

of a Cu∼-semigroup forms, when not empty, an absorbing abelian group. From there, we are able to

deﬁne a suitable notion of Cu∼-ideal. We will also use concepts from Domain Theory that we recall now

(see [7]).

Finally, we say that a Cu∼-semigroup Sis countably-based if there exists a countable subset B⊆S

such that for any pair a′≪a, there exists b∈Bsuch that a′≤b≪a.

3.1. Deﬁnition of a Cu∼ideal.

Deﬁnition 3.1. [7, Deﬁnition II.1.3] Let Sbe a Cu∼-semigroup. A subset O⊆Sis Scott-open if:

(i) Ois an upper set, that is, for any y∈S,y≥x∈Oimplies y∈O.