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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.