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ISSN 2074-1871 yfoi/iMCKi/m MaTeMaTi/iHecKi/m wypHaR. TOM 16. № 3 (2024). C. 118-129.
CATEGORICAL CRITERION FOR EXISTENCE OF UNIVERSAL C* ^ALGEBRAS
R.N. GUMEROV, E.V. LIPACHEVA, K.A. SHISHKIN
Abstract. We deal with categories, which determine universal C*-algebras. These categories are called the compact C*-relations. They were introduced by T.A. Loring. Given a set X, a compact C*-relation on X is a category the objects of which are functions from X to C*-algebras, and morphisms are *-homomorphisms of C*-algebras making the appropriate triangle diagrams commute. Moreover, these functions and *-homomorphisms satisfy certain axioms. In this article, we prove that every compact C*-relation is both complete and cocomplete. As an application of the completeness of compact C*-relations, we obtain the criterion for the existence of universal C*-algebras.
Keywords: compact C *-relation, complete category, universal C *-algebra.
Mathematics Subject Classification: 16B50, 46L05, 46M15
1. Introduction
The motivation for our work comes from the theory of universal C*-algebras generated bv sets of generators subject to relations (see |1|- |6|) and the study of limits for inductive systems consisting of universal C*-algebras and their *-homomorphisms in [ ]- [ ]. A categorical approach to relations that determine universal C*-algebras was developed by Loring [ ]. In the framework of this approach, one deals with categories called C*-relations. Given a set X, a C*-relation R on X is a category, the objects of which are functions from X to C*-algebras, and morphisms are *-homomorphisms of C*-algebras making the appropriate triangle diagrams commute. In addition, the objects and the morphisms of R satisfy certain axioms. The C*-relations determining universal C*-algebras are called compact. A necessary and sufficient condition for R to be compact is the existence of an initial object C*(R) in the categorv R [ ]. The universal C*-algebra for the compact C*-relation R is the initial object C*(R) of this
R
The called the relations associated with *-polynomial pairs were
studied in [ ]. A polynomial pair (X,P) ^^^^^^^s of a non-emptv set X and a non-empty subset P of the free ^^^^bra F(X) ^^^^rated by X over the field of complex numbers. The objects of the *-polynomial relation associated with (X, P) are all functions f from the set X to satisfying the property: the set P is contained in the kernel of the unique
*-homomorphism, which is an extension of / to the fr ee *-algebr a F (X). It was proved in [ ] that every C*-algebra is a universal Cdetermined bv a *-polynomial relation and every compact C*-relation is isomorphic to a *-polynomial relation.
R.N. Gumerov, E.V. Lipacheva, K.A. SnisiiKiN, A categorical criterion for the existence of
universal c*^algebras.
(c) Gumerov R.N., Lipacheva E.V., Shishkin K.A. 2024.
Submitted November 3, 2023.
In this article we continue the study of properties of the compact C*-relations initiated in [ ], We show that each compact C*-relation is both complete and cocomplete. To obtain this result, we use of the completeness and cocompleteness of the category of C*-algebras and their *-homomorphisms [ ], The completeness of every compact C*-relation together with the aforementioned equivalence between the compactness of a C*-relation R and the existence of an initial object in R yields the criterion for the existence of the universal C*-algebra C*(R). Namely, C* (R) exists if and only if the categorv R is complete.
The article is organized as follows. It consists of the Introduction and three sections. Section 2 contains needed notation, definitions and facts from the category theory and the theory of C*-relations. In Section we prove that every compact C*-relation is complete. As a consequence of this result, we obtain the criterion for the existence of universal C*-algebras, Section is devoted to the proof of the cocompleteness of all compact C*-relations,
In this section, we recall some necessary definitions and facts from the theory of categories and functors. For detail we refer the reader to book |17|,
Let C be a category and Z be a small category, A functor V: Z ^ C is called a diagram in C of shape Z.
A cone on the diagram V is a pair (V,t), where V: Z ^ C is a constant functor and t : V ^V is a natural transformation from V to V. Thus, the functor V sends each, object / of Z to a fixed object V in C and V(f) is the identitv 1y on V for each morphism /of Z. Moreover, one has a family of morphisms 77: V ^V(I) indexed by objects / of the categorv Z such that the diagram
commutes for every morphism f : / ^ J in Z,
A cone (V, t) ^n the diagram V is said to be universal if for every cone (V, r') on V there exists a unique morphism tp: V' ^ V in C such that t' = t o that is, the diagram
commutes for every morphism f : / ^ J in Z A universal cone on V is called a limit of the diagram V. A category is said to be complete if it has a limit for every diagram in this category. In what follows, two basic types of limits of diagrams are involved in our arguing. These are products and equalizers; let us recall the definitions.
Let A be a set. We denote by C the discrete category, the objects of which are the elements of A and all morphisms are the identities. Let (CaIaga be a family of objects in the category C.
2. Preliminaries
Consider the diagram V: C ^ C, which sends an object A of £ to the object C\ in C. A limit of the diagram V is called tie product of the family {Ca}aga- It is denoted by l H C\, {pa}aga ) ■
Vaga /
The object H C\ itself is often called the product of the family {Ca}agA- The morphisms px \e a
are called the projections of the product. Thus, the product possesses the following universal property. For each object Cm C and each A-indexed family of morphisms f\ : C ^ C\ in C there exists a unique morphism f : C ^ f! C\ such that for each y E A the diagram
is commutative. We say that a category has all products if every family of its objects indexed by a set has a product in this category.
Another basic limit is an equalizer, which is defined as follows. Let 8 be a category with two objects, say A and B, with two morphisms u,v: A ^ B, and with no other morphisms except for identities. Let f,g: C\ ^ C2 be morphisms of the сategorv C. We refer to pairs of morphisms like / and g as parallel morphisms. Consider the diagram "D in С of shape 8 such that V(u) = / and V{v) = g. A limit of this diagram V: 8 ^ С is called the equalizer of f and g. Thus, it is a pair (E,e), where E is an object of the category С and e: E ^ C\ is a morphism of С such that f о e = g о e and the following universal property holds:
every morphism h: H ^ C\ such th at f o h = g o h can be factorized uniquely through e, that is, there exists a unique morphism d: H ^ E such that e o d = h. In case each pair of parallel morphisms in a category C has an equalizer, we say that C has all equalizers.
The next result states that all limits can be built up from products and equalizers |IT, Ch, V, Sect, 2, Cor. 2|,
Lemma 2.1. A category is complete, if and only if it has all products and equalizers.
Using the duality principle, one obtains the dual notions, namely, a cocone, a universal cocone, a colimit, a coproduct, a coequalizer, a cocomplete category and the dual of Lemma 2.1. For details, we refer the reader to |IT, Ch. I, Sect, 1|.
We denote by C*-alg the category of all C*-algebras and *-homomorphisms between them. The trivial Cconsisting of single zero element is denoted by 0.
For a family {Aa | A E A} of objects in C*-alg indexed by a set A, we consider the direct product
Afc A
which is a C*-algebra with respect to the coordinatewise algebraic operations and the supremum norm.
С
H
Further, we give the definitions of categories from Loring's paper |5|, These categories are the main objects of investigation in the present article.
Given a set X, the null C*-relation on X is the categorv Tx, the objects of which are all functions of the form j : X ^ A, where A is a C*-algebra, For two objects j : X ^ A and k : X ^ B in Tx, a morphism from j to k is each *-homomorphism of C*-algebras p : A ^ B making the diagram
X
n fi : X ^ n
A---^B
commute, i.e., k = p o j.
A C*-relation on X is a full subcategory R of Tx satisfying the following axioms:
CI the function X ^ 0 is an object of R;
C2 if p : A ^ B is an injective *-homomorphism of C*-algebras, f : X ^ A is a function
and p o f is an object of R, then f is an object of R; C3 if ^ : A ^ 5 is a *-homomorphism of C*-algebras and f : X ^ A is an object of R, then
p o f is an object of R; C4f if fi : X ^ Ai is an object of R for everv i = 1,... ,n, n G N, then the function
n
i=l i=l
R
Objects of C*-relations are also called the representations.
A C*-relation R on a set X is said to be compact if, in addition, the following condition is fulfilled:
C4 for each non-empty set A, if fx : X ^ A\ is an object of R for every A G A, then the function
n A : X ^ n ^
AeA AeA
R
The following statement is a reformulation of Theorem 2,10 from |5| (see also |2, Prop, 1.3.6|, |3, Sect. 3,11 and |4, Sect. 1.4|).
Lemma 2.2. Let R be a C*-relation on a set X. Then R is compact if and only if there
R
In what follows, for a compact C*-relation R on a set X, we consider an initial object i : X ^ A of R The C*-algebra A is denoted by C*(R). Thus, for every representation j : X ^ B of R there exists a unique *-homomorphism of C*-algebras k : C* (R) ^ B such that the diagram
X
C*(R)~ -- -B
is commutative, i.e., j = k o i.
j
The object i : X m C*(R) is called the universal representation, and the C*-algebra C*(R) is called the universal C*-algebra to the compact C*-relation R,
Finally, we give three examples of C*-relations, which are denoted by R\, R2 and R3. Since every C*-relation must be a full subcategory in the null C*-relation Tx, we specify only objects for these categories. One can easily verify that Axioms CI, C2, C^d C4f hold in R1, R2, and R3, that is, these categories are C*-relations.
Example 2.1. Let X = {x} be an one-element set. We consider the category R\, the objects of which are all functions f : X M A, where A is a C*-algebra, and f (x) is a normal element of A.
We claim that R\ is not a compact C* -relation. Indeed, to see this, we fix a C*-algebra A and a non-zero normal element a G A. For each n G N, we consider the object fn of the category R\ defined as
fn : X M A : x M na.
Since supraeN = +<x>, Axiom C4 is not valid for R\. That is, the C*-relation R\ is not
compact, as claimed.
By Lemma , there is no initial object in the category R\, and the universal C* -algebra for R\ is not defined.
We note that the category R1 is a * -polynomial relation associated with the * -polynomial pair (X, {x*x — xx*}). This fact also guarant ees that R\ is a C * -relation [ , Prop. 2].
Example 2.2. Let X = {x}. objects of the category R2, we take all functions of the form f : X M A, where A is a unital C *-algebra a nd f (x) is a unitary eleme nt in A. It is straightforward to verify that Axiom C4 is satisfied in the C*-relation R2, hence, it is compact.
By Lemma , there exist the universal representation in R2 and the universal C* -algebra
C *R).
Using the continuous functional calculus, one can see that C*(R2) is isomorphic to the commutative C*-algebra C(Sconsisting of all continuous complex-valued functions on the unit circle S1 in the complex plane.
Example 2.3. Let n ^ 2 be an integer and X = {x1, ... ,xn} be a set consisting of n elements. We define R3 as the category, the objects of which are all functions of the form f : X M A, where A is a unital C *-algebra a nd f (x-\),... ,f (xn) are isometries with, pairwi.se orthogonal ranges. It is easy to see that Axiom C4 holds for the C*-relation R3, that is, R3 is compact.
Consequently, by Lemma , there is the universal representation i : X M C*(R3) in the category R3.
The universal C*-algebra C*(R3) is called the Toeplitz — Cuntz algebra for n generators. This algebra was defined and studied by Cuntz /14/, ¡15/. In particular, it was shown that the Toeplitz — Cuntz algebra contains a closed two-sided ideal, which is isomorphic to the compact operators on an infinite-dimensional separable Hilbert space, and the quotient of C*(R3) by this ideal is the Cuntz algebra [ J. In [ ], [ ], the universal property of C*(R3) is used for constructing the direct sequences of the Toeplitz — Cuntz algebras and studying properties of reduced semigroup C* -algebras.
3. Completeness of compact C*-relations
In this section we show that all compact C*-relations are complete. Our proof is based on the fact that the category C*-alg is complete [ ], More precisely, we explore explicit limit constructions in the category C*-alg from [ ], Using completeness of compact C*-relations
and Lemma , we obtain the criterion for the existence of universal С*-algebras for C*-relations.
Lemma 3.1. Every compact С*-relation R on a set X has all products.
Доказательство. Let (Д: X ^ Ax }лел be a family of objects of R indexed by elements of a set Л, Consider the function
ПА: X ^ П: ж ^ (А(х))^л, x e X.
Аел Аел
By Axiom C4, it is an object of the category R For each A e Л, we denote by p\ the natural
projection of the direct product of the С *-algebras П A^ onto th e С *-algebr a Ax. Obviously,
ме л
the *-homomorphism p\ is a morphism of R, We claim that the pair
Па, {Pa ■ П a
ß ^ ^A}Aeл I \Аел ^л /
A
is a product of this family in R, Indeed, to show that this pair satisfies the universal property, we take an object f : X ^ A and a family of morphisms [g\ : A ^ Aa}agA in the category R such that
g\ o f = fx whenever A G A. (3,1)
Since the pair ( Ax, {pa}a€A ) is a product [ , Thm. 2,9] of the family {AA}AeA in the vaga J
category of C*-algebras and their *-homomorphisms, there is a unique *-homomorphism
n#A : A ^ H A\ : a ^ (gx(a))XeA
Ae A
Ae л
such that
Pß о П = 9ß
(3.2)
Ае л
for each index ^ E A, that is, in the next diagram the bottom triangle is commutative:
X
Moreover, using (3.2), (3.1) and the commutativity of the triangle on the right-hand side of the diagram, we have the equalities
(p*o (n*o ^(x) =o f)(x) = = o na)(x)
\AeA / AeA
p
for every index ^ G A and to every x G X. Consequently, by the definition of an element of a product in category of C*-algebras, the triangle on the left-hand side of the diagram is commutative:
(Ша) o f = П/»
\лел / лел
that is, the *-homomorphism П g\ is a morphism оf the C*-relation R,
ле л
Thus, the required universal property is satisfied and the pair ( П Д, {рл}лел ) is a product
" \лел J
in the category R, as claimed. The proof is complete, □
To prove the following statement we use the fact that the category C*-alg has all equalizers 116, Lm. 2,5|,
Lemma 3.2. Every compact С*-relation R on a set X has all equalizers.
Доказательство. We take two objects f: X ^ ^d g: X ^ В and two parallel morphisms p: A ^ В and ф: A ^ В torn / to g in the categorv R,
Let us consider the C*-algebra E and the *-homomorphism e of С*-algebras defined as
E = {a G A | p(a) = ф(а)} , e: E ^ A: a^ a, a G E.
It is clear that
E---A ; В
-Ф
is an equalizer diagram in the category of C*-aIg,
Further, we define a function e: X ^ E such that the pair (e: X ^ E,e) is an equalizer of morphisms ^d ф in the category R, We show that this function is determined by the condition
e о e = f. (3.3)
Namely, we let
e(x):= f (x), x G X. (3.4)
First of all, we need to verify that the function e: X ^ E given by the rule ( ) is well-defined, that is,
f (x) G E whenever x G X. (3.5)
Since ^d ф are parallel morphisms from / to g in R, we have
p(f (^)) = g(x) = Ф(f (%)).
Hence, condition (3.5) holds, as required.
Since e: E ^ A is an injective *-homomorphism and f: X ^ A is an object of the category R, by Axiom C2, it follows from the equality ( ) that the function e is an object of R. Moreover, the equality ( ) implies that the *-homomorphism e is a morphism of R.
We claim that the pair (e: X ^ E,e: E ^ A) is an equalizer of the morphisms p: A ^ В and гф: A ^ В in R, Indeed, firstly, we have the equality
p о e = ф о e.
Secondly, we need to show that the pair (e,e) possesses the universal property in the category R To this end, we take a pair (h: X ^ C,x: С ^ A) consisting of an оbject h in R and a morphism ^ m R from h to f such that p о = ф о х- By the universal property of the
equalizer (E, e) in the categorv С*-alg, there exists a unique *-homomorphism т : С ^ E of С*-algebras making the triangle
A
В
commute, that is,
X = £ ° т. (3.6)
Since the *-homomorphism of С*-algebras x is a morphism of the category R, we have the equality
f = X ° h. (3.7)
Using the equalities (3.3), (3.7) and (3.6), we obtain
e ° e = f = x ° h = £ ° т ° h. (3.8)
Since the function e is a monomorphism in the category of sets and functions, the equality ( ) implies the equality e = т ° h. The latter means that the *-homomorphism т is a morphism in R torn h to e, Thus, the pair (e, e) is an equalizer of parallel morphisms ^d ф in R, as claimed. The proof is complete. □
Using Lemma 3.1, Lemma 3.2 and Lemma 2.1, we have
Theorem 3.1. Every compact С * -relation is a complete category.
As an application of Theorem 3.1, we obtain the criterion for the existence of universal С *-algebra.
Theorem 3.2. Let R be а С * -relation. Then the un iversal С *-algeb га С *(R) exists if and R
Доказательство. By Lemma , the category R has a universal representation i : X ^ С*(R) if and only if the C*-relation R is compact. By Theorem , every compact C*-relation is complete. Conversely, if the С*-relation R is complete, then R has all products and satisfies Axiom С4, as required. This completes the proof. □
■p
Ф
T
4. COCOMPLETENESS OF COMPACT C*-RELATIONS
In this section we show that every compact С*-relation is cocomplete. In our proof we employ colimit constructions in the category С*-aig (see [ ]),
Lemma 4.1. Each compact С*-relation R on a set X has all coproducts.
Доказательство. Let {fx: X ^ Ал}лел a of objects in the category R and the pair
(ЦАл, {U : Л ^ Ц
\лел ^ел /
* 1 СЛ 1П
be a coproduct of the family {AA}Ae^f С*-algebras in C*-aig (see [ , Lm. 2.3])
In the C *-algebra U A\, we consider the closed two-si ded ideal I generated by the differences Ae a
ix(fx(x)) — where x runs over X and E A:
I = ({i\(h(x)) — i„(f„(x)) | X E X, E A)).
We denote bv
P: H^A ^ HAx/1
iA ^ ^A AeA AeA
the canonical *-homomorphism between the C*-algebras. By the construction of the ideal I, we have
p ° i\ o fx = p o iM o f^
whenever E A. We let f = p o ix o fx for A E A. By Axiom C3, the function f is an object of the category R. Hence, the *-homomorphism p o ix is a morphism of R for everv A E A. We claim that the pair
if: X ^ ^Ax/1, [p o zx: Aa ^ ^A,/1 }Ae a )
V Ae A ^e A /
(4.1)
is a coproduct of the family [fx: X ^ in the category R, Indeed, we need to verify
that (4.1) satisfies the universal property. To this end, we take a pair
(h: X ^ C, [gx : Aa ^ C}Aea)
where h is an object of R and g\ is a morphism in R torn /A to h for every A E A. Let us show that there is a unique *-homomorphism
<P: HAx/1 ^ C
Ae A
such that p o f = h, that is, p is a morphism of R torn / to h, and gx = p o (p o ix) for every A E A.
To do this, for arbitrary index y E A, we consider the diagram
U^A
Ae A
^ / /
u V i
Ae A '
Since U Ax is a coproduct in the category C* alg, there is a unique *-homomorphism gx aga aga
making the central triangle in the above diagram commute.
For all v E A, we have
(H^a) o ^ o fp - iv o fv) = ((H^a) o ^ O f^ - ((H,a) o iv O f^ aga aga aga
= {gM o /J - (gv o U) = h - h = 0.
It follows that the kernel of gx contains the ideal I, and there is a unique *-homomorphism
aga
P: Ц Ах/1 ^ С
iA
Аел
such that the bottom triangle in the above diagram is commutative, that is,
V ° P = 11 9а.
Аел
It is easy to see that p ° f = h. Therefore, p is a morphism of V. Moreover, we have
gA = p ° (p ° iA) for each Л e Л.
Thus, the required universal property is satisfied, and the pair (4,1) is a coproduct in the category V, as claimed. The proof is complete, □
In the proof of the following statement we use the explicit construction of a coequalizer in the category C* alg (see [ , Lm, 2,5]),
Lemma 4.2. Every compact С*-relation V on a set X has all coequalizers.
Доказательство. We take two objects f: X ^ ^d g: X ^ В and two parallel morphisms p: A ^ В and ф: A ^ В torn / to g in the categorv V.
In the C*-algebra Б, we construct the closed two-sided ideal I generated by the differences p(a) — ф(а), where a runs over A:
I = {{p(a) - | a E A}).
Let C = ^^d n: B ^ C be the canonical surjection. It was shown in the proof of Lemma 2,5 in |16| that
A ; B-K-» C
p
is a coequalizer diagram in the category C*^alg.
To construct a coequalizer of the morphisms p and ty in the categ orv R, we use Axiom C3 and define the object c: X ^ C of R by
c := K o g, (4,2)
which guarantees that the *-homomorphism n is a morphism of the categorv R torn g to c.
We claim that the pair (c: X ^ C,n: B ^ C) is a coequalizer of the morphisms p: A ^ B and ty: A ^ 5 in R. Indeed, by the construction of the ideal I, we have the equality
K o p = ft o ty.
We need to prove that the pair (c, ft) has the universal property in the category R. To this end, we take a pair (h: X ^ D,x: B ^ D) consisting of an object h in R and a morphism ^ m R from g to h such that x o p = X o ty- By the universal property of the coequalizer (C, n) in the
category C*-alg, there exists a unique *-homomorphism r: C ^ D of C*-algebras making the triangle in the diagram
A
B
commute, that is,
X = t o n. (4.3)
It remains to show that the *-homomorphism of C*-algebras t is a morphism from c to h in the category R. Because the *-homomorphism of C*-algebras x'ismorphism of the category R, we have
h = X o g. (4-4) By the equalities (4.4), (4.3) and (4.2), we get
h = X o g = t o K o g = r o c,
which means that t is a morphism from c to h in the categorv R, as required. It follows that the pair (c, n) is a coequalizer of parallel morphisms p and ^ in R, as claimed. This completes the proof. □
As an immediate consequence of Lemma 4.1, Lemma 4.2, Lemma 2.1 and the categorical duality principle |IT, Ch. E, Sect, 1|, we obtain the following theorem.
Theorem 4.1. Every compact C* -relation is a cocomplete category.
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Lobachevskii Institute of Mathematics and Mechanics,
Kazan (Volga Region) Federal University,
Kremlevskava str. 35,
420008, Kazan, Russia
E-mail: Renat. Gumerov@kpfu. ru
Ekaterina Vladimirovna Lipacheva, Chair of Higher Mathematics, Kazan State Power Engineering University, Krasnoselskava str. 51, 420066, Kazan, Russia
Lobachevskii Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, Kremlevskava str. 35, 420008, Kazan, Russia E-mail: elipacheva@gmail.com
Kirill Andreevich Shishkin,
Lobachevskii Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, Kremlevskava str. 35, 420008, Kazan, Russia E-mail: keril911@gmail.com
