On a topology and limits for inductive systems of C ∗ -algebras over partially ordered sets
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On a topology and limits for inductive systems of C ∗ -algebras over partially ordered sets Gumerov R.N., Lipacheva E.V., GrigoryanT.A.
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Abstract
Motivated by algebraic quantum field theory and our previous workwe study properties of inductive systems of C ∗ -algebras over arbitrary partiallyordered sets. A partially ordered set can be represented as the union of the familyof its maximal upward directed subsets indexed by elements of a certain set. Weconsider a topology on the set of indices generated by a base of neighbourhoods.Examples of those topologies with different properties are given. An inductivesystem of C ∗ -algebras and its inductive limit arise naturally over each maximalupward directed subset. Using those inductive limits, we construct different typesof C ∗ -algebras. In particular, for neighbourhoods of the topology on the set ofindices we deal with the C ∗ -algebras which are the direct products of those in-ductive limits. The present paper is concerned with the above-mentioned topologyand the algebras arising from an inductive system of C ∗ -algebras over a partiallyordered set. We show that there exists a connection between properties of thattopology and those C ∗ -algebras. Keywords C ∗ -algebra · Inductive limit · Inductive system · Partially orderedset · Topology
R. GumerovChair of Mathematical Analysis, N.I. Lobachevsky Institute of Mathematics and Mechanics,Kazan (Volga Region) Federal University, Kremlevskaya 35, Kazan, Russian Federation, 420008E-mail: [email protected]. LipachevaChair of Higher Mathematics, Kazan State Power Engineering University, Krasnoselskaya 51,Kazan, Russian Federation, 420066Tel.: +7(843)519-42-84E-mail: [email protected]. GrigoryanChair of Higher Mathematics, Kazan State Power Engineering University, Krasnoselskaya 51,Kazan, Russian Federation, 420066E-mail: [email protected] Gumerov R.N., Lipacheva E.V., Grigoryan T.A.
The motivation for the present paper comes from algebraic quantum field theory[1]–[9] and our previous work on inductive systems of C ∗ -algebras [10]–[12].The general framework of algebraic quantum field theory is given by a covariantfunctor. Usually that functor acts from a category whose objects are topologicalspaces with additional structures and its morphisms are structure preserving em-beddings into a category describing the algebraic structure of observables. Thestandard assumption in quantum physics is that the second category consists ofunital C ∗ -algebras and unital embeddings of C ∗ -algebras. The basic tool of the al-gebraic approach to quantum fields over a spacetime is a net of C ∗ -algebras definedover a partially ordered set defined as a suitable set of regions of the spacetimeordered under inclusion [1]–[3].In the papers [7]–[9] the authors consider nets containing C ∗ -algebras of quan-tum observables for the case of curved spacetimes. The net that is constructed bymeans of semigroup C ∗ -algebra generated by the path semigroup for partially or-dered set is studied in [10]. The paper [11] contains results on limit automorphismsfor inductive sequences of Toeplitz algebras which are closely related to the factson the mappings of topological groups [13], [14]. In [12] the authors deal with anet of C ∗ -algebras associated to a net over a partially ordered set consisting ofHilbert spaces.In this paper we consider a covariant functor from a category associated withan arbitrary partially ordered set K into the category of unital C ∗ -algebras andtheir unital ∗ -homomorphisms. That functor is also called an inductive system over K . Using Zorn’s lemma, the set K can be represented as the union of the family { K i } of its maximal upward directed subsets indexed by elements of a set I . Weconsider a topology on the set I generated by a base of neighbourhoods. For everyset K i , i ∈ I , the original inductive system over K yields naturally the inductivesystem of C ∗ -algebras over K i and its inductive limit. Using those inductive limits,we construct different types of C ∗ -algebras. In particular, for neighbourhoods ofthe topology on the set of indices we deal with the C ∗ -algebras which are thedirect products of limits for inductive systems over the sets K i .The present paper is devoted to the study of properties of the above-mentionedtopology and the C ∗ -algebras. We show that there exists a connection betweentopological and algebraic structures.The paper consists of Introduction, three sections and Appendix. The firstsection contains preliminaries. In the second section we consider a topology on theindex set I and study its properties. Examples of those topologies with differentproperties are given. The third section deals with inductive limits. In this sectiona connection between topological and algebraic constructions is studied. Finally,Appendix contains the figures for the examples in the second section.A part of the results in this paper was announced without proofs in [15]. In what follows, we shall consider an arbitrary partially ordered set ( K, ≤ ) thatis not necessarily directed. The category associated to this set is denoted by thesame letter K . We recall that the objects of this category are the elements of the n a topology and limits for inductive systems of C ∗ -algebras 3 set K , and, for any pair a, b ∈ K , the set of morphisms from a to b consistsof the single element ( a, b ) provided that a ≤ b , and is the void set otherwise.Further, we consider a covariant functor F from the category K into thecategory of unital C ∗ -algebras and their unital ∗ -homomorphisms. As was men-tioned in Introduction, such a functor is called an inductive system in the cat-egory of C ∗ -algebras over the set ( K, ≤ ). It may be given by a collection( K, { A a } , { σ ba } ) satisfying the properties from the definition of a functor. Weshall write F = ( K, { A a } , { σ ba } ). Here, { A a | a ∈ K } is a family of unital C ∗ -algebras. We also suppose that all morphisms σ ba : A a −→ A b , where a ≤ b , areembeddings of C ∗ -algebras, i. e., unital injective ∗ -homomorphisms. Recall thatthe equations σ ca = σ cb ◦ σ ba hold for all elements a, b, c ∈ K satisfying the con-dition a ≤ b ≤ c . Furthermore, for each element a ∈ K the morphism σ aa is theidentity mapping.Considering the family of all upward directed subsets of the set ( K, ≤ ) andmaking use of Zorn’s lemma, one can easily prove Proposition 1
Let ( K, ≤ ) be a partially ordered set. Then the following equalityholds: K = [ i ∈ I K i , (1) where { K i | i ∈ I } is the family of all maximal upward directed subsets of ( K, ≤ ) .Moreover, for every i ∈ I and a ∈ K i the set { b ∈ K | b ≤ a } is a subset of K i . Now, for each i ∈ I , we consider the inductive system F i = ( K i , { A a } , { σ ba } )over the upward directed set K i .Throughout the paper, for a unital algebra A its unit will be denoted by I A .The simplest example of the inductive system F i is that in which { A a | a ∈ K i } is a net of C ∗ -subalgebras of a given C ∗ -algebra A . By this, one means that each A a is a C ∗ -subalgebra containing the unit I A , A a ⊂ A b and σ ba : A a −→ A b is theinclusion mapping whenever a, b ∈ K i and a ≤ b . Given such a net F i , the normclosure of the union of all A a is itself a C ∗ -subalgebra of A .We recall the definition and some facts concerning the inductive limits forinductive systems of C ∗ -algebras (see, for example, [16, Section 11.4]). The inductive limit of this system is a pair ( A K i , { σ K i a } ) where A K i is a C ∗ -algebra and { σ K i a : A a → A K i | a ∈ K i } is a family of unital injective ∗ -homomorphisms such that the following two properties are fulfilled [16, Proposition11.4.1]:1) For every pair elements a, b ∈ K i satisfying the condition a ≤ b the diagram A a σ ba / / σ Kia ! ! ❉❉❉❉❉❉❉ A bσ Kib } } ④④④④④④④ A K i is commutative, that is, the equality for mappings σ K i a = σ K i b ◦ σ ba . (2) Gumerov R.N., Lipacheva E.V., Grigoryan T.A. holds. Moreover, we have the following equality: A K i = [ a ∈ K i σ K i a ( A a ) , (3)where the bar means the closure of the set with respect to the norm topology inthe C ∗ -algebra A K i .2) The universal property. If B is a C ∗ -algebra, ψ a : A a −→ B is an injective ∗ -homomorphism for each a ∈ K i , and conditions analogous to those in (2) and(3) are satisfied, then there exists a ∗ -isomorphism θ from A K i onto B such thatthe diagram A a ψ a ❆❆❆❆❆❆❆❆ σ Kia } } ③③③③③③③ A K i ϕ / / B is commutative for every a ∈ K i , that is, the equality ψ a = ϕ ◦ σ K i a holds.The inductive limit ( A K i , { σ K i a } ) is denoted as follows:( A K i , { σ K i a } ) := lim −→ F i . The C ∗ -algebra A K i itself is often called the inductive limit.It is well known that the inductive limit can always be constructed for aninductive system in the category of C ∗ -algebras. For the convenience of the reader,we recall briefly the components of that construction (the details are contained inthe proof of Proposition 11.4.1 in [16]). We shall use them in our proofs.We consider the direct product of C ∗ -algebras Y a ∈ K i A a := (cid:26) ( F a ) (cid:12)(cid:12) k ( F a ) k = sup a k F a k < + ∞ (cid:27) . It is a C ∗ -algebra relative to the pointwise operations and the supremum norm. Ithas a norm-closed two-sided ideal Σ consisting of all elements ( F a ) in the directproduct for which the net {k ( F a ) k | a ∈ K i } converges to 0. When a ∈ K i the ∗ -homomorphism θ K i a : A a −→ Y b ∈ K i A b is defined at an element A ∈ A a as follows:[ θ K i a ( A )]( b ) = ( σ ba ( A ) , if a ≤ b ;0 , otherwise . (4)The mapping A −→ θ K i a ( A ) + Σ is a unital injective ∗ -homomorphism σ K i a from A a into the quotient C ∗ -algebra Q a ∈ K i A a /Σ . The family { σ K i a ( A a ) | a ∈ K i } is a net of C ∗ -subalgebras of Q a ∈ K i A a /Σ , and the norm closure of S σ K i a ( A a ) isthe inductive limit A K i which is a C ∗ -subalgebra of Q a ∈ K i A a /Σ .It is worth noting that below we shall denote by the same symbols Σ and θ K i a the corresponding ideals and mappings for distinct inductive systems. n a topology and limits for inductive systems of C ∗ -algebras 5 Finally, we construct the direct product for the inductive limits of the functors F i denoted by M F := Y i ∈ I A K i = (cid:26) ( a i ) (cid:12)(cid:12) k ( a i ) k = sup i k a i k < ∞ (cid:27) . For additional results in the theory of C ∗ -algebras we refer the reader, forexample, [17, Ch. 4, §
7] and [18]. Necessary facts from the theory of categoriesand functors are contained, for example, in [17, Ch. 0, §
2] and [19].
Throughout this section, K is an arbitrary partially ordered set that is not neces-sarily directed. By Proposition 1, we have equality (1).Now we endow the index set I with a topology. To this end, for every element a ∈ K we define the set U a = { i ∈ I : a ∈ K i } . Obviously, U a is a non-emptyset.Using the property of maximal upward directed sets K i from Proposition 1, itis straightforward to prove Lemma 1 If a, b ∈ K such that a ≤ b then U b ⊂ U a . We shall now show that the family of sets { U a | a ∈ K } generates a topologyon the index set I . Proposition 2
The family { U a | a ∈ K } is a base for a topology on the set I .Proof To prove the assertion we use Proposition 1.2.1 from [20]. According to it,we need to check two properties of a base.Firstly, for any U a and U b , a, b ∈ K , and every point i ∈ U a T U b there exists U c , c ∈ K , such that i ∈ U c ⊂ U a T U b . Indeed, since i ∈ U a T U b the elements a and b belong to the maximal upward directed set K i . Hence, there is an element c ∈ K i satisfying the conditions a ≤ c and b ≤ c . By Lemma 1, the set U c satisfiesthe required property.Secondly, it is clear that for every i ∈ I there exists an element a ∈ K suchthat i ∈ U a . ⊓⊔ We denote by τ the topology generated by the base { U a | a ∈ K } . Thus, τ isthe family of all subsets of I that are unions of subfamilies of { U a | a ∈ K } .Below we prove the propositions describing properties of the topological space( I, τ ) and give several examples.
Proposition 3
The topological space ( I, τ ) is a T -space.Proof Take any pair of distinct indices i, j ∈ I . The condition i = j implies that K i = K j . Hence, we can take an element a ∈ K i \ K j . Then we have the conditions i ∈ U a and j / ∈ U a . ⊓⊔ Corollary 1
For every index i ∈ I the equality T a ∈ K i U a = { i } holds. Gumerov R.N., Lipacheva E.V., Grigoryan T.A.
Corollary 2
For every index i ∈ I the one-point set { i } is closed. The next example shows that (
I, τ ) may not be a Hausdorff space.
Example 1
We consider the set of points with integer coordinates K := { ( x, y ) | x ∈ {−
1; 0; 1 } , y ∈ Z } . A partial order ≤ on the set K is defined in the following way:( x , y ) ≤ ( x , y ) := ( x , x ∈ {−
1; 1 } , x = x , y ≤ y ; x ∈ {−
1; 1 } , x = 0 , y < y . It is straightforward to check that the pair ( K, ≤ ) is a partially ordered set, whichis not upward directed.One has the representation of K as the union of maximal upward directed sets K i indexed by the set of all integers Z together with two symbols −∞ and + ∞ ,that is, I = Z ∪ {−∞ ; + ∞} : K = + ∞ [ i = −∞ K i , where K −∞ := { ( − , y ) | y ∈ Z } ; K + ∞ := { (1 , y ) | y ∈ Z } and K i := { (0 , i ) } [ { ( x, y ) | x ∈ {−
1; 1 } , y < i, y ∈ Z } for each i ∈ Z . A base { U ( x,y ) | x ∈ {−
1; 0; 1 } , y ∈ Z } for the topology τ on the index set I consists of the sets of three types, namely, U ( − ,y ) := {−∞}∪{ i ∈ Z | i > y } ; U (1 ,y ) := { + ∞}∪{ i ∈ Z | i > y } ; U (0 ,y ) := { y } . Since any two neighbourhoods of indices −∞ and + ∞ have a non-empty intersec-tion the topological space ( I, τ ) is not a Hausdorff space.
Proposition 4
For a ∈ K , the set K a is upward directed if and only if the neigh-bourhood U a consists of a single point.Proof Necessity. Assume that K a is an upward directed set. By Zorn’s Lemma,there exists an index j ∈ I such that K a is contained in the maximal upwarddirected set K j . Hence, a ∈ K j . Thus, by the definition of the neighbourhood U a ,we get the inclusion { j } ⊂ U a . (5)Next, we shall show the reverse inclusion. In order to obtain a contradiction,we suppose that there exists an index i ∈ I distinct from the index j such that i ∈ U a . Then, we have a ∈ K i and K i = K j .In this case we can take an element c ∈ K i \ K j . (6)Since a, c ∈ K i and K i is an upward directed set there exists an element d ∈ K i satisfying the conditions a ≤ d as well as c ≤ d . The first condition implies that d ∈ K a ⊂ K j . Obviously, the latter together with the maximality property of the n a topology and limits for inductive systems of C ∗ -algebras 7 upward directed set K j yields the inclusion c ∈ K j . This contradicts condition (6).Therefore, we obtain the inclusion that is reverse to (5). Thus, the equality U a = { j } (7)holds, as required. Sufficiency.
Let equality (7) be valid for some index j ∈ I . By the the definitionof the neighbourhood U a , the element a belongs to the unique maximal upwarddirected set K j .We claim that the following inclusion holds: K a ⊂ K j . (8)Indeed, take any b ∈ K a . Then a ≤ b and, by Lemma 1, U b ⊂ U a . Therefore, U b = { j } and b ∈ K j . Consequently, we have inclusion (8), as claimed.To show that K a is an upward directed set we take two elements b, c ∈ K a .By (8), we have b, c ∈ K j . Since K j is an upward directed set there exists d ∈ K j such that both the conditions b ≤ d and c ≤ d hold. Obviously, we have d ∈ K a .Thus, the set K a is upward directed, as required. ⊓⊔ Now we give examples of locally compact and discrete topological spaces.
Example 2
As the set K we consider the lower half-plane without the axis y = 0,that is, K = { ( x, y ) | x, y ∈ R , y < } . We define a partial order ≤ on K as follows. Let us fix a positive number a ∈ R . Then we put( x , y ) ≤ ( x , y ) := ( x = x and y = y ; y − y > a | x − x | . It is easily verified that the pair ( K, ≤ ) is a partially ordered set. Moreover, it isworth noting that this set is not upward directed.We have the representation of K as the union of maximal upward directed sets K i indexed by the set of all real numbers, that is, I = R : K = [ i ∈ R K i , where K i := { ( x, y ) ∈ K | − y > a | i − x |} . Taking a point ( x , y ) ∈ K , one can easy see that U ( x ,y ) = n i ∈ R | x + y a < i < x − y a o . Thus, in this example the topology τ coincides with the natural topology onthe set R wich is locally compact. Gumerov R.N., Lipacheva E.V., Grigoryan T.A.
Example 3
Here, we take as K the points with integer coordinates in the lowerhalf-plane including the axis y = 0, that is, K = { ( n, m ) : n, m ∈ Z , m ≤ } . We define a partial order ≤ on the set K by the following rule:( n , m ) ≤ ( n , m ) if and only if m − m ≥ n − n ≥ . The pair ( K, ≤ ) is a partially ordered set. It is not upward directed as well.We have the representation of K as the union of maximal upward directed sets K i indexed by the set of all integers, that is, I = Z : K = [ i ∈ Z K i , where K i := { ( n, m ) ∈ K | − m ≥ i − n ≥ } . For any point ( n , m ) ∈ K , in the space I we have the neighbourhood U ( n ,m ) = { i ∈ Z : n ≤ i ≤ n − m } . Since the equality U ( n, = { n } holds we see that every point in the space I = Z is isolated. Thus, we conclude that the topology τ is discrete. Let us consider an inductive system F = ( K, { A a } , { σ ba } ), where A a is an arbitraryunital C ∗ -algebra. Then for each index i ∈ I we can construct an inductive system F i and its inductive limit A K i .Further, we take any a ∈ K and consider the direct product of C ∗ -algebras B a := Y i ∈ U a A K i . Recall, by Lemma 1, for every pair a, b ∈ K satisfying the condition a ≤ b ,we have the inclusion U b ⊆ U a . Hence, the ∗ -homomorphism τ ba : B a −→ B b between C ∗ -algebras given by the rule τ ba ( f )( j ) = f ( j ) , where f ∈ B a and j ∈ U b , is well-defined.Obviously, we have the equality τ ca = τ cb ◦ τ ba whenever a, b, c ∈ K such thatthe condition a ≤ b ≤ c holds.Thus we have constructed the inductive system of C ∗ -algebras ( K, { B a } , { τ ba } ).Therefore, for each index i ∈ I we can consider the inductive system ( K i , { B a } , { τ ba } )and its inductive limit( B K i , { τ K i a } ) := lim −→ ( K i , { B a } , { τ ba } ) . For the direct product of these inductive limits we introduce the followingnotation: c M F := Y i ∈ I B K i . Further, we prove the theorems that show the connections between the struc-tures of C ∗ -algebras A K i , B K i , M F , c M F and the properties of the topologicalspace ( I, τ ). n a topology and limits for inductive systems of C ∗ -algebras 9 Theorem 1
For every index i in the set I the algebra A K i is isomorphic to asubalgebra of B K i .Proof To construct an injective ∗ -homomorphism from the algebra A K i into thealgebra B K i we proceed as follows.Take an arbitrary neighbourhood U a of the point i . For every index j ∈ U a weconsider the inductive limit( A K j , { σ K j b } ) = lim −→ ( K j , { A b } , { σ cb } ) . Using the family of injective ∗ -homomorphisms { σ K j a : A a −→ A K j | j ∈ U a } ,we define a ∗ -homomorphism of C ∗ -algebras σ U a a : A a −→ B a = Y j ∈ U a A K j by means of the formula [ σ U a a ( A )]( j ) := σ K j a ( A ), where A ∈ A a , j ∈ U a . Notethat the injectivity of the ∗ -homomorphisms σ K j a implies the injectivity of σ U a a .Moreover, the following equalities hold: k [ σ U a a ( A )]( j ) k = k σ K j a ( A ) k = k A k . (9)Now we take two inductive systems ( K i , { A a } , { σ ba } ) and ( K i , { B a } , { τ ba } ).For every pair of elements a, b ∈ K i satisfying the condition a ≤ b we have thediagram A aσ Uaa (cid:15) (cid:15) σ ba / / A bσ Ubb (cid:15) (cid:15) B a τ ba / / B b (10)It is commutative, that is, we have the equality for the compositions of morphisms σ U b b ◦ σ ba = τ ba ◦ σ U a a . (11)To show the validity of (11) let us take any element A ∈ A a . For every index j ∈ U b we have the following equalities:[ σ U b b ◦ σ ba ( A )]( j ) = σ K j b ( σ ba ( A )) = σ K j a ( A );[ τ ba ◦ σ U a a ( A )]( j ) = σ U a a ( A )( j ) = σ K j a ( A ) . Therefore, we obtain equation (11).The commutativity of (10) yields the commutativity of the diagram A a σ ba / / τ Kia ◦ σ Uaa ! ! ❉❉❉❉❉❉❉❉ A bτ Kib ◦ σ Ubb } } ③③③③③③③ B K i Indeed, we have the following equalities: τ K i a ◦ σ U a a = ( τ K i b ◦ τ ba ) ◦ σ U a a = ( τ K i b ◦ σ U b b ) ◦ σ ba . We claim that for every a ∈ K i the ∗ -homomorphism τ K i a ◦ σ U a a is injective.To see that we take two arbitrary distinct elements A and A in the C ∗ -algebra A a . For l = 1 , τ K i a ◦ σ U a a ( A l ) = θ K i a ( σ U a a ( A l )) + Σ. We need to show that the condition θ K i a ( σ U a a ( A − A )) / ∈ Σ (12)holds, that is, the net {k [ θ K i a ( σ U a a ( A − A ))]( x ) k | x ∈ K i } does not converge to0. By (4) for every element x ∈ K i one has[ θ K i a ( σ U a a ( A l ))]( x ) = ( τ xa ( σ U a a ( A l )) , if a ≤ x ;0 , otherwise . (13)In view of the commutativity of diagram (10) with x instead of b we have informula (13) the following: τ xa ( σ U a a ( A l )) = σ U x x ( σ xa ( A l )) . (14)It follows from (13), (14), (9) and the injectivity of the ∗ -homomorphism σ xa ) thatfor every element x ∈ K i we get k [ θ K i a ( σ U a a ( A − A ))]( x ) k = ( k A − A k , if a ≤ x ;0 , otherwise . (15)Really, on the right-hand part of (15) the first row is valid because we have theequalities: k σ U x x ( σ xa ( A − A )) k = sup n(cid:13)(cid:13)(cid:13) [ σ U x x ( σ xa ( A − A ))]( j ) (cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12) j ∈ U x o == sup n(cid:13)(cid:13)(cid:13) σ K j x ( σ xa ( A − A )) (cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12) j ∈ U x o == k σ xa ( A − A )) k = k A − A k . Hence, there exists ε >
0, for example, ε = k A − A k , satisfying the followingproperty: for every y ∈ K i there is an element x ∈ K i such that y ≤ x , a ≤ x and k [ θ K i a ( σ U a a ( A − A ))]( x ) k > ε. This means that condition (12) is satisfied, and the ∗ -homomorphism τ K i a ◦ σ U a a is an injection, as claimed.Thus, by the universal property of inductive limits, we get the injective ∗ -homomorphism of C ∗ -algebras A K i −→ B K i , as required. ⊓⊔ Corollary 3
The C ∗ -algebra M F is isomorphic to a subalgebra of c M F . n a topology and limits for inductive systems of C ∗ -algebras 11 Theorem 2
Let i ∈ I be an isolated point in the topological space ( I, τ ) . Thenthe C ∗ -algebras B K i and A K i are isomorphic.Proof Since the one-point set { i } is open in the topological space ( I, τ ) and thefamily of sets { U a | a ∈ K } constitutes a base for the topology τ there exists anelement a ∈ K i such that we have the equality U a = { i } . Fix that element a ∈ K i .By Lemma 1, the neighbourhood U b coincides with the one-point set { i } when-ever b ∈ K and a ≤ b . Note that such an element b lies in the set K i .Further, we take the upward directed set K ai := { b ∈ K i | a ≤ b } , which is a cofinal subset in K i .Then we consider the inductive system ( K ai , { B b } , { τ cb } ) over the set K ai . Forevery element b ∈ K ai we have the equality B b = B a . Note that the C ∗ -algebra B a consists of all the functions from the one-point set { i } into the C ∗ -algebra A K i . Obviously, the C ∗ -algebras B a and A K i are isomorphic. Moreover, in thissystem each bonding morphism τ cb is the identity mapping. Thus, one has theisomorphism of C ∗ -algebraslim −→ ( K ai , { B b } , { τ cb } ) ≃ A K i . (16)Further, we claim that there exists an isomorphism between C ∗ -algebraslim −→ ( K ai , { B b } , { τ cb } ) ≃ B K i . (17)The existence of such an isomorphism follows from the universal property for theinductive limits.To show this, firstly, for every b ∈ K ai we consider the ∗ -homomorphism τ K i b : B b −→ B K i . It is injective. Indeed, we take an arbitrary non-zero element f ∈ B b .Then we have τ K i b ( f ) = θ K i b ( f ) + Σ, where the ∗ -homomorphism θ K i b : B b −→ Q x ∈ K i B x is given by the formula[ θ K i b ( f )]( x ) = ( τ xb ( f ) = f, if b ≤ x ;0 , otherwise . Obviously, we have θ K i b ( f ) / ∈ Σ . Hence, τ K i b is an injective ∗ -homomorphism.It is clear that τ K i b = τ K i c ◦ τ cb whenever b, c ∈ K ai and b ≤ c .Secondly, we prove the following equality: B K i = [ b ∈ K ai τ K i b ( B b ) . (18)To this end, we recall that one has the equality B K i = [ x ∈ K i τ K i x ( B x ) . (19) Certainly, the right-hand part of (18) is contained in the right-hand part of(19). To obtain the reverse inclusion we fix an arbitrary element x ∈ K i . Since K ai is cofinal in K i there is b ∈ K ai such that x ≤ b . The commutativity of thediagram B x τ bx / / τ Kix " " ❉❉❉❉❉❉❉❉ B bτ Kib | | ③③③③③③③ B K i yields the following equality for sets τ K i b ( τ bx ( B x )) = τ K i x ( B x )which implies the desired inclusion for sets, namely, τ K i x ( B x ) ⊂ τ K i b ( B b ) . Consequently, the right-hand parts of (18) and (19) coincide. Thus, we haveproved equality (18).It follows from Proposition 11.4.1(ii) in [16] that there exists an isomorphism(17).Finally,combining isomorphisms (16) and (17), we conclude that the C ∗ -algebras B K i and A K i are isomorphic, as required. ⊓⊔ The following statement is an immediate consequence of Theorem 2.
Corollary 4
Let ( I, τ ) be a discrete topological space. Then the C ∗ -algebras M F and c M F are isomorphic. Theorem 3
Let i ∈ I be a non-isolated point with a countable neighbourhood base.Then the algebra B K i has a non-trivial center.Proof It is clear that in the space (
I, τ ) one can construct a countable neighbour-hood base { U a n | a n ∈ K i , n ∈ N } at the point i ∈ I satisfying the followingconditions: U a ⊃ U a ⊃ U a ⊃ . . . ; a ≤ a ≤ a ≤ . . . . Further, for each n ∈ N we consider the subset W n in I given by W n := U a n \ U a n +1 . Using these sets, we define the element f in the subalgebra Q i ∈ I CI A Ki of thealgebra M F . Namely, for every i ∈ I the value of the function f at the point i isdefined as follows: f ( i ) = , if i / ∈ U a ; I A Ki , if i ∈ W k − , k ∈ N ;0 , if i ∈ W k , k ∈ N . (20) n a topology and limits for inductive systems of C ∗ -algebras 13 Now for every element a n we consider two elements f a n and g a n in the subal-gebra Q i ∈ U an CI A Ki of the algebra B a n . We put f a n := f | U an , that is, the function f a n is the restriction of the function f to the neighbourhood U a n , and g a n := I B an − f a n .Together with the functions f a n and g a n we define two elements ˜ f and ˜ g in theinductive limit B K i by˜ f := τ K i a n ( f a n ) and ˜ g := τ K i a n ( g a n ) . (21)It is clear that these elements do not depend on the choice of the index a n .Obviously, we have the equality ˜ f · ˜ g = 0 as well as ˜ f + ˜ g = I B Ki . Thus theelements ˜ f and ˜ g are non-trivial projections in the C ∗ -algebra B K i . We claim that the elements ˜ f and ˜ g belong to the center of the algebra B K i . Tothis end, we take an element A = h + Σ of the C ∗ -algebra B K i , where h ∈ Q x ∈ K i B x .Let us show that the following equality holds:˜ f · A = A · ˜ f. (22)Indeed, for the left-hand part of (22) we have the expression:˜ f · A = τ K i a n ( f a n ) · A = ( θ K i a n ( f a n ) + Σ ) · ( h + Σ ) = θ K i a n ( f a n ) · h + Σ. (23)Analogously, for the right-hand part of (22) we get the representation A · ˜ f = h · θ K i a n ( f a n ) + Σ. (24)Further, for x ∈ K i we have[ θ K i a n ( f a n ) · h ]( x ) = ( τ xa n ( f a n ) · h ( x ) , if a n ≤ x ;0 , otherwise . (25)For the function τ xa n ( f a n ) · h ( x ) from the algebra B x we take its value at apoint j ∈ U x :[ τ xa n ( f a n ) · h ( x )]( j ) = [ τ xa n ( f a n )( j )] · [ h ( x )( j )] = f a n ( j ) · [ h ( x )( j )] . (26)Changing the order of the factors in (25) and (26), one gets the similar expres-sions for the summand h · θ K i a n ( f a n ) in the right-hand part of (24).By (20) and the definition of the function f a n , we obtain the equality f a n ( j ) · [ h ( x )( j )] = [ h ( x )( j )] · f a n ( j ) . Therefore the elements in the right-hand parts of (23) and (24) are the same.Hence, equality (22) is proved. Similarly one can prove equality (22) with ˜ g insteadof ˜ f .It follows that the elements ˜ f and ˜ g belong to the center of the algebra B K i ,as claimed. ⊓⊔ As a consequence of Theorem 3 and the definition of the C ∗ -algebra c M F wehave the following statement. Corollary 5
Let ( I, τ ) be a first-countable topological space without isolated points.Then the C ∗ -algebra c M F has a non-trivial center. Appendix: Figures for Examples
Example K y K y +1 K y +2 K y +3 ( − , y ) (1 , y ) K −∞ K + ∞ U (1 ,y ) = { + ∞} ∪ { y + 1 , y + 2 , . . . } U ( − ,y ) = {−∞} ∪ { y + 1 , y + 2 , . . . } Fig.1.
Non Hausdorff space
Example x ; y ) ( x ; y ) i K i y − y = − a ( x − x ) y − y = a ( x − x ) U ( x ; y ) ( x ; y ) xy O Fig.2.
Locally compact space n a topology and limits for inductive systems of C ∗ -algebras 15 Example n ; m ) ( n ; m ) K i U ( n ; m ) ( n ; m ) nmO i Fig.3.
Discrete space
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