K-theory on Buildings

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1 K-theory on Buildings Johan Konter 15th of May

2 Contents Preface 2 1 Introduction Equivalence relations Cuntz-Krieger algebras & Graphs The Cuntz-Krieger algebras The graphs Boundary operator algebra Conclusion Buildings Chamber systems Coxeter complexes Buildings The boundary Buildings of type à Boundary operator algebra Higher rank Cuntz-Krieger algebras The general case The second order Cuntz-Krieger algebra The transitive case Projective planes Restricting Γ The Cayley graph M 1, M 2, ˆM1 and ˆM The class of the identity On three vertices Polygonal presentation ˆM1 and ˆM The class of the identity Conclusion 46 1

3 Preface About a year ago I was looking for a subject for my bachelor thesis. There weren t any problems I had in mind yet, so loving all of the broad perspective of mathematics I turned to Gunther Cornelissen whether he had something for me to work on. I was presented with several subjects and I started making myself more comfortable with modular forms, monstrous moonshine, the shape of a drum and a particular problem in graph theory. After some elaboration I decided to go for the graph theory, which involved C -algebras and some K-theory, for I thought that topic needed the least reading in advance. I think I was right about that, but it was easily compensated for during the rest of the time. Now the thesis is about classifying C -algebras one can define on buildings, which are generalizations of graphs. We do this with the help of K-theory. I want to thank Alina Vdovina for arranging my stay in Newcastle and the very fruitful discussions we had. I want to thank Het A.F. Monna Fonds for making this financially possible. And of course I want to thank my supervisor Gunther Cornelissen for his continuous support and useful remarks. 2

4 1 Introduction In this thesis we will construct two different C -algebras on buildings. Most of the time these buildings will be of type Ã2. It is shown that these algebras coincide and that they can be classified with K-theory. To be more precise they will be classified by their K 0 - group and the order of the identity in this group. These groups will be investigated and we will give upper and lower bounds for the order of the identity. The first section is a bit of a warm up to get more feeling for the subject. It deals with the one dimensional case of which we know almost everything now. The next two sections are about defining new theoretical setups to take the problem discussed in the first section to a higher dimension. The theorems and discussions in these two section will be applicable throughout the rest of the paper. Then there are two sections that have to be read separately, they are both playing with are particular type of the buildings discussed in general. For simplicity they use the same notation, but some elements, like Γ, have really different characteristics. Before we start there is one last thing. In this thesis we will be talking about classifying C -algebras up to certain equivalence relations. I will make quite a lot remarks about the combinatorics we encounter, because they are at the core of most proofs. About C - algebras however I will tell you nothing but how they are defined and how we will classify them. We will hardly work with the C -algebras because the results on how to classify them are there and the proofs of these results would be to much to include. The three equivalence relations we will be dealing with are based on isomorphism, stable isomorphism and strong Morita equivalence. 1.1 Equivalence relations We will encounter quite a lot of isomorphisms. They preserve all structure of the objects we are discussing at that moment. For example, when we are talking about isomorphic C -algebras, we are of course talking about -isomorphic. If A is isomorphic to B, we will denote this by A B. Definition Two C -algebras A and B are stably isomorphic if A K and B K are isomorphic, where K is the algebra of compact operators on a separable Hilbert space. Two rings R and S are Morita equivalent if and only if there are two bimodules M and N, more precise M is a left-r- and a right-s-module, N the other way around, such that R M N and S N M as bimodules. But C -algebras have more structure than rings so a slightly different notion is developed. Definition Two C -algebras A and B are called strong Morita equivalent if there exists an A-B-equivalence bimodule [10]. 3

5 An A-B-equivalence bimodule X is an A,B-bimodule with A- and B-valued inner products,, A and, B respectively. Equipped with these inner products X should be a left Hilbert A-module and a right Hilbert B-module, respecting ˆ x, y A z = x y, z B for all x, y, z X, ˆ X, X A spans a dense subset of A and X, X B spans a dense subset of B. We will be talking only about (strong) Morita equivalence between C -algebras, thus we will omit the adjective strong. When we are talking about Morita equivalence we mean strong Morita equivalence between C -algebras. Theorem 1.1 (Brown, Green, Rieffel [3]) If two C -algebras are isomorphic, then they are stably isomorphic. If two C -algebras are stably isomorphic, they are Morita equivalent. The converse of the last statement is also true if both algebras have a strictly positive element, or equivalently both contain a countable approximate identity. 4

6 2 Cuntz-Krieger algebras & Graphs 2.1 The Cuntz-Krieger algebras As one may guess Cuntz-Krieger algebras were introduced by J. Cuntz and W. Krieger [8]. These C -algebras originate from their relation to topological Markov chains about thirty years ago. But they also occur as C -algebras on directed graphs, and this will be our main interest. Definition The Cuntz-Krieger algebra O A associated with a nondegenerate n n matrix A is the universal C -algebra generated by n partial isometries {s i } n i=1 fulfilling the conditions s i s i s j s j = 0 for i j, and s i s i = n A ij s j s j. (1) Lemma 2.1 One needs not to distinguish between the notions of stably isomorphic and Morita equivalent Cuntz-Krieger algebras. Proof: The first condition in (1) is equivalent to s 1 s 1 + s 2 s s n s n = 1 ( ). This can be seen by applying s i s i to the left side of ( ) and keeping in mind that s i s i s i s i = s i s i for the s i are partial isometries. If (1) is true the outcome is clearly s i s i, hence ( ) is true. If ( ) is true we find that ( j i s js j)s i s i = 0, and because of the independence of the terms of the sum this implies the first condition in (1). So O A is unital and has a countable approximate identity. Hence one needs not to distinguish between the notions of stably isomorphic and Morita equivalent Cuntz-Krieger algebras, due to the Theorem 1.1. Definition A square {0, 1}-matrix that has exactly one entry 1 in each row and column is called a permutation matrix. A square matrix is called irreducible if it is not similar to a block upper triangular matrix via a permutation, i.e. there does not exist a permutation matrix such that conjugation with this permutation matrix gives a block upper triangular matrix. Remark If a matrix A is irreducible, then for each i and j there exists some k N such that the entry (A k ) ij of the matrix A k is strictly positive. Theorem 2.2 (Theorem 2.14 of [8]) The C -algebra O A is simple if and only if the matrix A is not a permutation matrix but still irreducible. Theorem 2.3 (Theorem 6.5 of [12] and Theorem 5.3 of [8]) Simple Cuntz-Krieger algebras are classified up to stable isomorphism by K 0 (O A ), and even up to isomorphism by the K 0 - group together with the position of the equivalence class of the unit of O A. Furthermore K 0 (O A ) Z n /(1 A t )Z n. This gives us the tools to classify Cuntz-Krieger algebras, so let s look at a particular example. 5 j=1

7 e ē 2.2 The graphs Figure 1: T (e) Now we are going to look at graphs, which we will assume to be finite and connected. We denote a certain graph with m edges by X and we define X + as the directed graph whose vertices equal the vertices of X and whose edges consist of all edges of X with both possible orientations. The two edges of X + that the edge e of X is contributing to we will denote by e and ē. By o(e) and t(e) we denote the origin resp. the terminal vertex of an oriented edge e. Let EX + denote the set of edges of the oriented graph and let Z EX+ denote the free Z-module spanned by this set. Definition We consider the linear operator T : Z EX+ Z EX+ defined on the basis elements e EX + by T (e) = e ē, t(e)=o(e ) as considered by G. Cornelissen, O. Lorscheid and M. Marcolli in [7]. The operator T is represented by a matrix A, indexed by the oriented edges, and the Cuntz-Krieger algebra O A is considered. Then the two following theorems are proved. Theorem 2.4 (Theorem 1 of [7]) It is shown for the K 0 -group of this C -algebra that K 0 (O A ) Z g Z/(g 1)Z, where g 1 is the first Betti number of the graph. Theorem 2.5 (Theorem 2 of [7]) For g 2 we have that the image of the unit of O A in K 0 (O A ) has exactly order g 1 gcd(g 1, V X ), where V X is the number of vertices of X. This is extremely useful because there is an automorphism of the group Z g Z/(g 1)Z that maps an element of Z/(g 1)Z to an other exactly when those elements have the same order. These results hence show that O A is classified up to stable isomorphism by the first Betti number g of the graph, and up to isomorphism by the first Betti number and gcd(g 1, V X ). 6

8 Figure 2: The boundary of a certain infinite tree. 2.3 Boundary operator algebra A different approach was pursued by G. Robertson in [14]. Definition Let X be a graph and let be the universal covering tree of X. The fundamental group Γ of X is a finitely generated free group that acts on freely and cocompactly. Freely means that no element of Γ other than the unit element fixes any element and cocompactly means that Γ\, that is under the equivalence relation that identifies points in the same orbit, is a finite graph, which may be seen as the graph of orbits. Remark Actually we might as well begin with a locally finite tree, instead of with X. To do so we endow the automorphism group Aut( ) with the compact-open topology, which makes it a locally compact topological group. Let Γ be a discrete subgroup of Aut( ) which acts freely and cocompactly on. Now Γ is referred to as a free uniform tree lattice. Then Γ is a uniform lattice in Aut( ) by acting cocompactly, and a finitely generated free group by acting freely. We find the graph by the isomorphism X Γ\. We define a equivalence relation on the semi-geodesics (i.e., lines extending to infinity at one end,) of by calling two of them equivalent if they contain a sub-semi-geodesic. Definition The boundary of is defined as the set of equivalence classes of semigeodesics in. There is a natural induced action of Γ on and there is a natural compact totally disconnected topology on. This allows us to define a full crossed product algebra. Consider the commutative C -algebra C( ) of continuous complex valued functions on. Definition If we fix a unitary representation π of Γ, the full crossed product algebra C( ) Γ is the universal C -algebra generated by C( ) and π(γ) satisfying the covariant defining relation f(γ 1 ω) = π(γ)fπ(γ) 1 (ω) for f C( ), γ Γ and ω. This algebra we also call the boundary operator algebra. 7

9 Remark The left hand side is the pullback of f C( ) under γ Γ. If we forget about the technical details a bit this relation just imposes that γ(f) = γfγ Conclusion The connection between the boundary operator algebra and the Cuntz-Krieger algebras is given by A. Kumjian and D. Pask. They found the following theorem. Theorem 2.6 (Kumjian, Pask [9]) The C -algebras O A and C( ) Γ are Morita equivalent. And then we have one of the main result of G. Robertson in [14], where we consider A, the matrix in of [7]. That is, A is a matrix of order 2g, with g again the first Betti number of the graph, and this matrix is indexed by generators of Γ in a logical order with { A 1 if y x 1, (x, y) = 0 if y = x 1. Theorem 2.7 (Theorem 1 of [14]) With the notation introduced above K 0 (C( ) Γ) Z g Z/(g 1)Z and the order of the the equivalence class of the identity is exactly g 1. Furthermore, C( ) Γ is isomorphic to the simple Cuntz-Krieger algebra O A. This leads to a nice conclusion of this section which we summarize in a theorem. Theorem 2.8 The Cuntz-Krieger algebras O A and O A they are isomorphic if and only if gcd(g 1, V X ) = 1. are always stably isomorphic and Proof: The first result is found by just combining Lemma 2.1, Theorem 2.6 and the second part of Theorem 2.7, which say together that O A is stably isomorphic to O A C( ) Γ. This may also be deduced from in [7]. To find the bonus that they are isomorphic if and only if gcd(g 1, V X ) = 1, we have combine Theorem 2.3, Theorem 2.5 and the first part of Theorem 2.7. Here we see a nice interaction between the graph-theoretical and the algebraic approach. This is what we will try to do in the rest of this paper as well. 8

10 3 Buildings A covering tree may be seen as a affine building of type Ã1, which has the Coxeter group of type (, Section 4.3 of [2]). And a special type of graphs, the generalized m-gons, m may be seen as buildings of type A m with Coxeter group of type, because the corresponding Coxeter complex is just a normal m-gon(, Prop. 3.3 of [21]). This indicates that it might be nice to look at buildings, as a generalization of graphs. These generalized m-gons will appear later on in the text as well. When we meet them next time I will give a more detailed survey. First we will give a survey of buildings. We will concentrate on a very straightforward method, only giving the definitions and using the things we need. For a more complete overview or more examples we refer the reader to [21], which is our main guide. Another good reference is [2]. 3.1 Chamber systems Definition We start with a set C and an index set I which together form a chamber system. Every element i I determines a partition of C and if x, y C lie in the same part, then they are called i-adjacent. This is obviously an equivalence relation and we ll denote this with x i y. For a subset J I a J-gallery is a sequence (c 0,..., c n ) such that every c i and c i+1 are j i -adjacent for some j i J. Such a gallery is said to be of type j 0 j 1 j n 1. The J-connected components are just what one thinks they are, the components in which every two chambers can be connected by a J-gallery, and they are also called J-residues. The residue is said to be of type J and the rank of a J-residue is defined as J, the number of elements of J. For a J-residues R and a K-residue S we say that S is a face of R if R S and J K. It is always useful to have a picture in your head of what you are doing. To help you with that one can draw a geometric realisation. It also clarifies some names we ve given as adjacent and face, which might have seem a bit awkward. This realisation has dimension I 1. We start by representing every residue of corank 1 with an vertex. Then every residue R of cotype {i, j}, and of corank 2, is represented by an edge in such a way that the endpoints are the representations of faces of R. So they are of cotype i and j to start with. Now we glue the residues of corank 3, represented by 2-simplices, etc. such that the faces of every residue R represent the faces of the simplex representing R. Example Take C = {x, y}, I = {1, 2, 3} and suppose x and y are 1, 2 and 3 adjacent. Because I = 3 the geometric realisation will be of dimension 2. We start with three vertices, the residue {x, y} of cotype {1}, the residue {x, y} of cotype {2} and the residue {x, y} of cotype {3}. Then they are all connected by edges, the residue {x, y} of cotype {1, 2}, the residue {x, y} of cotype {2, 3} and the residue {x, y} of cotype {3, 1}. Finally two triangles are adjoined to the three edges, the residue {x} of cotype {1, 2, 3}, the residue {y} of cotype {1, 2, 3}. The resulting image is homeomorphic to the sphere with three points on the equator, which we see in Figure 3. 9

11 Example Now consider the same C and I but suppose x and y are only 1 adjacent. This time we start with four vertices, the residue {x} of cotype {1}, the residue {y} of cotype {1}, the residue {x, y} of cotype {2} and the residue {x, y} of cotype {3}. Now we get five edges, the residue {x, y} of cotype {2, 3}, the residue {x} of cotype {1, 2}, the residue {x} of cotype {1, 3}, the residue {y} of cotype {1, 2} and the residue {y} of cotype {1, 3}. And we get again two triangles adjoined to three of the edges. We get the picture of a diamond with one diagonal, as seen in Figure 3 {2},{x,y} I,{x} {1},{x,y} I,{y} {3},{x,y} {1},{x} {2},{x,y} {1,2},{x} {1,2},{y} {2,3} {1},{y} I,{x},{x,y} I,{y} {1,3},{x} {1,3},{y} {3},{x,y} Figure 3: A picture of the first two examples. Example Another way to construct a building, with a geometric realisation of dimension n, is to start with an n-dimensional vector space over a field k. Then we take the maximal nested sequences V 1 V 2... V n, with V i a subspace of dimension i, as chambers. The chambers V 1 V 2... V n and W 1 W 2... W n are i-adjacent if V j = W j for all j i. This means that, if J = i 1,..., i m, a residue of cotype J corresponds to a sequence V i1 V i2... V im and it s chambers are the maximal nested sequences W 1 W 2... W n, such that W i = V i for i I. If we take n = 2 and k = F 2, we find that the residues of cotype {1} correspond to the lines (1 : 0 : 0),(0 : 1 : 0),(0 : 0 : 1),(1 : 1 : 0),(1 : 0 : 1),(0 : 1 : 1) and (1 : 1 : 1). The residues of cotype {2} correspond to planes spanned by pairs of these lines. But in every three plane there are exactly three lines so there are ( 7 2) /3 = 7 of these planes. And as indicated pairs of a line and a plane such that the line is embedded in the plane corresponds to a residue of corank 2, of which there will be 21. Drawing the residues of corank 1 as points and the residues of corank 2 as lines between them we get Figure Coxeter complexes Definition Let I be the index set again and define a set of m ij by M = {m ij : m ii = 1, m ij = m ji 2 for all i, j I with i j}. 10

12 Figure 4: The third example Most of the time we will denote M by it s diagram. This diagram is drawn by starting with nodes labelled by the elements of I and by representing the elements of M with edges in the following way: if m ij = 2, if m ij = 3, if m ij = 4, m if m ij = m 5. The Coxeter group of type M is the group generated by r i given by W = r i : r 2 i = (r i r j ) m ij = 1 for all i, j I. To see this as a chamber system we take the elements of W as the chambers and i-adjacency between two chambers is defined by w i wr i. The geometric realisation of this chamber system is called the Coxeter complex of type M. If we look at the definition of i-adjacency and the fact that ri 2 = 1, we might think of r i as mirroring in the face between w and wr i. And we see immediately that rank 1 residues consists of 2 chambers, every chamber belongs to rank 2 residues consists of 2m ij chambers. Example The first example is one we already mentioned in the introduction of this section. The Coxeter group of type Ã1 has the diagram. The Coxeter complex of this group can be represented by R with vertices on Z. Example Another example, which we will become of particular interest to us, is the Coxeter group of type Ã2. It has the diagram. The residues of corank 1, hence of rank 2, all consist of 6 chambers. (This can be seen directly in the definition of W.) These become our vertices in the Coxeter complex. The residues of corank 2, the edges, all consist of 2 chambers. (This also can be seen directly in the definition of W.) The Coxeter complex becomes the tessellation of R 2 by equilateral triangles. 11

13 Figure 5: Part of the Ã2 Coxeter complex. Definition If f = i 1 i 2... i n is a word in the free monoid generated by I, then r f = r i1 r i2 r in is a gallery in W. We say that an f-gallery between x and y exist iff y = xr f. Two words in the free monoid generated by I are homotopic if one can be transformed into the other by alterations of the form f 1 p(i, j)f 2 to f 1 p(j, i)f 2, where p(i, j) =... ijij a word of length m ij. A word is called reduced if it is not homotopic to a word of the form f 1 iif 2. If f is a reduced word, then we call r f a minimal gallery. The terminology minimal gallery is very suggestive as is often the case in the terminology of buildings. And, as it is in at least most of these cases, this is not without reason. In the geometric realisation a minimal gallery becomes really what the term suggests. This will made precise with a metric in a few moments. 3.3 Buildings Definition We start with W, a Coxeter group of type M over an index set I. Then we say that is a building of type M if is a chamber system over I, such that every rank 1 residue contains at least two chambers, and having a distance function δ : W, such that if f is reduced, then δ (x, y) = r f W if and only if x and y can be joined in by a f-gallery. Things that can be deduced quickly from this definition are for example that is connected, δ maps onto, δ (y, x) = δ (x, y) 1 and an f-gallery is minimal (in ) if and only if f is reduced. Example A nice example to start with is the Coxeter complex W itself. From the definition of f-galleries and reduced words in W we find immediately that δ W (x, y) = x 1 y. This is a slightly dangerous example though. It might send people wondering why we do not define δ (x, y) = x 1 y all the time. The observation to be made is of course that x 12

14 and y are not elements of W. We are working with a word f in the free monoid of I, a word r f in the Coxeter group W and a minimal f-gallery in the building. But because of this example we can talk about isometric images between W and, because they both have a metric now. Definition An apartment of is defined to be an isometric image α(w ). Theorem 3.1 (Corollary 3.7 in [21]) Any two chambers lie in a common apartment. This theorem is very useful. We see the statement coming back in the next theorem, which is another powerful one. It provides us with a more convenient way to check wether a chamber complex is a building. Theorem 3.2 (Theorem 3.11 in [21]) Let C be a chamber system containing subsystems (called apartments) isomorphic to a given Coxeter complex (over the same index set I), and such that any two chambers lie in a common apartment. If we have furthermore that any two given apartments A and A containing a common chamber x and a chamber or panel y are isomorphic via an isomorphism fixing x and y, then C is a building. 3.4 The boundary Now choose your favorite apartment and fix a chamber c in this apartment and a vertex s of this chamber. This apartment is isomorphic to W. Definition In W a reflection r is defined as a conjugation of some r i by some w W, so r = wr i w 1. The wall corresponding to r is the subset of W left unchanged by the action of r acting from the left on W. We see that r 2 = 1 and r interchanges w and wr i. Because w i wr i the chambers w and wr i share a face and this face, which corresponds to an {i}-residue, is part of the wall. Theorem 3.3 (Lemma 2.1 in [21]) A minimal gallery cannot cross a wall twice. And given chambers x and y the number of times a gallery crosses a certain wall mod 2 is independent of the gallery. Because of this theorem a wall divides W in two complementing parts. Definition (Chapter 9.1 and 9.3 of [21]) These two parts are called roots. Every face of c having s as a vertex determines a root containing c and their intersection is called a sector in this apartment. This sector is said to have s as base point. A sector of a building is defined to mean a sector of an apartment of this building. A sector of a building is a sector of every apartment containing it, and given two sectors S and T there are two subsectors S S and T T such that S and T lie in a common apartment, Chapter VI.8 of [2]. We already encountered sectors. The sectors of a tree are namely the semi-geodesics we were talking about in the rank 1 case. 13

15 Definition We define the boundary of the building to be the set of equivalence classes of sectors. Two sectors are called equivalent if their intersection contains a sector. We need a topology on to define a full crossed product later on. To do this we fix a origin vertex O in. For any x there is a unique sector, which we will denote with [O, x), in the class of x with base point O (Theorem 9.6 in [21] or Lemma VI.9.2 in [2]). Theorem 3.4 (Section 7 of [16]) The boundary is a totally disconnected compact Hausdorff space, with a base for the topology given by sets of the form with y running through. y = {x : y [O, x)} In the case of the semi-geodesics we did not use a base point. Also in the present case the base point is only used for convenience. The topology is independent of the base point. 3.5 Buildings of type Ã2 So far we have been talking about general buildings. This is a vast subject so we attempt to solve a more modest case first. From now on we only look at locally finite thick affine buildings of type Ã2. The Coxeter complex of type Ã2 we ve already seen in an example above and it is the tessellation of R 2 by equilateral triangles. So all apartments are isomorphic to this triangulation of R 2, which implies that the building is affine. From this point of view the chambers are the triangles and adjacency is just defined by sharing an edge(, not only a vertex). Furthermore the building is thick, all edges have a valency of three or more, i.e. every edge faces at least three triangles. And the building is locally finite, all vertices and edges have finite valency. Notation Let B denote a building with above properties. Theorem 3.5 It is possible to label the vertices of the building B with only three types. Proof: In every apartment, a triangulation of R 2, we might label all vertices with three labels, say 0, 1 and 2, in such a way that the vertices of every triangle have three different types, as shown in Figure 6. By Theorem 3.2 we can continue this labeling to the whole building. Let s put this labeling to use and lat s make life easier by restricting Γ to be of a special kind. Definition A type rotating isometry is an isometry with respect to the metric on B which is well defined as a map between the classes of vertex types, and sends a vertex of type i to a vertex of type i + j mod 3 for some fixed j. An isometry obviously sends neighboring vertices to neighboring vertices, so this is also intuitively an isometry. 14

16 Figure 6: Labeling the vertices of the Ã2 Coxeter complex. Definition Let Γ denote a group of type rotating isometries from B to B which acts freely on the vertex set and with finitely many orbits. We have labelled all vertices in the building with three types in such a way that every triangle has three different labelled vertices. This allows us to label the edges of the triangles in such a matter as well. They get the label of the vertex that it is not incident with. An edge between two points of type 1 and 2 would get labelled 0 for example. Definition The edges are directed, in the sense that a edge of type i has an origin of type i + 1 and an terminal point of type i 1. Now the edges are labelled in such a way that every triangle has three different labelled edges. This has a nice consequence which is best stated in the language of links. Every vertex of an Ã2 building has a link. It is a directed graph which tells us about the structure of the building near that vertex, say v, in a comprehensible and transparant way. Definition To create the link of v, a 0-cell, we first draw a vertex for every edge in the building, a 1-cell, incident to v and then a edge between those vertices if the 1-cells in the building corresponding to these vertices belong to a common triangle, a 2-cell, directed according to the third 1-cell of this 2-cell. The link might intuitively be seen as the intersection of a very small sphere around v with the building B. Theorem 3.6 Every link of B is bipartite, i.e the set of vertices can be partitioned in two subsets such that no two vertices of one subset share an edge. Proof: For a fixed vertex v we can arrange vertices of the link by whether the 1-cells corresponding to them have v as origin or as terminal vertex. The only edges drawn in the link are between different types of vertices, because we have already seen that every 2-cell has three different kind of 1-cells. 15

17 3.6 Boundary operator algebra In the remark at the beginning of Section 2.3 we looked at the correspondence between X and {, Γ}. Every graph produces a covering tree and a fundamental group, and on the other hand every discrete subgroup Γ of the automorphism group of a locally finite tree which acts freely and cocompactly on, mods out to a graph X Γ\. This works in the one dimensional setting because Γ is a lattice in the automorphism group of. But in the higher dimensional case, which we are treating now, this correspondence is broken and in general Γ isn t a lattice in the automorphism group of B. For example, there are some Ã2 groups that cannot be embedded as co-compact lattices in PGL(3, k) with k a local field. This is shown in section 8 of [5], at the end under the header Application to the Buildings T. Examples of such groups are the ones that are labelled 2.1 an 4.1 in [5]. There is an induced action of Γ on B and because of the topology on the sectors we introduced in Section 3.4 we may again define the full crossed product algebra. Definition The boundary operator algebra is the full crossed product algebra C( B) Γ, the universal C -algebra generated by C( B) an π(γ) satisfying the covariant defining relation f(γ 1 ω) = π(γ)fπ(γ) 1 (ω) for f C( B), γ Γ and ω B and a fixed unitary representation π of Γ. As we concluded at the end of Section 2.4 we want to play with the two different viewpoints, the full crossed product algebra viewpoint and the more combinatorial viewpoint of Cuntz- Krieger algebras. We now have defined the first view, but before playing we have to define the second half as well. Remark More information on the topology on B for a building B of type Ã2 can be found in section 2 of [6]. There the topology is created with help of a base point, and in Lemma 2.5 of [6] it is then shown that it does not depend on this base point. 16

18 4 Higher rank Cuntz-Krieger algebras Let us start with a full generalization of the Cuntz-Krieger algebras we encountered in Section 2.1. These are called the higher rank Cuntz-Krieger algebras. Later on we will turn to the second order Cuntz-Krieger algebra, as one might have guessed from the section on buildings, because calculations in the general case seem nearly impossible. 4.1 The general case The higher rank Cuntz-Krieger algebras are defined analogously to the rank 1 Cuntz- Krieger algebra, but not by J. Cuntz and W. Krieger. They are first seen in [16] written by R. Robertson and T. Steger. For m, n Z with m n let [m, n] denote the set {m, m + 1,..., n}. For m, n Z r with m i n i for all 1 i r, which we will denote by m n, we define [m, n] = [m 1, n 1 ] [m r, n r ]. Let 0 denote the zero vector of Z r and e j the j th standard unit basis vector. Definition Fix a finite set, an alphabet, A, choose {0, 1}-matrices M 1, M 2,..., M r of size A A and denote their elements by M j (a, b) for a, b A. For m 0 elements of Z r define the set of words as W m = {w : [0, m] A; M j (w(l + e j ), w(l)) = 1 whenever l, l + e j [0, m]}. The matrices are acting on Z A the free Z-module spanned by A. For the generators of Z A we write {e a ; a A}. We can identify W 0 with A through the map that sends w to w(0). And if we view the matrix M j as the matrix which tells us wether two letters are neighbors in the j th direction, a word can be seen as [0, m]-array with a letter assigned to each vertex such that neighboring vertices have letters that are neighbors in the same direction. We have depicted that in Figure 7. For r = 1 these are indeed what we usually call words in mathematics, with some conditions on them. t(w)=w((4,2)) w((0,1)) o(w)=w((0,0)) Figure 7: If every dot is an element of A, this picture is a word of shape σ(w) = (4, 2). Definition Set W = m Z r + W m. We define the shape σ(w) of a word w W m as σ(w) = m and we identify W 0 with A via the natural map w w(0). We also define the origin and the terminal vertex of a word via the maps o : W m A and t : W m A given by o(w) = w(0) and t(w) = w(m). 17

19 We assume that the matrices to satisfy the following conditions. (H0): Each M i is a nonzero {0, 1}-matrix. (H1a): M i M j = M j M i. (H1b):For i < j, M i M j is a {0, 1}-matrix. (H1c): For i < j < k, M i M j M k is a {0, 1}-matrix. (H2): The directed graph with vertices a A and the directed edges (a, b) whenever M i (b, a) = 1 for some i, is irreducible. (H3): For any nonzero p Z r there exists a w W m which is not p-periodic, i.e. there exists a l such that w(l) and w(l + p) are both defined but not the same. The condition (H0) tells that there are non-trivial words. The next three conditions,(h1a)- (H1c), imply that we can define a product in the following way. Let u W m and v W n with t(u) = o(v). Then there is a unique w W m+n such that w [0,m] = u and w [m,m+n] = v, where w [0,m] is the restriction of the map w to [0, m], Lemma 1.4 of [16]. Notation We denote this product with w = uv. Then (H2) adds that we can always find a word with a given origin and terminal vertex. The condition (H3) is one with less immediate results. It is used to show that there exist words lacking certain periodicities. And that is used to show that the algebra we will construct now is simple, Theorem 5.9 of [16]. Definition The rank r Cuntz-Krieger algebra O M is the universal C -algebra generated by a family of partial isometries {s u,v ; u, v W and t(u) = t(v)} satisfying some relations analogously to (1). This time of course there are more, that is s u,v = s v,u (2) s u,v = s uw,vw for 1 j r (3) w W ; σ(w) = e j, o(w) = t(u) = t(v) s u,v s v,w = s u,w (4) s u,u s v,v = 0 for u, v W 0, u v. (5) Remark Note that in [16] a more general algebra is considered. There the definition of this algebra is extended with a set of countable decorations D together with a map δ : D O M and this extended algebra is denoted with A D. However it is was shown in Corollary 5.15 and Lemma 5.12 of [16] respectively that A D K A A N O M K and it is this stabilization that is used for the calculation of the K-theory, which we are interested in. The construction of O M K is exactly what this set of decorations is useful for. Apart from that we can just look at O M instead of A D. 18

20 Remark Of course the one-dimensional case can also be viewed in this more general way, by taking r = 1. To see how this works one can for example take a look in section 4 of [15]. Theorem 4.1 Morita equivalence and stable isomorphisms provide the same notion of equivalence for higher rank Cuntz-Krieger algebras as well. And O M is classified up to isomorphism by its K 0 -group, its K 1 -group and the class of the identity in K 0 (O M ). Proof: Remark 6.15 of [16] tells us that O M is a simple, unital, nuclear, purely infinite and separable algebra which satisfies the Universal Coefficient Theorem. So Morita equivalence and stable isomorphisms provide the same notion of equivalence. Furthermore we know, from a result in [11], that O M is classified up to isomorphism by its K-groups and the class of the identity in K 0 (O M ). So we have a new algebra to study and we know quite a lot about it already. Calculating the K-groups and keeping track of the identity turns out not to be easy however. As indicated at the beginning of this chapter we turn to the lowest case we didn t solve yet, the second order Cuntz-Krieger algebra. 4.2 The second order Cuntz-Krieger algebra We follow G. Robertson and T. Steger in [16] by defining some matrices to define a second order Cuntz-Krieger algebra. For that we will work on a building B and with an automorphism group of type rotating isometries Γ as defined in Section 3.5. We fix an apartment a, an origin O of type 0 in a and a sector of a with base point O. We put coordinates on the vertices of a by setting the two edges through O defining the walls of the sector as unit vectors. Definition We define p m to be a model parallellogram of shape m = (m 1, m 2 ) if the corners of p m are given by (0, 0), (m 1 + 1, 0), (m 1 + 1, m 2 + 1) and (0, m 2 + 1) and we denote the model parallellogram of shape (0, 0) with t. Let us denote the set of type rotating isometries p : p m B with P m, and set W m = Γ\P m. Furthermore we take A = W 0 = Γ\P 0. Notice that P 0 is the set of type rotating isometric injections p 0 : t B and keep in mind that we take A as alphabet with A letters for our Cuntz-Krieger algebra of rank 2. This structure already looks like the general set up of these C -algebras and we will continue this similarity by defining o(p) : t B and t(p) : t B for all p P m, very much alike the origin and the terminal vertex we encountered earlier on. Definition For every p P m we define o(p)(t) = p(t) and t(p)(t) = p(t + m). So o(p), t(p) P 0 and the tile o(p)(t) lies in the corner at p(0, 0) of the parallellogram p(p m ) in B and the tile t(p)(t) lies in the opposite corner. 19

21 p((m 1 +1,m 2 +1)) t(p)(t) p(p m) 0(p)(t) p((0,0)) Figure 8: A parallellogram with its original and terminal tile. Definition For two letters a, b A in our alphabet we define that M 1 (b, a) = 1 if and only if there exists an isometry p P (1,0) such that the classes of isometries a and b can be written as a = Γo(p) and b = Γt(p). And similarly we say for c, a A that M 2 (c, a) = 1 if and only if there exists a p P (0,1) such that a = Γo(p) and c = Γt(p). In all other cases M 1 (b, a) = 0 and M 2 (c, a) = 0. t(p)(t) o(p)(t) t(p )(t) o(p )(t) Figure 9: M 1 (b, a) = 1 resp. c = Γt(p ). M 2 (c, a) = 1, with a = Γo(p) = Γo(p ), b = Γt(p) and It is immediate that M 1 and M 2 are {0, 1}-matrices. And they are indeed the matrices that we want to use to define the rank 2 Cuntz-Krieger algebra O M. Remember that we 20

22 already know that the rank two Cuntz-Krieger algebra is classified up to isomorphism by K 0, K 1 and the identity in K 0, by Theorem 4.1. Theorem 4.2 (Theorem 7.7 of [16]) If (H2) is satisfied, then O M C( B) Γ. Proof: It is shown in [16] that (H0), (H1), i.e. (H1a)-(H1c), and (H3) are satisfied, given that Γ acts more generally on the vertex set freely and with finitely many orbits, as we assume it does right now. In Theorem 7.7 of [16] we find that, if (H2) is also satisfied, O M, which is now well defined, and C( B) Γ are isomorphic. Remark It is shown in Corollary 7.11 of [16] that (H2) is satisfied if Γ is a lattice in PGL 3 (K), with K a local field of characteristic zero. And in [19] the result is improved by showing that (H2) is satisfied if we demand only that Γ acts freely on the vertex set and with only one orbit. For the rest of this section we will assume that (H2) is indeed satisfied. Theorem 4.3 The current assumptions for B and Γ imply K O (O M ) K 1 (O M ) Z 2r T, where r is the rank of coker(i M 1, I M 2 ) and T is the torsion part of coker(i M 1, I M 2 ). Proof: If Γ acts on the vertex set freely and with finitely many orbits, Proposition 4.13 and Lemma 5.1 of [18], due to G. Robertson and T. Steger, already prove that K O (O M ) K 1 (O M ) Z 2r T, where r is the rank of coker(i M 1, I M 2 ) and T is the torsion part of coker(i M 1, I M 2 ). Remark It might seem awkward to have restrictions on Γ, but they are necessary. Remember that Γ is involved in creating our alphabet. And observe that we actually said that coker(i M 1, I M 2 ) = Z A / im(i M 1, I M 2 ) is isomorphic to Z r T. Also the problem calculating coker(i M 1, I M 2 ) is attacked in [18]. This is done by introducing slightly different matrices and showing they generate the same cokernel. Definition We define a model triangle ˆt which is the top of our model tile, i.e. ˆt has the vertices (1, 0), (1, 1) and (0, 1). Now we construct I as the set of type rotating isometric injections i : ˆt B and  = Γ\I. Then ˆM 1 and ˆM 2 are constructed out of  in the same way that M 1 and M 2 are constructed out of A, shown in Figure 10. Again  is going to serve as alphabet. Note that in this Figure â and ˆb are only indicating about which triangles we are talking. Strictly speaking the triangles they are written on are the images of ˆt under one of there representations. 21

23 ˆb â ĉ â Figure 10: How ˆM 1, ˆM2 are derived from M 1, M 2. The matrices M 1 and M 2 act on Z A and that ˆM 1 and ˆM 2 act on ZÂ. This is convenient notation now because we may write coker(i M 1, I M 2 ) = e a : e a = b A M 1 (b, a)e b, e a = b A M 2 (b, a)e b, which is isomorphic to coker(i ˆM 1, I ˆM 2 ) = eâ : eâ = ˆb  ˆM 1 (ˆb, â)eˆb, eâ = ˆb  ˆM 2 (ˆb, â)eˆb. Of course every type rotating isometry p 0 : t B restricts to a type rotating isometry i : ˆt B by i = p 0 ˆt. This allows us to produce a surjective map ˆπ : A  by ˆπ(Γp 0) = Γi. Theorem 4.4 (Lemma 6.1 of [18]) The surjection ˆπ induces a canonical map from Z A onto Z which send generators to generators. This map induces an isomorphism coker(i M 1, I M 2 ) coker(i ˆM 1, I ˆM 2 ). The new matrices ˆM 1, ˆM2 are not only smaller, they are nicer to work with. Be careful however, ˆM1 and ˆM 2 cannot be used to define a second order Cuntz-Krieger algebra. By replacing tiles with triangles we got more flexibility. So much that, for fixed â and ĉ, there may be more then one ˆb such that ˆM 1 (ˆb, â) = 1 and ˆM 2 (ĉ, ˆb) = 1. Hence ˆM 1 ˆM2 need not to be a {0, 1}-matrix. Theorem 4.5 If ˆM1 and ˆM 2 happen to satisfy (H0), (H1), (H2) and (H3), then O ˆM O M. 22

24 ĉ ˆb â Figure 11: ˆM1 ˆM2 does not need to have entries in {0, 1}. Proof: Because ˆM 1 and ˆM 2 now satisfy (H0), (H1), (H2) and (H3) the other algebra O ˆM is also well defined. Theorem 4.1, Theorem 4.3 and Theorem 4.4 already imply that O ˆM and O M are stably isomorphic. It is argued in Theorem 8.3 of [18] that the order of [1] K 0 (O M ) is equal to the order of a A e a coker(i M 1, I M 2 ). Analogously we have that the order of [1] K 0 (O ˆM) is equal to the order of â Â e â coker(i ˆM 1, I ˆM 2 ). But due to Theorem 4.4 the generators of Z A are mapped onto the generators of ZÂ, so these orders should be equal. 23

25 5 The transitive case I want to remind you that B is still a locally finite thick affine building of type Ã2. The vertices as well as the edges are labelled with three types in such a way that every chamber has three different labelled vertices and three different labelled edges. The group of type rotating isometries Γ acts freely on the vertex set and with finitely many orbits. Before going on we take a moment to consider the useful concept of projective planes. 5.1 Projective planes Definition A finite projective plane is a combinatorial structure of two finite sets called points and lines, with a relation called incidence. There are several equivalent sets of rules this structure has to obey. One of these sets is ˆ There is exactly one line incident with any two distinct points. ˆ There is exactly one point incident with any two distinct lines. ˆ There are four points such that there is no line incident with more than two of them. The first two rules suggest a duality between points and lines. The third law seems to break this duality, but in fact it doesn t. Theorem 5.1 These rules yield that for every finite projective plane we can find a q N, such that the plane has q 2 +q +1 points, the plane has q 2 +q +1 lines, every line is incident with q + 1 points and every point is incident with q + 1 lines. If q is power of a prime there does exist such a plane and for all known finite projective planes q is indeed a power of a prime. Definition Given a finite projective plane we will draw the incidence graph. In the incidence graph, the points as well as the lines are represented by vertices. Two vertices are joined by an edge if the point and the line corresponding to these vertices are incident, whence the name. It is immediate that this graph is bipartite graph. The two sets of vertices are the points resp. the lines and edges are only drawn between points and lines. Example The smallest and simplest finite projective plane is the Fano plane, with q = 2 and seen in Figure 12. The incidence graph of the Fano plane is the Heawood graph, which we find drawn in Figure 13. We have already seen this graph in an example of Section 3.1. Another useful concept for this story is the idea of generalized m-gons. Definition A finite connected graph is called a generalized m-gon if the maximum distance between two points and half the length of its smallest loop both equal m. 24

26 Figure 12: The Fano plane. Figure 13: Figure of the bipartite and circular representation of the Heawood graph. Such a graph is the union of cycles of 2m edges, the normal m-gons, and such a cycle is m exactly a Coxeter complex belonging to the diagram. It can be shown that any two edges belong to a common cycle and, if two cycles have a common edge, there is an isomorphism between the cycles fixing all vertices in their intersection. This means that we can view these generalized m-gons as buildings of type A m, which have Coxeter diagram m. Finite projective planes and generalized m-gons are very interesting on their own, but there is more. Theorem 5.2 The incidence graph of a finite projective plane is a generalized 3-gon. This allows us to use all kind of results from the theory of finite buildings as well as the nice properties of finite projective planes. 5.2 Restricting Γ Instead of finitely many orbits, from now on Γ may have only one orbit of vertices. One says that Γ acts transitively on the vertex set of B. And because Γ also acts freely it is said that Γ acts simply transitively on the vertex set of B. This means that for every a, b vertices of B there is a unique γ Γ such that γ(a) = b. Again this doesn t necessarily make Γ a lattice, for we are only talking about the vertices. Theorem 5.3 If Γ acts simply transitively on the vertex set, then O M C( B) Γ (cf. Theorem 4.2). 25

27 Below, I will now present two theorems which holds now Γ acts simply transitively on the vertex set B. Theorem 5.4 All links of B are isomorphic to the same incidence graph of a finite projective plane with some fixed order q. Proof: For every two vertices v 1, v 2 in the vertex set of B there is a γ Γ such that γ(v 1 ) = v 2, because Γ acts transitively. But γ is an isometry. This implies that the links of all vertices in the building are isomorphic. And Theorem 3.1 of [4] says the link is isomorphic to an incidence graph of a finite projective plane with some order q, so the theorem follows. Remark We just saw that the incidence graph of a finite projective plane is bipartite so Theorem 5.4 is coherent with the structure already in place due to Theorem 3.6. And we just saw that our restriction on Γ actually has consequences for the structure of B, and hence also for the second order Cuntz-Krieger algebra O M. Theorem 5.5 Consider the alphabets A and  as defined in Section 4.2. There is a bijection between A and the tiles having O as one of its vertices. And there is a bijection between  and the triangles having O as one of its vertices. Proof: Every element a A is defined as an equivalence class Γp 0, with p 0 : t B. Because Γ is acting simply transitively this class can be uniquely represented by an element p a sending to t to a tile having O as one of its vertices with p((0, 0)) = O. Then one can identify a with that tile. Furthermore, if we fix a tile having O as one of its vertices, the equivalence class of the injection sending t to the tile while sending (0, 0) to O will be identified with that tile. So the identification is surjective. And if Γa and Γb are identified with the same tile they contain a p a resp. a p b which send t to the same tile with p a ((0, 0)) = p b ((0, 0)) = O. Hence p a = p b and Γa = Γb. So the identification is injective. Thus there is a bijection between A and the tiles having O as one of its vertices. The proof for  is completely analogous. Every element â  is defined as an equivalence set Γi, with i : ˆt B. Because Γ is acting simply transitively this set can be represented with an element iâ sending to ˆt to a triangle with iâ((0, 0)) = O. Now one identifies â with that triangle. And surjectivity and injectivity may again be explicitly be checked. As a corollary every element of  is uniquely represented with a directed edge in the link of O. Figure 17 shows how we may read of the matrices ˆM 1 and ˆM 2 out of the incidence graph of the finite projective plane. If we let E denote the set of edges in the link and Z E the free Z-module spanned by this set, then ˆM 1 and ˆM 2 might be defined as acting on Z E as well. In this view ˆM 1 (ˆb, â) = 1 or ˆM 2 (ˆb, â) = 1 iff there is an ĉ E which has its vertices common with â E and ˆb E. Whether it is ˆM 1 or ˆM 2 depends on the orientation of ĉ in the link. 26

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