University of Twente. Faculty of Mathematical Sciences. Algebras related to posets of hyperplanes. University for Technical and Social Sciences

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1 Faculty of Mathematical Sciences University of Twente University for Technical and Social Sciences P.O. Box AE Enschede The Netherlands Phone: Fax: memo@math.utwente.nl Memorandum No. 55 Algebras related to posets of hyperplanes G.A.M. Jeurnink November 000 ISSN

2 Algebras related to posets of hyperplanes G.A.M. Jeurnink September 000 Abstract We compare two noncommutative algebras which are related to arrangements of hyperplanes. For three special arrangements the induced approximately finite dimensional C -algebra and the graded Orlik-Solomonalgebra are investigated. MSC 99 : 47L40, 6W50, 9K4 Keywords : AF-algebra, OS-algebra, hyperplanes Introduction Within Mathematical Physics there is a lot of activity in the field of non-commutative mathematics. This is a consequence of the fact that classical commutative methods are not applicable in modern quantummechanics. In approximating a continuum topological space, G. Landi has made a study of partially ordered sets (posets, for short). Such a poset is endowed with a non-hausdorff topology and it can be realized as the structure space of a noncommutative C -algebra (AF-algebra). In their book Arrangements of hyperplanes the authors P. Orlik and H. Terao connect to an arrangement also a non-commutative algebra (OS-algebra). Indeed, such a configuration of hyperplanes can be regarded as a poset. We want to compare the AF-algebra and the OS-algebra for some arrangements. As we will see, both algebras differ a lot in spite of the fact that they are related to the same partially ordered set. AF-algebras In this report we only deal with algebras A over the field C of complex numbers. In general the multiplication is not commutative, nevertheless we will assume A to have a unit. Definition. A is called a C -algebra if it admits an involution A A, a a and a norm A R,a a such that A becomes a Banach space with the properties a = a, (ab) = b a, (αa+ βb) = αa + βb, besides ab a b and a a = a for every a, b A and α, β C. Examples of C -algebras are C(M), the space of continuous functions on a compact Hausdorff space M, with complex conjugation (involution) and the supremum norm, and B(H), the space of bounded linear operators on a Hilbert space H with involution given by the adjoint together with the operator norm. Remark that C(M) is commutative whereas B(H) fails to be.

3 AF-ALGEBRAS Definition. A norm closed subalgebra B of a C -algebra is called a C - subalgebra if a B implies a B. Any C-linear map π : A A between two C -algebras A and A, for which in addition π(ab) = π(a)π(b) and π(a ) = π(a) hold for every a, b A is called a -morphism. A -representation of a C -algebra A is a pair (H, π) with H a Hilbert space and π : A B(H) a -morphism. The representation is called irreducible if the only closed subspaces of H, which are invariant under the action of π(a),are the trivial subspaces {0} and H. The kernel of an irreducible representation is called a primitive ideal. The famous Gelfand-Naimark theorem states that every commutative C -algebra A is isomorphic to C(Prim(A)), whereprim(a) is the space of kernels of irreducible -representations of A, endowed with the hull-kernel topology. A non-commutative C -algebra will be thought of (see []) as the algebra of continuous functions on some virtual non-commutative space. Indeed, any poset can be thought of as being Prim (A) for some C -subalgebra of operators on a separable Hilbert space. Definition.3 (see also [] and [3]). The C -algebra A is said to be approximately finite dimensional (AF) if there exists an increasing sequence A A A 3 A n A n+ of finite dimensional C -subalgebras A n of A such that A is the closure of A n ; the maps i n : A n A n+ are supposed to be injective -morphisms. We also say that A is the inductive limit of the sequence (A n,i n ). A useful way to represent an AF-algebra, or better the sequence (A n,i n ),isby means of a Bratteli-diagram (see []). Many properties of the AF-algebra can be picked from such a diagrammatic representation. Recall that every finite dimensional C -algebra A n is a direct sum of matrixalgebras, A n = k n (C), and every embedding A n A n+ can be split k= M d (n) k up in such a way that M (n) d j N kj 0, where N kj d (n) j j (C) is embedded in M (n+) d (C) with multiplicity k = d (n+) k (see [5]). In a Bratteli-diagram one draws horizontal rows of vertices, the n-th row is representing A n and consists of k n vertices, one for each block M(C) labeled with the corresponding dimensions d (n),...,d(n) k n.ifm (n) d (C) is embedded in j (C) with multiplicity N kj > 0, one draws a labeled line segment: M d (n+) k d j (n) N kj (n+) d k Whenever N kj =, we simply draw a line segment without the label N kj. For shortness we will write M n instead of M n (C). Example. Let H be the infinite dimensional separable Hilbert space l.the C -subalgebra A = K(H) + C of B(H), withk(h) the class of compact operators on H and the identity, is an AF-algebra. It can be viewed,up to isomorphisms, as the inductive limit of the sequence M M M M M 3 M...

4 AF-ALGEBRAS 3 with embeddings M n M ([Λ], λ) ( [ [Λ] λ ],λ) M n+ M (for details, see [3]). The algebra A is therefore associated with the following Bratteli-diagram 3 Figure : Bratteli-diagram K(l ) + C In general an AF-algebra may have a rather complicated ideal structure. Working with partially ordered sets, G. Landi has derived (see [3]) a one-to-one correspondence between non-commutative lattices P and a certain class of Bratteli-diagrams in such a way that the resulting AF-algebra A has a primitive ideal space Prim (A) which is homeomorphic to P. Herewesuppose Prim (A) to be endowed with the hull-kernel topology and partially ordered by set-inclusion. The relationship between the topology and the order on P is given by declaring the collection {y P y x} with x P, tobeasubbasis of open sets. Example. The algebra A = K(H) + C from the previous example has two irreducible representations, namely π : A B(H), k + λ k + λ and π : A C, k+ λ λ with kernels I = ker(π ) ={0} and I = ker(π ) = K(H) (see [3]). Therefore, Prim (A) is a two-points-poset {I,I } with I I. The hull-kernel topology is defined through its basis ({I }, {I,I }), so the open sets in Prim (A) are, {I } and Prim (A). Example.3 Take H = H H the direct sum of two copies of l. Consider the C -subalgebra A = CP + K(H) + CP of B(H), wherep i denotes projection on H i for i =,. This algebra has three irreducible representations, π : A B(H), a a, π : A C, λ P + k + λ P λ and π 3 : A C, λ P + k + λ P λ with kernels I ={0},I = K(H) + CP and I 3 = CP + K(H) respectively. Here Prim (A) is a three-points-poset {I,I,I 3 } with I I and I I 3. After identification, A is the inductive limit of M M M M M M M 4 M...

5 FROM POSET TO AF-ALGEBRA 4 with M M n M (λ,[λ], λ ) (λ, λ [Λ],λ ) M M n M λ In figure. the corresponding Bratteli-diagram is shown. 4 Figure : Bratteli-diagram CP + K(l l ) + CP In the next section we will go the other way around. Starting with a poset we examine the induced AF-algebra. From poset to AF-algebra In [3] the rather complicated construction of a Bratteli-diagram (AF-algebra), starting with a finite poset P, is given explicitely. We are especially interested in what will be the outcome of this procedure in case P is the poset associated with an arrangement of hyperplanes in R n, which are ordered by reverse setinclusion. Such a poset always has a unique minimal element (the whole space R n ) and in case the intersection of all hyperplanes in the arrangement is nonempty, it also has a unique maximal element (that intersection). Example. Consider the Hasse-diagram (a pictorial representation, see Figure 3) of the poset P ={x,x,x 3,x 4,x 5 } with partial order given by the relations: x <x,x <x 3,x <x 4,x <x 5,x <x 5,x 3 <x 5 and x 4 <x 5. As we will see in section 3, this poset corresponds to an arrangement of three incidental lines in R (represented by the points x,x 3 and x 4 ) Figure 3: Hasse-diagram of three incidental lines Following the Landi construction (the recursive calculations are omitted here), the AF-algebra AF(P), which corresponds to P, is the inductive limit of the sequence (A n,i n ) with for n 5 A n = M 3 n 5 n+4 M n M n M n 4 M

6 FROM POSET TO AF-ALGEBRA 5 and i n (B,C,D,E,λ)= ( B C D E λ [ C, λ ] [ D, λ ] [ E, λ ],λ) for all B M 3 n 5 n+4, C M n, D M n,e M n 4 and λ C. In Figure 4 we show the stable part of the Bratteli-diagram of AF(P). From level 5 on, this part repeats at every level in the diagram. 3/ n 5/ n + 4 n n n 4 level n 3/ n 9/ n + 8 n n n 3 level n+ Figure 4: Bratteli-diagram of three incidental lines Although AF(P) is determined by now, we can also realize this algebra as a C -subalgebra of B(H) for some separable Hilbert space H (see the remark at the end of this section). Take H = H 5 H H 53 H 3 H 54 H 4,wherein every H ij is a copy of the separable Hilbert space l. Then the algebra AF(P) can be given as a subspace of B(H), namely AF(P) K(H) + K(H 5 ) P H + K(H 53 ) P H3 + K(H 54 ) P H4 + C, where K denotes the class of compact operators and P stands for projection. (The indices ij reflect to the elements x i and x j of the poset P.) Example. With any arrangement of three lines in general position in the plane, the 7-points poset P from Figure 5 is associated. In a Hasse-diagram one arranges the points of the poset at different levels and connects them by using the rules. if x<y,thenx is at a lower level than y, and. if x<yand there is no z such that x<z<y,thenx is at the level immediately below y and these two points are linked. The Bratteli-diagram belonging to P stabilizes from the seventh level on. Therefore we can derive that AF(P) is the inductive limit of the sequence (A n,i n ) with A n = M 3n 34n+36 M n 0 M n M n 3 M M M

7 FROM POSET TO AF-ALGEBRA 6 and i n (B,C,D,E,λ,µ,ν)= ( B C D E λ µ ν, C λ µ, D λ ν, E µ,λ,µ,ν) ν for all B M 3n 34n+36,C M n 0,D M n,e M n 3 and λ, µ, ν C Figure 5: Hasse-diagram of three lines in general position 3n 34n+36 n 0 n n 3 level n 3n 8n+05 n 8 n 9 n level n+ Figure 6: Bratteli-diagram of three lines in general position Let H be the following direct sum of six copies of l l, indicated as H = H 5 H H 53 H 3 H 6 H H 64 H 4 H 73 H 3 H 74 H 4. Now AF(P) can be realized as the following algebra of bounded operators on H: AF(P) K(H) + K(H 5 H 6 ) P H + K(H 53 H 73 ) P H3 + K(H 64 H 74 ) P H4 + CP H5 H H 53 H 3 + CP H6 H H 64 H 4 + CP H73 H 3 H 74 H 4 Example.3 In Figure 7 we have drawn the Hasse-diagram of a central arrangement of nine hyperplanes in R 3. This so-called B3-arrangement has defining polynomial xyz(x + y)(x y)(x + z)(x z)(y + z)(y z), i.e. the nine planes are indicated by taking one of the factors in this polynomial equally zero. The poset P of non-empty intersections of hyperplanes, ordered by reverse inclusion, consists of 4 elements, including the maximal element 0 and the minimal element V = R 3. From level 4 on, the corresponding Bratteli-diagram stabilizes. Its stable part can be constructed following the next procedure. Denote P ={x := V; x := x = 0;... ; x 3 := y = 0,z = 0; x 4 := 0} and let

8 FROM POSET TO AF-ALGEBRA 7 0 x=y y=z x= y y= z x=z x=y x= y y= z y=z x= z x=y y= z x= y y=z x=y x= y x=z x= z y= z y=z V Figure 7: Hasse-diagram of the B3-arrangement S k = cl0{x k } be the smallest closed subset of P containing x k (remember that the collection {y P y x} with x P is a basis of open sets in P). For every n 4 the vertices (n, j) and (n+,k)are connected if and only if x j S k (see [3]). In drawing the Bratteli-diagram (see Figure 8) it is helpfull to summarize the S k for k =,...,4: S = {x,x,...,x 3,x 4 } S = {x,x,x 7,x 8,x 9,x 4 } S 3 = {x 3,x,x,x 5,x,x 4 } S 4 = {x 4,x,x 3,x 6,x,x 4 } S 5 = {x 5,x,x 3,x 4,x 7,x 4 } S 6 = {x 6,x,x 4,x 0,x 3,x 4 } S 7 = {x 7,x 7,x 0,x,x,x 4 } S 8 = {x 8,x 3,x 8,x,x 3,x 4 } S 9 = {x 9,x,x 9,x,x 3,x 4 } S 0 = {x 0,x 5,x 6,x 7,x 3,x 4 } S k = {x k,x 4 } for k =,,...,3 S 4 = {x 4 } Suppose H is the Hilbert space consisting of 36 sommands, each sommand is a copy of l l l and corresponds to a maximal chain in P. Then(see[3]) AF(P) is up to an isomorphism the class of operators on H which is generated by all B(x k ) for k =,...,4, whereb(x k ) acts by compact operators on the Hilbert subspace determined by the points which follow x in the Hasse-diagram and by multiples of the identity on the Hilbert subspace determined by the points which precede x (see Figure 7) Remark. In example. we already went on ahead to this construction of AF(P). In Figure 3 we see three maximal chains, namely x 5 x x,x 5 x 3 x and x 5 x 4 x. If we take H = H 5 H H 53 H 3 H 54 H 4 then AF(P) K(H) + K(H 5 ) P H + K(H 53 ) P H3 + K(H 54 ) P H4 + C.

9 3 ORLIK-SOLOMON ALGEBRAS S S S S S S S S S S S S S S S S S S S S S S S S Figure 8: Bratteli-diagram of the B3-arrangement 3 Orlik-Solomon algebras Definition 3. For l an l-arrangement of hyperplanes is a finite collection of codimension one subspaces in R l. The arrangement is called central whenever after choosing appropriate coordinates the intersection of all hyperplanes contains {0}, otherwise the arrangement is called affine. In [4] the authors associate with every l-arrangement A an anticommutative graded algebra OS(A), the so-called Orlik-Solomon algebra. This algebra, a quotient of an exterior algebra by a homogeneous ideal, can be constructed only using the intersection - poset of the arrangement (ordered by reverse setinclusion). Notice that R l is the intersection of the empty collection and is therefore the minimal element of this poset. In this section special attention is paid to the three arrangements from the examples.,. and.3. Let A be a central arrangement of n hyperplanes in R l and E = E(A) the exterior algebra of H A Ce H. So E is the graded algebra E = n p=0 E p with E 0 = C, E = H A Ce H and generally, for p =,...,n the space E p is spanned over C by all e H...e Hp with H k Afor k =,...,p. Remark that the following rule holds in E: e H e K = e K e H (anticommutivity) k= Define the linear map : E E by = 0, e H = and (e H...e Hp ) = p ( ) k e H...ê Hk...e Hp for all H,...,H p A. For a p-tuple of hyperplanes S = (H,...,H p ) we write e S = e H...e Hp further on. Definition 3. S = (H,...,H p ) is called dependant if the codimension of p k= H k is less than p. LetI be the homogeneous ideal of E generated by e S for all dependent S, the quotient E/I is called the Orlik-Solomon algebra of A, denoted by OS(A). The Orlik-Solomon algebra is, like E(A), a graded algebra, OS(A) = n p=0 OS p (A), wherein the elements e H + I and e S + I are denoted by a H and a S respectively. Proposition 3. If H S then e S = e H e S Proof From e H e S = 0 it follows that 0 = (e H e S ) = e S e H e S. Example 3. The central -arrangement with defining polynomial xy(x + y) consists of three incidental lines: H : x = 0 H : x + y = 0 and H 3 : y = 0.

10 3 ORLIK-SOLOMON ALGEBRAS 9 In order to compute the Orlik-Solomon algebra of this arrangement we want to recognize the dependent p-tuples of {H,H,H 3 }. Of course, every -tuple (H i ) is independent. The only dependent -tuples are S = (H i,h i ) for i =,, 3, but here we have e S = e Hi e Hi = 0. So, the elements a i := a Hi are linearly independent over C, they generate a basis of OS. Of course, for p> every p-tuple S is dependent and by proposition 3. we have a S = 0. Writing e i = e Hi we see that (e e e 3 ) = e e 3 e e 3 + e e belongs to I, therefore OS is already spanned by a a 3 and a a 3. These two elements are linearly independent: suppose α a a 3 + α a a 3 = 0, thenα e e 3 + α e e 3 I and since = 0 we have (α e e 3 + α e e 3 ) = α (e 3 e ) + α (e 3 e ) = 0 and therefore α = α = 0. Weconcludethat OS(xy(x + y)) = C Ca Ca Ca 3 Ca a 3 Ca a 3 is a 6-dimensional algebra with multiplication indentities: a i a j = a j a i (i, j =,, 3) and a a = a a 3 a a 3. By copying the above proof we have (see also [4]). Theorem 3. For every central -arrangement A = {H,...,H n } the corresponding Orlik-Solomon algebra has dimension n and is given by OS(A) = C n p= Ca p n k= Ca ka n with multiplication identities a i a j = a j a i and a i a j + a j a k + a k a i = 0 for all i, j, k =,...,n We also want to define an OS-algebra for an affine arrangement. definition is almost a copy of definition 3.. The next Definition 3.3 Suppose A is an affine l-arrangement of hyperplanes. We call a p-tuple S = (H,...,H p ) of hyperplanes from A dependent whenever p k= H k and has codimension less than p. As in 3., E denotes the exterior algebra of H A Ce H.LetIbe the ideal generated by {e S S = } { e S S is dependent }, then the quotientalgebra OS(A) = E/I is called the Orlik-Solomon algebra corresponding to A. Example 3. The affine -arrangement with defining polynomial xy(x+y ) consists of three planar lines in a general position: H : x = 0, H : x + y = 0 and H 3 : y = 0. Here the resulting Orlik-Solomon algebra is 7-dimensional OS(xy(x + y ) = C Ca Ca Ca 3 Ca a Ca a 3 Ca a 3 with multiplication rule a a a 3 = 0 (see [4]). Example 3.3 Now we will examine the central 3-arrangement with defining polynomial xyz(x y)(x + y)(x z)(x + z)(y z)(y + z), the so-called B3- arrangement. As indicated in definition 3., the Orlik-Solomon algebra is a direct sum OS(B3) = 3 p=0 OS p(b3) with OS 0 (B3) = C and OS (B3) = 9 i= Ca i. The nine sommands Ca i are reflecting to the nine hyperplanes H : x = 0, H : x = y, H 3 : x = y, H 4 : x = z, H 5 : y = 0 H 6 : x = z, H 7 : y = z, H 8 : y = z and H 9 : z = 0

11 3 ORLIK-SOLOMON ALGEBRAS 0 In order to determine OS (B3) and OS 3 (B3) we should realize there are sixteen dependent 3-tuples (H i,h j,h k ) with i<j<k, denoted by ijk, for short, namely 3, 5, 35, 46, 49, 69, 35, 48, 67, 347, 368, 469, 578, 579, 589 and 789. As a consequence we have sixteen relations in OS (B3); for example a 3 a 3 + a = 0 as a result of the fact that e 3 = e 3 e 3 + e belongs to the ideal I. Truely there are only thirteen independent relations in OS (B3), sowecan skip 3 elements in the collection of 36 generators {a ij i < j 9} of OS (B3). In our choice for a basis of OS (B3) we want to have every a i9 in order to get nice calculations for OS 3 (B3). basis OS (B3) : a,a 3,a 5,a 7,a 8,a 9,a 4,a 6,a 7, a 8,a 9,a 34,a 36,a 37,a 38,a 39,a 45,a 49,a 56, a 59,a 69,a 79,a 89. Of course, every 4-tuple S = (H i,h j,h k,h l ) is dependent. It follows that a jkl a ikl + a ijl a ijk = 0 in OS 3 (B3), so{a ij9 i<j 8} will be a spanning set for OS 3 (B3). Because of the thirteen relations in OS (B3) we derive that {a ij9 i<j 8} is not a basis of OS 3 (B3). For example, a 3 = a 3 a in OS (B3) and therefore a 39 = a 39 a 9 in OS 3 (B3); because of a 79 = a 59 a 57 in OS (B3) it even follows that a 579 = 0. Through deleting thirteen elements from {a ij9 i<j 8}, the remaining fifteen create a basis of OS 3 (B3); linear independency can be proved in the same way as in example 3., making use of the above mentioned basis of OS (B3). Therefore we have basis OS 3 (B3) : a 9,a 39,a 59,a 79,a 89,a 49, a 69,a 79,a 89,a 349,a 369,a 379,a 389, a 459,a 569. We conclude that the algebra OS(B3) = OS 0 (B3) OS (B3) OS (B3) OS 3 (B3) has dimension = 48. Remark 3. In [4] there is given a construction for a basis of OS(A) as a free C-module, the so-called broken circuit basis. Of course, such a basis tells us what the vector space dimension of OS(A) will be, but it ignores the multiplication. Remark 3. There is no possibility to reconstruct the arrangement poset P from the Orlik-Solomon algebra OS(P). The two 3-arrangements A (defining polynomial xyz(x z)(y z)(y x + z)) and A (defining polynomial xyz(x z)(y x +z)(y +x z)) generate inequivalent posets (there is no order preserving bijection, see Figure 9) while the associated algebras OS(A ) and OS(A ) are isomorphic as graded C-algebras (see [4]).

12 3 ORLIK-SOLOMON ALGEBRAS 0 x=y y=z y= z x=z y=z x=z x=z y=z y=z x=3z x=z H H x=z H 3 y=z H 4 y=o H 5 y= x z H 6 IR 3 0 x=y x= y y= z y=z x=z y=z x=z x=z y= z x=z H x=z y= y= H H 3 H H H x z x+z IR 3 Figure 9: Two inequivalent posets with isomorphic OS-algebras

13 4 ALGEBRAS FOR -ARRANGEMENTS 4 Algebras for -arrangements In this section we summarize all the inequivalent configurations of n lines in R in case n = and 3. We mention the associated AF- andos-algebra induced by such an arrangement, but we will see that both algebras cannot be compared in some algebraic sense. As before, every H is a copy of the separable Hilbert space l, K denotes the compact operators and P denotes projection on H. Case n = two intersecting lines A={H,H } with H : x = 0andH : y = 0 OS(A) = C Ca Ca Ca a AF(A) = K(H 4 H H 43 H 3 ) + K(H 4 ) P + K(H 43 ) P 3 + C. two parallel lines A={H,H } with H : x = 0andH : x = OS(A) = C Ca Ca with a a = 0, so dim OS(A) = 3 AF(A) = K(H 3 H 3 ) + CP 3 + CP 3. Case n = 3 Three concurrent lines A={H,H,H 3 } with H : x = 0,H : x + y = 0andH 3 : y = 0 OS(A) = C Ca Ca Ca 3 Ca a 3 Ca a 3 with a a + a a 3 + a 3 a = 0, so dim OS(A) = 6 AF(A) = K(H 5 H H 53 H 3 H 54 H 4 ) + K(H 5 ) P + K(H 53 ) P 3 + K(H 54 ) P 4 + C. three lines in general position A={H,H,H 3 } with H : x = 0,H : x + y = andh 3 : y = 0 OS(A) = C Ca Ca Ca 3 Ca a Ca a 3 Ca a 3 with a a a 3 = 0, so dim OS(A) = 7 AF(A) = K(H 5 H H 53 H 3 H 6 H H 64 H 4 H 73 H 3 H 74 H 4 ) + K(H 5 H 6 ) P + K(H 53 H 73 ) P 3 + K(H 64 H 74 ) P 4 + CP CP CP three lines, two of them are parallel A={H,H,H 3 } with H : x = 0,H : x = andh 3 : y = 0 OS(A) = C Ca Ca Ca 3 Ca a 3 Ca a 3 with a a = 0, so dim OS(A) = 6. AF(A) = K(H 5 H H 53 H 3 H 63 H 3 H 64 H 4 )+ K(H 5 ) P + K(H 53 ) P 3 + K(H 63 ) P 3 + K(H 64 ) P 4 + CP CP three parallel lines A={H,H,H 3 } with H : x = 0,H : x = andh 3 : x = OS(A) = C Ca Ca Ca 3 with a i a j = 0 for all i, j =,, 3, so dim OS(A) = 4 AF(A) = K(H H 3 H 4 ) + CP + CP 3 + CP 4.

14 5 K-THEORY 3 As we have already seen in the examples of section and 3, the two algebras associated with an arrangement of hyperplanes are hard to compare with each other. Also in the case of four lines in R there is no algebraic resemblance between the OS-algebras and the AF-algebras of the eight inequivalent arrangements (the dimensions of the associated OS-algebras varies from 5, if we deal with four parallel lines, to in case we have four lines in general position). 5 K-theory Algebraically, the analogue of vector bundles over a noncommutative lattice P are projective modules of finite type over the corresponding algebra AF(P). The K-theory groups of AF(P) classifies equivalent classes of such projective modules (see [3]). In this section we compute the ordered K-group of AF(A) for three special arrangements A from the examples.,. and.3. From [6] it is known that K 0 behaves continuously with respect to inductive limits and moreover, K 0 (C) K 0 (M n (C)) Z whereas K 0+ (C) K 0+ (M n (C)) N. So we are able to compute the ordered group (K 0,K 0+ ) in case we are dealing with AF-algebras. Indeed, for AF-algebras K vanishes identically (see [6]). Example 5. As in. we take xy(x + y) as the defining polynomial for the central arrangement A of three lines in R. We have derived AF(A) to be the limit of the sequence of C -algebras A n = M 3 n 5 n+4 M n M n M with an embedding A n A n+ which can be described by the matrix T = wherein T ji denotes the multiplicity of the i-th sommand M pi sommand M qj of A n+. of A n in the j-th By direct computation we see that T is invertible over the integers. Following the definition of inductive limit we have K 0 (AF(A)) ={(k n ) k n K 0 (A n ) and N0 such that k n+ = T(k n ) for n>n 0 } is isomorphic as a group to Z 5 = Z Z Z Z Z. The positive cone of K 0 (AF(A)), being the limit of T m (N N N N N) for m, can be deduced from the matrix It follows that T m = m m m 3 m + m m m m K 0+ (AF(A)) = Z Z Z Z N + Z N N N + 0 Z N N + N 0 Z N + N N 0 N

15 5 K-THEORY 4 or in set theoretical notation K 0+ (AF(A)) = {(a,a,a 3,a 4,a 5 ) Z 5 (a 5 < 0) or (a 5 = 0 and a,a 3,a 4 N with a + a 3 + a 4 > 0) or (a = a 3 = a 4 = a 5 = 0 and a N)} Example 5. A defining polynomial for the affine arrangement A of three lines in R is xy(x + y ). InthiscaseAF(A) is the limit of the sequence (A n ) with A n = M 3n 34n+36 M n 0 M n M n 3 M M M ; the embedding A n A n+ can be described by (see.) T = Also this matrix is invertible over the integers and for m>0 we have T m = m m m m m m m m m 0 m m m Therefore we conclude K 0 (AF(A) Z 7 and K 0+ (AF(A)) = = {(a,a,a 3,a 4,a 5,a 6,a 7 ) Z 7 (a 5,a 6,a 7 N with a 5 a 6 + a 6 a 7 + a 5 a 7 > 0) or (a 0,a 7 > 0 and a 5 = a 6 = 0) or (a 3 0,a 6 > 0 and a 5 = a 7 = 0) or (a 4 0,a 5 > 0 and a 6 = a 7 = 0) or (a,a 3,a 4 N with a + a 3 + a 4 > 0 and a 5 = a 6 = a 7 = 0) or (a N and a = a 3 = a 4 = a 5 = a 6 = a 7 = 0)} Example 5.3 We summarize our K-theory calculations for the B3-arrangement with defining polynomial xyz(x + y)(x y)(x + z)(x z)(y + z)(y z) by describing the ordered K 0 -group. Again, K 0 (AF(B3)) is a direct sum of copies of Z; inthiscasewehavek 0 (AF(B3)) Z 4. Its positive cone consists of all (a,a,...,a 0,a,...,a 3,a 4 ) Z 4 with a 4 N + or (a 4 = a 3 =...= a 3 = a = 0 and a N) or (a 4 = a 3 =...= a = 0 and a 0,...,a N with a a > 0) or (a 4 = 0; a 3,...,a N and if x a k = 0 then a j N k Sj k j for j =, 3,...,0) Here the S j are the subsets of P ={x,...,x 4 } as described in example.3.

16 REFERENCES 5 References [] O. Bratteli, Inductive limits of finite dimensional C -algebras, Trans. Amer. Math. Soc. 7 (97), pp [] A. Connes, Noncommutative Geometry, Academic Press, 994. [3] G. Landi, An introduction to noncommutative spaces and their geometry, Springer Verlag, 997. [4] P. Orlik, H. Terao, Arrangements of hyperplanes, Springer Verlag, 99. [5] M. Takesaki, Theory of operator algebras I, Springer Verlag, 979. [6] N. Wegge-Olsen, K-theory and C -algebras, Oxford University Press, 993.