On n-homogeneous c*-algebras over two-dimensional oriented compact manifolds Текст научной статьи по специальности «Математика»

YAK: 517.9

MSC2010: Primary 46L05, Secondary 19K99

ON N-HOMOGENEOUS C*-ALGEBRAS OVER TWO-DIMENSIONAL ORIENTED COMPACT MANIFOLDS

© M. V. Shchukin

Belarusian national technical university ul. Hmelnizkogo 9, Minsk, 220013, Belarus e-mail: mvshchukin@bntu.by

On ^-homogeneous C*-algebras over a two-dimensional compact oriented connected manifold.

Shchukin M. V.

Abstract. We consider the n-homogeneous C*-algebras over a two-dimensional compact oriented connected manifold. Suppose A be the n-homogeneous C*-algebra with space of primitive ideals homeomorphic to a two-dimensional connected oriented compact manifold P(A). It is well known that the manifold P(A) is homeomorphic to the sphere Pk glued together with k handles in the hull-kernel topology. On the other hand, the algebra A is isomorphic to the algebra r(E) of continuous sections for the appropriate algebraic bundle E. The base space for the algebraic bundle is homeomorphic to the set Pk. By using this geometric realization, we described the class of non-isomorphic n-homogeneous (n > 2) C*-algebras over the set Pk. Also, we calculated the number of non-isomorphic n-homogeneous C*-algebras over the set Pk.

Keywords: C*-algebra, primitive ideals, base space, algebraic bundle, operator algebra, irreducible representation

Introduction

In [1] I. Gelfand and M. Naimark proved that for any C*-algebra A there exists a Hilbert space H such that A is isomorphic to the algebra B(H) of bounded operators on H. Furthermore, let A be a commutative C*-algebra. Thus there exists a Hausdorff space M such that A is isomorphic to the algebra C(M) of all continuous functions on M. Let n be an irreducible representation of the commutative C*-algebra A. Hence the dimension of n(A) equals 1. Moreover, let A be a non-commutative C*-algebra. Consider an irreducible representation n of the algebra. If there exists an integer n such that for any n the dimension of n(A) equals n then the algebra A is said to be n-homogeneous. In [2, 3] J. Fell, I. Tomiyama, M. Takesaki proved that a n-homogeneous C*-algebra A is isomorphic to the algebra of all continuous sections for the appropriate algebraic bundle £.

In [4] F. Krauss and T. Lawson described the set of algebraic bundles over the spheres Sk.

In [5] A. Antonevich and N. Krupnik described the difference between bundles and algebraic bundles over the sphere Sk. On the other hand, in that paper they introduced some operations on the classes of algebraic bundles over Sk. In [6] S. Disney and I. Raeburn described the set of algebraic bundles over the torus T2 and T3. In the present paper we describe the set of algebraic bundles over two-dimensional compact oriented manifolds. Let us remind that a triple (E,B,p) is called bundle if the following conditions hold:

(I) E and B are topological spaces.

(II) p : E ^ B is a continuous surjection.

The space E is called bundle space, the space B is said to be base space. The surjection p is called projection. The set F = p-1(x) is the fiber over a point x E B. For example, consider the product-bundle E = B x F, where B and F are topological spaces. By p denote the projection B x F ^ B to the first multiplier. The bundle £ is said to be the trivial bundle if it is isomorphic to a product-bundle. On the other hand, consider the Mobius tape M. Note that the Mobius tape M is a non-trivial bundle. The circle S1 is the bundle space. The interval I is the fiber. However M is not isomorphic to the product-bundle S1 x I. At the same time M is locally trivial.

A G-bundle £ = (E,B,p) is called algebraic bundle if the following conditions hold:

(I) The fiber Fx is the algebra Mat(n) = Cnxn of square matrices of order n.

(II) The group G is the group Aut(n) of all automorphisms for the algebra Mat(n). Two bundles £1 = (E1, B1,p1) and £2 = (E2, B21,p2) are said to be isomorphic if there

exists a homeomorphism 7 : E1 ^ E2 such that 7 (Fx) = Fa(x). Here a : B1 ^ B2 is a homeomorphism of the bases, the set Fa(x) = p-1 (a(x)) is the fiber over the point a(x) E B2.

1. Algebraic bundles over the compact connected two-dimensional

oriented manifolds

Proposition 1 ([7]). If M is a compact connected two-dimensional oriented manifold, then M is homeomorphic to the sphere S2 with k handles.

We denote it by Pk (k can be equal 0). Let £ be an algebraic bundle (E,Pk,p) over Pk. Let n be the order of fiber F = Mat(n) for £. Cut out the part D from Pk, where D is homeomorphic to the disk D2. Further, we consider the set Pk as the union (Pk\D) U D, where (Pk\D) П D = S1. The set D is contractible. Therefore, the restriction £D of £ to D is trivial. Thus, the restriction £D is isomorphic to Mat(n) x D.

Lemma 1. The restriction £pk\D of the bundle £ to the set Pk\D is trivial.

Доказательство. The proof is by induction on number k of handles. Note that the case B = P0 was conside red in [5]. First consider the case B = Pi, where P1 is homeomorphic to the torus. Now we realize P1 as torus. Cut out the set D from the set Pi. Now let us prove that the restriction £Pl\D of the bundle £ to the set P1\D is trivial.

We realize the torus P1 as the square /2 with conditions of gluing u on its border:

u(1,y) = u(0,y); u(x, 0) = u(x, 1)(0 < x < 1;0 < y < 1).

Let /02 5 denotes the square with the side equals 0.5. Suppose that the set /0.5 has the same center as /2. Cut out the set /^ from the square /2. The set /2\/0;5 is homeomorphic to the set P1\D. The homotopic class of P1\D is the same as the homotopic class of the border 8(/2) with the functions of gluing u. It is homeomorphic to two circles; these two circles contain a common point. Every algebraic bundle over two circles is trivial [5]. Hence the restriction of £ to P1 \D is trivial. The base of the induction is proved.

The induction hypothesis. Let us suppose that for any integer m < k any algebraic bundle £ over Pm\D is trivial.

The step of the induction.

Let us show that the restriction of the algebraic bundle £ to Pk+1 \D is trivial. Indeed, cut out a handle P1\D1 from the set Pk+1\D. Here we realize the handle as the set P1 without the set D1. Let L be the intersection of Pk+1\D and P1\D1. Thus the set L is homeomorphic to the unit interval /.

Now we have two sets: Pk\D2 and P1\D1 with the gluing function v : L ^ Aut(n) of the bundle £. The restrictions of the bundle £ to the sets Pk\D2 and P1\D1 are trivial by the induction hypothesis. The class of the bundle £ is determined by the homotopic class of the mapping 7 : L ^ Aut(n)[5]. Since the set L is contractible, it follows that the mapping y is homotopic to the constant mapping. □

Lemma 2. Let f be a continuous mapping from S1 to Aut(n), where S1 = 8 (Pk\D). The identity [f] = 0 is a necessary and sufficient condition for the mapping f to have a continuous extension to f * : Pk\D ^ Aut(n). Here [f ] denotes the class of f from the group n1 (Aut(n)).

Доказательство. The proof is by induction on the number of handles k.

1. The base of induction. The set P0 is homeomorphic to the sphere S2. In this case we can extend the mapping f : S1 ^ Aut(n) to the disk D if and only if [f] = 0([5]). Moreover, we shall to prove the statement for the set P1 = T2. The set P1 is homeomorphic to the torus T2. On the other hand, the set P1 is homeomorphic to the unit square /2 with the rule u of gluing on the border: u(1,y) = u(0,y),u(x, 0) = u(x, 1)(0 < x < 1,0 < y < 1). Let us cut out the square

10.5 from the set 12. The square I^g has the same center as 12. The side of I^g is equal 0.5. The set 1 Vo2 5 is homeomorphic to the set Pl\D. Now we can consider the function f as the function on the border of Z^. Let U(x,t) = f (x(1 + t), (y(1 + t))) be a homotopy such that u(0) = f(x,y),u(1) = f*(x,y), where (x,y) G i(12). Here 2) denotes the border of 12. Consider the side a = (x, 0), (0 < x < 1) of the square 12. The opposite side c = (x, 1), (0 < x < 1) is glued with a. When we move on the border #/2 we move on the side c in opposite direction. In addition, the same statement is true for two other sides. Therefore, [f*] = [f] = 0.

Otherwise let f be a mapping f G C^(Z^), Aut(n)) and [f ] = 0. This yields that we can extend the mapping f to all of [5]. Let f * be the extension of f to the square /025: f * G C(12.5, Aut(n)). Denote by f2 G C(PfcAut(n)), fi G C(Pi\Di, Aut(n)) the restrictions of f * to the sets Pk\D2 and P1\D1. Consider any point y G 12. Since y G 12, we have y = r ■ x, where x G i (Z^) , r G [0; 2]. For all r such that r G [0; 1] we have r ■ x G By definition, put f *(r ■ x) = f *((2 — r) ■ x) for r G [1; 2]. Therefore, the mapping f * is well defined with respect to the function u of gluing for the square 12.

The assumption of the induction. Suppose the lemma is true for all m < k.

The step of the induction. Consider the set Pk+1\D. Cut out one handle P1\D1 from the set Pk+1\D. Let the set L1 be the intersection of Pk+1\D and P1\D1. The set L1 is homeomorphic to the unit interval 1. Now consider the set Pk+1\D as a union Pk\D2 U P1\D1 ((Pk\D2) n (PAD) = L1). Denote by f * G C(Pk+1 \D, Aut(n)) an extension of f to Pk+1\D, where f G C (i(Pk+1\D), Aut(n)). Let a1 : 1 ^ S1 U L1 be a parametrization of i(P\\D1) such that a1 ([0; 2]) = S1 ,a1 ([ 1; 1]) = L1, «2 ([0; 1 ]) = L2,«2 ([ 1 ;1]) = S2.

Denote by g(x) the element from the class [f1] + [f2] such that

f1(«1(2x)) ,x G [0;1 ] g( ) ^ f2 M2x — 1),x G [ 1; 1]) '

Now we define the homotopy by the next rule:

F(t x) = i g(t+1 ),xg[0; 1]

F (t,x) I g (f+) ,x G [ 1; 1] •

Thus F(0,x) = g(x), F(1; x) = < g (2+-,\x G ^0;r 1 - = f (x), because

I g m,x G [2a]

g (2) = f (x) for all x G [0; 2] and g (xx+1) = f (x) for all x G [1 ; 1]. Note that «1 fa; 1 ]) = S1,«2 (r 1; 1]) = S2. This implies that [f] = f + [f2].

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Otherwise consider a mapping f e C Aut(n)) such that [f] = 0. In this case we extend it to the set Li. Let Si(t) (t e [0; 1]) be a parametrization of the set Si = D\D2. Suppose Li(t) be a parametrization of Li = Di n D2. Now we define the mapping f* by the rule: f * (Li(t)) := f (Si(t)). It follows in the standard way that [f *] = 0 on the sets Di = Si U Li and D2 = S2 U Li.

By the inductive hypothesis extend the mapping f* to the set Pk\D2, because [f*/£D2] = 0. Finally, extend the mapping f * to the set Pi\Di. □

Suppose £1 and £2 be two algebraic bundles with fiber Mat(n) over the set (Pk\D) UD. Let n12 = Y-Ч be the gluing function for the bundle £1 and ^12 = u-1u1 be the gluing function for the bundle £2. Denote by v1 the map from £1/(Pk\D) ^ (Pk\D) x Mat(n) and by v2 the map from £1/D to D x Mat(n). These maps are well defined by lemma 1. For any point x G 8(D) the image of the fiber Fx from the bundle £1/(Pk\D) is the fiber (F1)x from the bundle £1/D. Thus the map n12 generates an automorphism (y1)x of the algebra Mat(n) over every point x G 8(D). The mapping y3(x) = (Y1)x : S1 ^ Aut(n) is continuous, because the restriction of the bundle £1 to 8(D) is trivial. By the same argument, the gluing function ^12 generates the mapping y4 G C (S 1,Aut(n)). Denote by [y3] the class of mapping y3 in the group n1(Aut(n)) = Z/nZ. In this notation, let в : n1(Aut(n)) ^ Z/nZ be the corresponded isomorphism. Suppose — [y4] be the element в- ( — в ([Y4])).

Theorem 1. A necessary and sufficient condition for the bundles £1 and £2 to be isomorphic is [y3] = ± [y4] .

Доказательство. Denote by y : £1 ^ £2 the isomorphism of the bundles. Let a : Pk ^ Pk be the corresponded homeomorphism of the bases for the bundles. Suppose П12 = Y-Ч be the gluing function for the bundle £1 over (Pk\D) U D. Denote by ^12 = u-1u1 the gluing function for the bundle £2 over (Pk\a(D)) Ua(D). In this notation, u1 : £1/(Pfc\a(D)) ^ (Pfc\a(D)) x Mat(n), u2 : £2/(a(D)) ^ (a(D)) x Mat(n).

Let в be a homeomorphism Pk ^ Pk such that a(D) = D and a(8(D)) has the same orientation as 8(D).

Denote by в1 the extension of в to the isomorphism of the bundles: в1 : £2/ (Pfc\a(D)) ^ £2/ (Pk\D). Let us remark that it is possible, because в(D) = D and в (Pk\D) = Pk\D. Denote by ^12 : £2/8(Pk\D) ^ £2/8(D) a mapping such that the next diagram is commutative:

£2/(Pk\a(D)) £2/ (a(D))

I в1 I в2

£2/(Pk\D) -4 £2/D

In this case, the bundle в1 (£2/(Pk\a(D))) U в2 (£2/a(D)) is isomorphic to the bundle £2.

Let в3 be the isomorphism of the bundles.

Suppose a be a homeomorphism a : Pk ^ Pk for the bases such that в ◦ a (8(D)) = 8(D). The restriction of the bundle £1 to the set Pk\D is trivial. Further, let вб E C (Pk\D, Aut(n)) be a mapping defined by the isomorphism в3 0 7. The image of a fiber Fx under the mapping вб is a fiber в3 ◦ 7(Fx) for any point x E Pk\D. In addition, let вб E C(D, Aut(n)) be a mapping defined by the isomorphism в3 ◦ 7. The image of a fiber Fx under the mapping вб is a fiber в3 ◦ 7 (Fx) for any point x E D. We have the next commutative diagram:

£1/(Pk\D) £1/D

I в3 о 7 I в3 о 7

£2/(Pk\D) -4 £2/D The image of a fiber Fx under the mapping в3 ◦ 7 is a fiber Fs0a(x). Therefore we have 74 (в о a(x)) вб(x) = вб о 73(x). Further,

[74(в о a(x))] + [вб(x)] = [вб(x)] + [73(x)] (1)

The mappings вб^) and вб(x) are defined on the sets Pk\D and D correspondingly. Therefore we have ^5(x)] = [вб(x)] = 0 by the lemma 2. Using (1), we get

[74 (в о a(x))] = [73(x)] (2)

Let the mapping в о a changes the orientation of the circle 8 (D). Therefore

[74(в о a(x))] = - [74(x)]. _

Otherwise, let the mapping в о a do not changes the orientation of the circle 8(D). In this case, [74(в о a(x))] = [74(x)]. Actually we obtain [74] = ± [73].

On the other hand, let [74] = ± [73]. Suppose we have [74] = — [73]. Denote by a a homeomorphism Pk ^ Pk such that a (D) = D. Let the homeomorphism a changes the orientation of the circle S1 = 8 (D). In this case, denote by 71 the isomorphism v-1 о u1 from £1/(Pk\D) ^ £2/(Pk\D). Here, v1 is the isomorphism of the bundles £2/(Pk\D) ^ (Pk\D) x Mat(n) such that restriction of v1 to Pk\D equals a. We

see that the isomorphism u1 : £1/(Pk\D) ^ (Pk\D) x Mat(n) produces the identity homeomorphism I for the bundle bases. Note that the isomorphism y1 produces a mapping Y5 G C(Pk\D,Aut(n)). Thus the mapping (Y4(ax)) Ys(x) (y3(x))-1 G C (S 1,Aut(n)) produces the isomorphism n12 ◦ Y1 ◦ : &/8D ^ £2/8D for the trivial bundles. In addition, the homeomorphism a changes the orientation for the circle S1 = 8D. Now we get [Y4(ax)] = — [y4(x)]. We obtain [y5(x)] = 0 by the lemma 2. In addition, the next equality has a place: [(y3(x))-1] = — [y3(x)]. Denote by y7 the extension of Y4(ax) о y5(x) о (y3(x))-1 to D by lemma 2. This means that y7 G C (D, Aut(n)). Define the isomorphism y2 : £1/D ^ £2/D by the rule (x,y) ^ (a(x), y7(x) ■ y) , (x G D,y G Fx). The isomorphism is well-defined with respect to the functions of gluing for the bundles

and £2. Further, define a map y : ^ £2 by the next rule: < Y1 on^1/(_fc\ ) .

I Y2 on £1/D

This implies that the map y is an isomorphism of bundles. On the other hand, let [y4] = [y3]. Define the homeomorphism a as identity I : Pk ^ Pk. Arguing as before, we get the map Y4(ax) о y5(x) о (y3(x))-1 G C (S 1,Aut(n)) such that |j4(ax) ■ Y5(x) ■ (Y3(x))-^ = [Y4(ax)] + [Y5(x)] + [^(x))"1] = Ы — [Y3] = 0. Therefore, we can extend the map Y4(ax) ■ y5(x) ■ (Y3(x))-1 to a map y7 G C (D, Aut(n)). The map y7 produces an isomorphism y2 : £1/D ^ £2/D that is coordinated with gluing functions for the bundles and £2. At the same time the map y7 is coordinated with the isomorphism Y1. These isomorphisms y1 and y2 produce an isomorphism y : ^ £2. П

Theorem 2. Suppose n = 2/ or n = 2/ + 1(Z G N). Then there are l + 1 non-isomorphic algebraic bundles with fiber Mat(n) over the set Pk.

Доказательство. Let n = 2l. In other notation, we need to find number of classes in Z/nZ with respect to the equality l = —l. We have the next classes {0} , {1, 2l — 1} , {2, 2l — 2} , {3, 2l — 3} ,..., {l — 1, l + 1} , {l}. There are exactly l +1 such classes.

Further, let n = 2l + 1. In this case we have the next classes: {0} , {1, 2l} , {2, 2l — 1} , {3, 2l — 2} , {4, 2l — 3} ,..., {l — 1, l + 2} , {l, l + 1}. There are l + 1 such classes. □

Conclusion

In the work we described the class of non-equivalent algebraic bundles with base space homeomorphic to the two-dimensional compact oriented connected manifold. We calculated the number of non-isomorphic n-homogeneous C*-algebras with space of primitive ideals homeomorphic to the two-dimensional compact connected oriented manifold. Further, it is interesting to know the structure of n-homogeneous C*-algebras

with its space of primitive ideals homeomorphic to more complicated manifolds, for example, 3-dimensional manifolds and other.

Acknowledgements The author would like to thank professor Anatolii Antonevich for useful discussions.

Описок литературы

1. NAIMARK M. A. (1968) Normirovannie kolca. Moscow.

2. FELL J. M. G. (1961) The structure of fields of operator fields. Acta Mathematica Vol. 106. No. 3-4, P. 233-280.

3. TOMIYAMA J., TAKESAKI M. (1961) Application of fiber bundle to certain class of C*-algebras. Tohoku Math. J. Vol. 13. No 2, P. 498-522.

4. KRAUSS F., LAWSON T. (1974) Examples of homogeneous C*-algebras. Memoirs of the American mathematical society, Vol. 148. P. 153-164.

5. ANTONEVICH A., KRUPNIK N. (2000) On trivial and non-trivial N-homogeneous C*-algebras. Integral Equations and Operator Theory. Vol. 38. P. 172-189.

6. DISNEY S., RAEBURN I. (1985) Homogeneous C*-algebras whose spectra are tori. Jornal of the Australian mathematical society (Series A). Vol. 38. P. 9-39.

7. MASSEY W. (1977) Algebraic topology: An introduction. Springer, 292 p.