Morfismos, Vol 19, No 1, 2015

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VOLUMEN 19 NÚMERO 1 ENERO A JUNIO DE 2015 ISSN: 1870-6525

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VOLUMEN 19 NÚMERO 1 ENERO A JUNIO DE 2015 ISSN: 1870-6525

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Morfismos

Contents - Contenido Real projective space as a space of planar polygons Donald M. Davis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Motion planning in tori revisited Jes´ us Gonz´ alez, B´ arbara Guti´errez, Aldo Guzm´ an, Cristhian Hidber, Mar´ıa Mendoza, and Christopher Roque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Toeplitz operators with piecewise quasicontinuous symbols Breitner Ocampo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Morfismos, Vol. 19, No. 1, 2015, pp. 1–6 Morfismos, Vol. 19, No. 1, 2015, pp. 1–6

Real projective space as a space of planar ∗ polygons Real projective space as a space of planar polygons Donald M. Davis

∗

Donald M. Davis Abstract We describe an explicit homeomorphism between real projective space RP n−3 and the space MAbstract n,n−2 of all isometry classes of ngons in the plane with one side of length n−2 and all other sides of We describe an explicit homeomorphism between real projective length 1. This makes the topological complexity of real projective of all isometry classes of nspace RP n−3 and the space M space more relevant to robotics. n,n−2 gons in the plane with one side of length n−2 and all other sides of length 1. This makes the topological complexity of real projective 2010 Mathematics Subject Classification: 58D29, 55R80, 70G40, 51N20. space more relevant to robotics.

Keywords and phrases: Topological complexity, robotics, planar polygon spaces. 2010 Mathematics Subject Classification: 58D29, 55R80, 70G40, 51N20.

1

Keywords and phrases: Topological complexity, robotics, planar polygon spaces.

Introduction

The1topological complexity, TC(X), of a topological space X is, roughly, Introduction the number of rules required to specify how to move between any two points X ([4]). complexity, This is relevant to robotics if X is space the space all The of topological TC(X), of a topological X is, of roughly, configurations of a robot. the number of rules required to specify how to move between any two A celebrated theorem subject that, for isreal points of X ([4]). Thisinisthe relevant tostates robotics if X theprojective space of all n with n = 1, 3, or 7, TC(RP n ) is 1 greater than the dimension space RP configurations of a robot. n can be immersed ([5]). of the A smallest Euclidean space in which celebrated theorem in the subjectRP states that, for real projective n n This is of interest to algebraic topologists because of the huge space RP with n = 1, 3, or 7, TC(RP ) is 1 greater than the amount dimension n of work that has been invested during the past 60 years in studying ([5]). of the smallest Euclidean space in which RP can be immersed thisThis immersion question. See, e.g.,topologists [6], [9], [1],because and [2].of In popular is of interest to algebraic thethe huge amount article [3], this highlighted an unexpected application of work thattheorem has beenwas invested duringasthe past 60 years in studying of algebraic topology. this immersion question. See, e.g., [6], [9], [1], and [2]. In the popular ∗article [3], this theorem was highlighted as an unexpected application Invited paper. of algebraic topology. ∗

Invited paper.

1 1

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Donald M. Davis

But, from the definition of RP n , all that TC(RP n ) really tells is how hard it is to move efficiently between lines through the origin in Rn+1 , which is probably not very useful for robotics. Here we show explicitly how RP n may be interpreted to be the space of all polygons of a certain type in the plane. The edges of polygons can be thought of as linked arms of a robot, and so TC(RP n ) can be interpreted as telling how many rules are required to tell such a robot how to move from any configuration to any other. Let Mn,r denote the moduli space of all oriented n-gons in the plane with one side of length r and the rest of length 1, where two such polygons are identified if one can be obtained from the other by an orientation-preserving isometry of the plane. These n-gons allow sides to intersect. Since any such n-gon can be uniquely rotated so that its r-edge is oriented in the negative x-direction, we can fix vertices x0 = (0, 0) and xn−1 = (r, 0) and define (1)

Mn,r = {(x1 , . . . , xn−2 ) : d(xi−1 , xi ) = 1, 1 ≤ i ≤ n − 1}.

Here d denotes distance between points in the plane. Most of our work is devoted to proving the following theorem. Theorem 1.1. If n − 2 ≤ r < n − 1, then there is a Z/2-equivariant homeomorphism Φ : Mn,r → S n−3 , where the involutions are reflection across the x-axis in Mn,r , and the antipodal action in the sphere. Taking the quotient of our homeomorphism by the Z/2-action yields our main result. It deals with the space M n,r of isometry classes of planar (1n−1 , r)-polygons. This could be defined as the quotient of (1) modulo reflection across the x-axis. Corollary 1.2. If n − 2 ≤ r < n − 1, then M n,r is homeomorphic to RP n−3 . These results are not new. It was pointed out to the author after preparation of this manuscript that the result is explicitly stated in [8, Example 6.5], and proved there, adapting an argument given much earlier in [7]. The result of our Corollary 1.2 was also stated as “well known” in [10]. Nevertheless, we feel that our explicit, elementary homeomorphism may be of some interest.

Real projective space

2

3

Proof of Theorem 1.1

In this section we prove Theorem 1.1. Let J m denote the m-fold Cartesian product of the interval [−1, 1], and S 0 = {±1}. Our model for S n−3 is the quotient of J n−3 × S 0 by the relation that if any component of J n−3 is ±1, then all subsequent coordinates are irrelevant. That is, if ti = ±1, then (t1 , . . . , ti , ti+1 , . . . , tn−2 ) ∼ (t1 , . . . , ti , t i+1 , . . . , t n−2 )

(2)

for any t i+1 , . . . , t n−2 . This is just the iterated unreduced suspension of S 0 , and the antipodal map is negation in all coordinates. An explicit homeomorphism of this model with the standard S n−3 is given by (t1 , . . . , tn−2 ) ↔ (x1 , . . . , xn−2 ), with xi = ti

i−1 1 − t2j ,

j=1

i−1 xi ti = if x2i < 1. 2 2 1 − x1 − · · · − xi−1 j=1

Then ti = ±1 for the smallest i for which x21 + · · · + x2i = 1. Let P ∈ Mn,r be a polygon with vertices xi as in (1). We will define the coordinates ti = φi (P) of Φ(P) under the homeomorphism Φ of Theorem 1.1. For 0 ≤ i ≤ n − 2, we have (3)

n − 2 − i ≤ d(xi , xn−1 ) ≤ n − 1 − i.

The first inequality follows by induction on i from the triangle inequality and its validity when i = 0. The second inequality also uses the triangle inequality together with the fact that you can get from xi to xn−1 by n − 1 − i unit segments. The second inequality is strict if i = 0 and is equality if i = n − 2. Let i0 be the minimum value of i such that equality holds in this second inequality. Then the vertices xi0 . . . , xn−1 must lie evenly spaced along a straight line segment. Let C(x, t) denote the circle of radius t centered at x. The inequalities (3) imply that, for 1 ≤ i ≤ i0 , C(xn−1 , n − 1 − i) cuts off an arc of C(xi−1 , 1), consisting of points x on C(xi−1 , 1) for which d(x, xn−1 ) ≤ n − 1 − i. Parametrize this arc linearly, using parameter values −1 to 1 moving counterclockwise. The vertex xi lies on this

4

Donald M. Davis

arc. Set φi (P) equal to the parameter value of xi . If i = i0 , then φi (P) = ±1, and conversely. The following diagram illustrates a polygon with n = 7, r = 5.2, and i0 = 5. We have denoted the vertices by their subscripts. The circles from left to right are C(xi , 1) for i from 0 to 4. The arcs centered at x6 have radius 1 to 5 from right to left. We have, roughly, Φ(P) = (.7, .6, .5, −.05, 1). .. ..... ..... .... ..... ...... . .... . . . . ...... . ... ...... ... ..... ... . .. . . . . . . . .... ... ... ..... .. ............... ............................... .. ............ ....................................... . ................ ............ .... ............ .. ....... . . . . . . . . . . . . . . . . . . . . . . . . . . ..... ..... .. ... .......... .... ... .... ...... ........ .... ... ...... ..... ........... .......... ...... ... ..... ...... ..... .... .... .. .... ..... ................. ............................ ... .. .... .. .. . .. ....... ..... ... ..... ...... ...... .... .. .. .... ...... ... . ..... . . . . . . . . . . . . . . . . . . . . . . ... .... ..... ....... ...... ... ... ... .... ... ... ... ... ... ... ... ..... ... ..... ..... ... ... ... ........................................................................... ... .... . ........................................... .... . ... ........ .... . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ ... ... ........ . . ......................... ... ... . ... . ...... . . . . . . . . . . . . . . . . . . . . .......... .. . . .... .......... . ...... ... ... ..... . . ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................... ... . ............... . ... ... .... . ... ... . . . . . . . . . . . ... . . . . . . . . . . . . ... ... ... . ... . .... . .... . . ....... . .... ......... . . .. . . . . . . . . . . . . . . ... ... ... ... ..... . .. .... .. ... ... ..... ..... .. ... .... ..... ... ... ... ... ...... ..... ............ .... ...... ... ... ..... ......... ... .. ... ... ... ........ .. ... ............ ... ..... ..... ........ ...... .. . . ...... .... ... ... ... .......... ....... .. ....... .. .................................................................................................................................................................................................................................................................................................................. ..... ..... ..... . .. ... ......... . . . . . . ... . . . . . ... ..... ...................................... ... ... ...... . . ... . . . . . . . . . . ....... .. ... ... . ... ....... .... ............ ... .... ... ... ... ... ......................... ... .. ... ... ... ... ... . ... ... ... ... ... ... ... . . ..... . . . . . . . ... ... ... . ... ..... ... .......... . ...... . ... ... ... . ........ . . . . . . ... ... ... ...................................... ... . ... ... ... ... ... ... ... ..

•2

•1

• 3

• 4

5•

• 0

•6

Here is another example, illustrating how the edges of the polygon can intersect one another, and a case with i0 < n − 2. Again we have n = 7 and r = 5.2. This time, roughly, Φ(P) = (.2, −.4, .4, 1, t5 ), with t5 irrelevant. Because i0 = 4, we did not draw the circle C(x4 , 1). . ... ... ... ..... .. ..... ... ... ............................... ..... .. ....... ... ........... ... ...................................... . . . ............ . . ...... . . . . . . . . ...... .... . ..... ..... ...... .... ... .... .......... .... ................................................ ....... ... .... . ... ........ . .. .. .. ... .. ........... ...... ...... ............... .. ................................ ... ..... .. .... ..... ........ ................ . ..... . . . . . . . . . . . . . . . ... .... ........... ... .... .............. ... .... .... .............. . .. ... . . ... . . . . . . ........ ..... ... ....... ... . ........ . ... ... ... . .. ... . . . .. . . . . . . . . . . . . . . . . . ... .. ... ..... .. ... ... .. . . ... ... ....... ......... ... ... ... ... .............. ..... .... . .......... .. ........ ........ .. ... .. ..... ... ... ... ... ... ........... ......... .. ........ ... ...................... .... ..... .............. . ... ... ... ... ... ... . ....... ... ................................................................................................................................................................................................................................................................................................... ..... . ... ... ... ... ... .. ... ... ........ ... .... ........... ... . . . ... . ... .. ... .. ... ......... ........... .. . . ... . . . . . . . . ............. ...... ... ... .. ... . ... .... ... ... .... ... .. ... ... .... ... ... ... ... ... ........ .... ... .. ... ......... . ... ... .. .. ..... ........ ... ........ .. . ... ...... ..... .. ............. ... ............ ......... ........ .. . . ..... . . . . . . . . . . . . . . . . . . . . . . . .......... ... . .................. ... .... ... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... . ... ..... ... ... ... ............. ... ........ . ... ... .... .......................................... ..... .... ........ ... ..... .. ... ...... ... ... ...... .... ......... . . . . . ... . . . .................................. ... ... ... ... .

0•

4•

1 •

• 2

•3

•5

•6

That Φ is well defined follows from (2); once we have ti = ±1, which happens first when i = i0 , subsequent vertices are determined and the values of subsequent tj are irrelevant. Continuity follows from the fact that the unit circles vary continuously with the various xi , hence so do the parameter values along the arcs cut off. Bijectivity follows from the

Real projective space

5

construction; every set of ti ’s up to the first ±1 corresponds to a unique polygon, and ±1 will always occur. Since it maps from a compact space to a Hausdorff space, Φ is then a homeomorphism. Equivariance with respect to the involution is also clear. If you flip the polygon, you flip the whole picture, including the unit circles, and this just negates all the ti ’s. We elaborate slightly on the surjectivity of Φ. The arc on C(x0 , 1) cut off by C(xn−1 , n − 2) is determined by n and r. Given a value of t1 in [−1, 1], the vertex x1 is now determined on this arc. Now the arc on C(x1 , 1) cut off by C(xn−1 , n − 3) is determined, and a specified value of t2 determines the vertex x2 . All subsequent vertices of an n-gon are determined in this manner. Donald M. Davis Department of Mathematics, Lehigh University Bethlehem, PA 18015, USA, dmd1@lehigh.edu

References [1] Davis D. M., Immersions of projective spaces: a historical survey, Contemp Math Amer Math Soc 146 (1993) 31–38. [2]

, Immersions of real projective spaces, www.lehigh.edu/∼dmd1/imms.html.

[3]

, Algebraic topology: there’s an app for that, Math Horizons 19 (2011), 23–25.

[4] Farber M., Invitation to topological robotics, European Math Society (2008). [5] Farber M,; Tabachnikov S.; Yuzvinsky S., Topological robotics: motion planning in projective spaces, Intl Math Research Notices 34 (2003) 1853–1870. [6] Gitler S., Immersions and embeddings of manifolds, Proc Symp in Pure Math Amer Math Soc 22 (1971) 87–96. [7] Hausmann J. C., Sur la topologie des bras articuls, Lecture Notes in Math, Springer, 1474 (1991) 146–159.

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[8] Hausmann J. C., Rodriguez E., The space of clouds in Euclidean space, Experiment. Math. 13 (2004) 31–47. [9] James I. M, Euclidean models of projective spaces, Bull London Math Soc 3 (1971) 257–276. [10] Kamiyama Y., Kimoto K., The height of a class in the cohomology ring of polygon spaces, Int Jour of Math and Math Sci (2013) 7 pages.

Morfismos, Vol. 19, No. 1, 2015, pp. 7–18 Morfismos, Vol. 19, No. 1, 2015, pp. 7–18

Motion planning in tori revisited∗ ∗ Motion planning tori Jes´ us Gonz´ alez, B´ arbara Guti´ein rrez, Aldorevisited Guzm´an, Cristhian Hidber, Mar´ıa Mendoza, Christopher Roque Jes´ us Gonz´alez, B´arbara Guti´errez, Aldo Guzm´an, Cristhian Hidber, Mar´ıa Mendoza, Christopher Roque Abstract The topological complexity (TC) of the complement of a complex hyperplane arrangement, which is either linear generic or Abstract else The affinetopological in general complexity position, has(TC) beenofcomputed by Yuzvinsky. the complement of a comThisplex is accomplished by noticing that efficient homotopy mod- or hyperplane arrangement, which is either linear generic els for such spaces are given by skeletons of Cartesian powers of else affine in general position, has been computed by Yuzvinsky. circles. Soon after, Cohen and Pruidze noticed that the topologiThis is accomplished by noticing that efficient homotopy modcal complexity the complement of the corresponding redundant els for suchofspaces are given by skeletons of Cartesian powers of subspace arrangement, as well as of right-angled Artin groups, can circles. Soon after, Cohen and Pruidze noticed that the topologibe obtained by considering general subcomplexes of cartesian powcal complexity of the complement of the corresponding redundant ers ofsubspace higher dimensional spheres. Cohen-Pruidze’s arrangement, as well Unfortunately as of right-angled Artin groups, can TC-calculations are flawed, and our work describes and thepowbe obtained by considering general subcomplexes of mends cartesian problems in order to validate the extended applications. In addiers of higher dimensional spheres. Unfortunately Cohen-Pruidze’s tion,TC-calculations we generalize Farber-Cohen’s of the topological are flawed, andcomputation our work describes and mends the complexity of oriented surfaces, now to the realm of Rudyak’s problems in order to validate the extended applications. In addihigher topological complexity. tion, we generalize Farber-Cohen’s computation of the topological complexity of oriented surfaces, now to the realm of Rudyak’s 2010 Mathematics Subject Classification: 20F36, 52C35, 55M30. higher topological complexity.

Keywords and phrases: Topological complexity, motion planner, zero2010cup-length. Mathematics Subject Classification: 20F36, 52C35, 55M30. divisior Keywords and phrases: Topological complexity, motion planner, zerodivisior cup-length.

1

Introducci´ on

Michael proposedoin 1 Farber Introducci´ n [5, 6] a topological model to study the continuity instabilities of the motion planning problem in robotics. FolMichael Farber proposed in [5, algorithm 6] a topological modelplanner) to study P the=conlowing Farber, a motion planning (or motion tinuity instabilities of the motion planning problem in robotics. Fol∗ This work is the result of the activities of the authors in the student workshop enFarber, a at motion planning motion planner) titledlowing “Applied Topology ABACUS: Motionalgorithm Planning in(or Robotics” held in AugustP = 2013. The ∗ authors thank all participants of the workshop for useful discussions, and This work is the result of the activities of the authors in the student workshop enkindly acknowledge the financial support received ABACUS through CONAtitled “Applied Topology at ABACUS: Motion from Planning in Robotics” held in August CyT 2013. grant The EDOMEX-2011-C01-165873. authors thank all participants of the workshop for useful discussions, and kindly acknowledge the financial support received from ABACUS through CONACyT grant EDOMEX-2011-C01-165873. 7

7

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Gonz´ alez et al

{Fi , si }i=1,...,k for a space X consists of a collection of k pairwise disjoint subsets Fi of X Ă— X, each admitting a continuous section si : Fi → X [0,1] for the end-points evaluation map Ď€ : X [0,1] → X Ă— X, Ď€(Îł) = (Îł(0), Îł(1)), such that {Fi }i is a covering of X Ă— X by ENR’s. The sets Fi and the maps si are respectively called the local domains and the local rules of P. The motion planner is said to be optimal when the number of local domains is minimal possible. The topological complexity of X, TC(X), is one less than the number of local domains in any optimal motion planner for X. A lower bound for TC(X) is described in Proposition 1.1 below through the concept of the zero-divisor cup-length of X with respect to a cohomology theory with products h∗ . An h∗ -zero-divisor of X is an element in the kernel of the induced map (1)

Ď€ ∗ : h∗ (X Ă— X) → h∗ (X [0,1] ).

The h∗ -zero-divisor cup-length of X, denoted by zclh∗ (X), is the maximal number of h∗ -zero-divisors whose product in h∗ (X Ă—X) is non-zero. The “zero-divisorâ€? adjective comes from the fact that Ď€ : X [0,1] → X Ă—X is a fibrational substitute for the diagonal map X → X Ă—X. Thus, if the strong form of the K¨ unneth formula holds for h∗ , the kernel of (1) can be identified with the kernel of the cup-product map h∗ (X) ⊗ h∗ (X) → h∗ (X). Proposition 1.1 ([5, Theorem 7]). The topological complexity of X is bounded from below by the h∗ -zero-divisor cup-length of X, i.e. zclh∗ (X)) ≤ TC(X).

Instead of the usual upper bound for TC(X) given by homotopy theory (see [7, 8, 11]), the novel ingredient in Yuzvinsky’s [16] and Cohen-Pruidze’s [4] works relies on the explicit construction of motion planners whose optimality is then guaranteed by Proposition 1.1. The relevant spaces arise as follows. Fix a positive integer k and consider the standard (minimal) cellular structure in the k-dimensional sphere S = S k = e0 âˆŞ ek . Here e0 is the base point, which we denote by e. Then take the product cell decomposition in (2)

¡ ¡ Ă— S = Sn = S Ă— ¡ n times

eJ ,

J

where the cells eJ , indexed by subsets J ⊆ [n] = {1, . . . , n}, are defined by eJ = ni=1 edi with di = 0 if and only if i ∈ J. Explicitly, eJ =

Motion planning in tori revisited

9

(x1 , . . . , xn ) ∈ Sn | xi = e0 if and only if i ∈ / J . Cohen and Pruidze’s main result is stated next. Theorem 1.2. For a subcomplex X of the cell decomposition (2), 1. TC(X) =

2 dim(X) k

for even k;

2. TC(X) = zclH ∗ (X) = max {|J| + |K| : J ∩ K = ∅, eJ and eK cells of X} for odd k. It is illustrative to compare Theorem 1.2 to its Lusternik-Schnirelmann category counterpart (in terms of the polyhedral power notation). F´elix and Tanr´e prove in [10] the equality cat((S k , )L ) = cat(S k )(dim(L) + 1). Here L is an abstract simplicial complex with vertices in [n], and cat denotes the (reduced) Lusternik-Schnirelmann category of X. For even k, this corresponds to the equality TC((S k , )L ) = TC(S k )(1 + dim(L)) in item 1 of Theorem 1.2. However, for an odd k, the answer TC((S k , )L ) = zclH ∗ ((S k , )L ) in item 2 of Theorem 1.2 has a value which is arbitrarily lower than that in item 1, as we explain next. Item 2 in Theorem 1.2 yields the calculations in [4, 16] of the topological complexity of complements of complex hyperplane arrangements (either linear generic, or affine in general position), and of EilenbergMacLane spaces K(π, 1) for π a right-angled Artin group. It is also interesting to notice that, while the value of TC(X) in item 1 of Theorem 1.2 is maximal possible (see [6, Theorem 5.2]), item 2 in Theorem 1.2 gives instances where the actual value of TC(X) can be arbitrarily lower than the dimension-vs-connectivity bound. In fact item 2 in Theorem 1.2 implies that the general estimate “cat ≤ TC ≤ 2 cat” in [5, Theorem 5] can reach any possible combination1 —besides the standard facts that TC(X) = cat(X) for H-spaces ([12, Theorem 1]), and 1 The authors learned of this fact at Dan Cohen’s lecture during the student workshop Applied Topology at ABACUS: Motion Planning in Robotics, that took place a week after the 2013 Mathematical Congress of the Americas, in M´exico.

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TC(X) = 2 cat(X) for closed simply connected symplectic manifolds ([9, Corollary 3.2]). Indeed, for any positive integers c and t with c ≤ t ≤ 2c, there is a space X with cat(X) = c and TC(X) = t. In detail, for a fixed non-negative odd integer k, let X = Sc ∨ St−c . Since cat(Sc ) = c ≥ t − c = cat(St−c ), we see cat(X) = max cat(Sc ), cat(St−c ) = c. On the other hand, item 2 in Theorem 1.2 yields TC(X) = t. (The case k = 1 is treated in [14] with different techniques.) We noticed that the inequality TC(X) ≤ 2 dim(X) in item 1 of Thek orem 1.2 is standard. Cohen and Pruidze assert to have constructed an explicit motion planning algorithm realizing this upper bound, but as described in the next section, their construction is flawed on several fronts. Similar problems hold in item 2 of Theorem 1.2, but in this case the situation is critical because the needed upper bound is not available by other means. The first main goal of this paper, addressed in the next section, is to fix the problems in [4]. Then, in Sections 3, we compute the higher topological complexity of oriented surfaces.

2

Correction of gaps in Cohen-Pruidze’s work

The simplest motion planner on Sn holds for k odd, assumption which will be in force in this section until further notice. When n = 1, the motion planner has two local domains described as follows: Let F1 ⊂ S × S and s1 : F1 → S[0,1] be given by F1 = {(x, −x)|x ∈ S}

and, for a fixed nowhere zero tangent vector field ν on S, s1 (x, −x) is the path from x to −x at constant speed along the semicircle determined by the tangent vector ν(x). The second local domain is given by the complement of F1 , F0 = S × S − F1 ,

with local rule s0 : F0 → S[0,1] where s0 (x, y) is the path from x to y at constant speed along the shortest geodesic arc. It is elementary to see that zclH ∗ (S) = 1, so TC(S) = 1 and the above motion planner is optimal. The corresponding product motion planner in Sn (described in [5, Theorems 11 and 13], and simplified in [6, p. 24]) is recalled in Proposition 2.1 below. The needed preliminaries go as follows: For a subset

Motion planning in tori revisited

I ⊂ [n], let

11

FI = {(x, y) ∈ Sn × Sn | xi = −yi iff i ∈ I}

and define sI : FI → (Sn )[0,1] using the maps s1 and s0 defined above, namely (3)

sI (x, y) = (t1 (x1 , y1 ), . . . , tn (xn , yn ))

/ I. (Here and below we use where ti = s1 if i ∈ I, and ti = s0 if i ∈ the shorthand x = (x1 , . . . xn ), y = (y1 , . . . , yn ), etc.) The sets FI ’s are conveniently separated as FI ∩ FJ = ∅ for I J. In particular, FI ∩ FJ = ∅ = FI ∩ FJ when |I| = |J| with I = J. This allows us to set (4)

Wj =

|I|=n−j

FI ∼ =

FI

|I|=n−j

for j = 0, 1, . . . , n, and define a local rule σj : Wj → (Sn )[0,1] by σj |FI = sI . Proposition 2.1 ([6, p. 24]). For k odd, the subsets Wj ⊂ Sn × Sn and maps σj : Wj → (Sn )[0,1] , j = 0, 1, . . . , n, determine an optimal motion planner for Sn , thus TC(Sn ) = n. Still assuming k is odd, let X be a subcomplex of the cell decomposition (2) and, for J ⊂ [n], let TJ denote the subcomplex of Sn generated / J}. If J ∩ K = ∅, TJ ∪ TK by eJ , i.e. TJ = eJ = {x ∈ Sn | xi = e0 if i ∈ n sits inside S as the wedge union TJ ∨ TK . Therefore the term on the right of the second item in Theorem 1.2 takes the form z(X) := max { |J| + |K| | J ∩ K = ∅ and TJ ∨ TK ⊆ X } .

Cohen-Pruidze’s critical assertion TC(X) = z(X) in [4, Theorem 3.4] is argued by (i) constructing a motion planner for X with z(X) + 1 local domains, and then (ii) showing that the H ∗ -zero-divisor cup-length of X is at least z(X). Their proof of (ii) is correct and straightforward, but their construction in (i) is flawed. Explicitely, the authors assert that the local rules in the product motion planner for Sn constructed in Proposition 2.1 restrict to give local rules for any subcomplex X of Sn . But such an assertion is false in most of the cases. We exhibit an explicit (but typical) counterexample (Example 2.2), and then show how the combinatorics of the cell decomposition of X need to be taken into consideration to fix the construction—and, therefore, Cohen-Pruidze’s proof of Theorem 1.2.

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Example 2.2. Take n = 2 and k = 1, so Sn = T 2 , the 2-torus. Let X = S 1 ∨ S 1 be the 1-dimensional skeleton in the minimal cell structure of T 2 , so X has the cell decomposition X = e∅ ∪ e{1} ∪ e{2} , and z(X) = 2. The local domain decomposition proposed in [4] is X ×X =

2

j=0

(X × X) ∩ Wj

with local rules given by the restricted sections σj (X) = σj |Wj ∩(X×X) . Now let us focus attention on the local domain (5)

(X × X) ∩ W0 = {(−1, 1, 1, −1)} ∪ {(1, −1, −1, 1)}

with corresponding local rule given by

σ0 (X)(x, −x, −x, x) = (s1 (x, −x), s1 (−x, x)).

The authors of [4] claim that the local rule σ0 (X) lands in X [0,1] — rather than in (T 2 )[0,1] . But such an assertion is clearly false. In fact, for any p ∈ (X × X) ∩ W0 , the path σ0 (X)(p) takes values in X only at t = 0, 1. The assertion in [4] that (X × X) ∩ Wj = ∅ for j < n − z(X) is true (and easy to verify). As illustrated above, the problem comes with the claim that the restrictions σj (X) of σj to (X × X) ∩ Wj , where (n − z(X) ≤ j ≤ n), give local motion planners in X. The gap noted in Example 2.2 is typical and can be corrected by taking into account the combinatorial properties of the cell structure in X. For instance, in the explicit situation considered in Example 2.2, rather than insisting on performing the motion in both coordinates in a “parallel” way, one should move in two halves; the first part of the motion should be on the coordinate different to 1, keeping the other coordinate fixed. Only when this part of the motion is complete, and we have arrived to the base point (1, 1), it will be safe to move the missing coordinate. For instance, the motion planning algorithm from (−1, 1) to (1, −1) —the first of the two “tasks” represented in (5)— is depicted by the thick curve in

(−1, 1) •

• (1, −1).

The correct general motion planner is described next.

Motion planning in tori revisited

13

Fixed motion planner for item 2 in Theorem 1.2 (k odd ). The description simplifies by normalizing S so to have great semicircles of length 1/2. For x, y ∈ S, we let d(x, y) stand for the length of the shortest geodesic between x and y over S, so d(x, −x) = 1/2. Likewise, the local rules s0 and s1 for S defined at the beginning of the section need to be adjusted. For i = 0, 1 and (x, y) ∈ Fi with x = y, set  1 s (x, y) t , i d(x,y) Si (x, y)(t) = y,

0 ≤ t ≤ d(x, y); d(x, y) ≤ t ≤ 1

(if x = y, Si (x, y)(t) = x = y for all t ∈ [0, 1]). Thus, Si reparametrizes si so to perform the motion at speed 1, keeping still at the final position once it is reached—which happens at most at time 1/2. In view of the homeomorphism in (4), it suffices to define a local rule on each (X Ă— X) ∊ FI taking values in X [0,1] . Replace (3) by the map SI : FI ∊ (X Ă— X) → (Sn )[0,1] defined by SI (x, y) = (T1 (x1 , y1 ), . . . , Tn (xn , yn )) where Ti (xi , yi ) : [0, 1] → S is the path Ti (xi , yi )(t) = Here txi =

1 2

xi , 0 ≤ t ≤ txi , ÎŁi (xi , yi )(t − txi ), txi ≤ t ≤ 1.

− d(xi , 1), and ÎŁi = S1 if i ∈ I while ÎŁi = S0 if i ∈ / I.

It is clear that SI is a continuous section on FI ∊ (X Ă— X) for the end-points evaluation map Ď€ : (Sn )[0,1] → Sn Ă— Sn . We only need to check that SI takes values in X [0,1] . With that in mind, note that the motion described by the local rule SI , from an “initial coordinateâ€? xi to the corresponding “final coordinateâ€? yi , is executed according to the relevant instruction Sj (j ∈ {0, 1}), except that the movement is delayed a time txi ≤ 1/2. The closer xi gets to 1, the closer the delaying time txi gets to 1/2. It is then convenient to think of the path SI (x, y) as happening in two stages. In the first stage (t ≤ 1/2) all initial coordinates xi = 1 keep still, while the rest of the coordinates (eventually) start traveling to their corresponding final position yi . Further, at the time the second stage starts (t = 1/2), any final coordinate yi = 1 will already have been reached. As a result, SI never leaves X. In more detail: Let eJ , eK ⊂ X be cells of X. For (x, y) = ((x1 , . . . , xn ), (y1 , . . . , yn )) ∈ FI ∊ (eJ Ă— eK ), coordinates corresponding to indexes i not in J, keep their initial position xi = 1 through time t ≤ 1/2. Therefore SI (x, y)[0, 1/2] stays within TJ ⊆ X. On the

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Gonz´ alez et al

other hand, by construction, Ti (xi , yi )(t) = yi = 1 whenever t ≼ 1/2 and i ∈ / K. Thus, SI (x, y)[1/2, 1] stays within TK ⊆ X.

The motion planning algorithm constructed above is a variation of the one carefully described in [16] for skeletons of the minimal cell decomposition in the n-th Cartesian power of the circle. In the current case, we are considering all possible subcomplexes of the minimal CWcomplex (S k )n for any odd k.

Proof of Theorem 1.2 (an additional fixing). It remains to check that zclH ∗ (X) is bounded from below by 2 dim(X)/k when k is even, and by z(X) when k is odd. The case for odd k is addressed correctly in [4, Proposition 3.7], however the case for even k requires some tune-up. We thus assume in this proof that k is even. Say dim(X) = k . Cohen and Pruidze’s argument starts by noticing that X must contain a copy of S as a subcomplex, and that the inclusion Κ : S → X induces an epimorphism Κ∗ : H ∗ (X) → H ∗ (S ). From this, they infer (6)

zclH ∗ (X) ≼ zclH ∗ (S ),

and obtain the desired conclusion from the well-known equality zcl(H ∗ (S )) = 2 . The subtlety here is that the surjectivity of the ring morphism Κ∗ is not enough to deduce (6). One actually needs to know that each factor in a non-zero product of zero-divisors realizing zcl(H ∗ (S ))—as the one described in the proof of Proposition 6.2 in [4]—is the image of a zerodivisor in zcl(H ∗ (X)). But such a property does hold in the current situation, as Proposition 3.6 in [4] holds true also for k even (cf. [1, Theorem 2.35]). Alternatively, the composition of the inclusion X → Sn with the obvious projection Sn → S gives a retraction Ď for the inclusion Κ : S → X and, evidently, both Κ and Ď are compatible with diagonal inclusions. Remark 2.3. The problem noted in Example 2.2 (for k odd) also holds in [4] for k even. The new issue is more subtle, and this is reflected in part by noticing an additional gap in the proof of [4, Theorem 6.3]. Here we illustrate the new error (and some of the subtleties needed to sort it out), so we assume in this remark that the reader is familiar with the notation set in the final section of [4] (where k is even). For X =

Motion planning in tori revisited

15

S 2 ∨ S 2 ⊂ S 2 × S 2 , the first paragraph in the proof of [4, Theorem 6.3] asserts that (X ×X)∩Wj = ∅ for j = 0, 1. In particular (AJ ×AK )∩Fα would have to be empty for J = {1}, K = {2}, and α = (1, 0). However ((−e, e), (e, −e)) clearly lies in the latter intersection. Of course, Cohen and Pruidze’s gap in the argument of their Theorem 6.3 comes from their assertion (in the second paragraph of their proof) that there should be some index in {1, 2} missing J ∪ K.

As part of her Ph.D. studies, the second author of this paper has managed to construct an optimal motion planner for any subcomplex of Sn when k is even. The construction, carried over in more general terms (for Rudyak’s higher TC and any parity of k), will be discussed elsewhere.

3

Higher TC of oriented surfaces

In [13] Yuli B. Rudyak introduced the concept of the higher topological complexity of a path connected space X, denoted by TCs (X). In this section we extend Farber and Cohen’s calculation of TC2 (Σg ) in [3] to the realm of higher topological complexity. Here Σg stands for an oriented surface of genus g. Let Js , s ∈ N, denote the wedge of n closed intervals [0, 1]i , i = 1, . . . , s, where the zero points 0i ∈ [0, 1]i are identified. Consider X s (the s-th cartesian product of X) and X Js , where X is a path connected space. There is a fibration (7)

es : X Js −→ X s ,

es (f ) = (f (11 ), . . . , f (1s ))

where 1i ∈ [0, 1]i . Recall that the s-th topological complexity of X, denoted by TCs (X), is defined as the reduced Schwarz genus of es . Note that (7) is a fibrational substitute of the iterated diagonal map s dX s : X −→ X . Hence TCs (X) coincides with the Schwarz genus of s X dX s : X −→ X . Using the iterated diagonal map ds and allowing cohomology with local coefficients we have the following standard definition: Definition 3.1. Given a space X and a positive integer n, we denote by zcls (H ∗ (X)) the maximal length of non-zero products of elements in ∗ the kernel of the map induced in cohomology by dX s . Thus, zcls (H (X)) is the largest integer m for which there exist cohomology classes ui ∈ H ∗ (X s , Ai ) with dX s (ui ) = 0, i = 1, . . . , m, and 0 = u1 ⊗ · · · ⊗ um ∈ H ∗ (X s , A1 ⊗ · · · ⊗ Am ).

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Gonz´ alez et al

Note that zcl2 (H ∗ (X)) recovers zclH ∗ (X). The following result bounds TCs (X) from below by zcls (H ∗ (X)) and from above by a number which involves the homotopy dimension of X, hdim(X) (the smallest dimension of CW complexes having the homotopy type of X), and the connectivity of X, conn(X). Theorem 3.1. For any path-connected space X we have: zcls (H ∗ (X)) ≤ TCs (X) ≤

s hdim(X) conn(X) + 1

For a proof of Theorem 3.1 see [15, Theorems 4 and 5]. Proposition 3.2. For g, s ≥ 2, the s-th higher topological complexity of Σg is TCs (Σg ) = 2s. This should be compared to the facts that TCs (Σ0 ) = s and TCs (Σ1 ) = 2(s − 1)

proved in [2, Corollary 3.12] (see also [13, Section 4]). In addition, it should be noted that Proposition 3.2 was also mentioned by Ibai Basabe during his talk at the conference “Applied Algebraic Topology” held at the Centro Internacional de Encuentros Matem´ aticos on July 2014. Proof of Proposition 3.2. We use cohomology with rational coefficients. Let ai , bi , i = 1, . . . , g, be the generators of H 1 (Σg ) which satisfy ai bj = ai aj = bi bj = 0 for i = j, a2i = b2i = 0 and ai bi = ω for any i, where ∗ ⊗s ω generates H 2 (Σg ). Let HΣg = H ∗ (Σ×s g ) = [H (Σg )] . For each i = 2, . . . , s, consider the elements αi = a1 ⊗ 1 ⊗ · · · ⊗ 1 − 1 ⊗ · · · ⊗ a1 ⊗ · · · ⊗ 1, βi = b1 ⊗ 1 ⊗ · · · ⊗ 1 − 1 ⊗ · · · ⊗ b1 ⊗ · · · ⊗ 1,

where the factor a1 (resp. b1 ) in 1 ⊗ · · · ⊗ a1 ⊗ · · · ⊗ 1 (resp. 1 ⊗ · · · ⊗ b1 ⊗ · · · ⊗ 1) appears in the i-th tensor coordinate, and γ1 = a2 ⊗ 1 ⊗ · · · ⊗ 1 − 1 ⊗ a2 ⊗ 1 ⊗ · · · ⊗ 1,

γ2 = b2 ⊗ 1 ⊗ · · · ⊗ 1 − 1 ⊗ b2 ⊗ 1 ⊗ · · · ⊗ 1

of HΣg . These elements lie in the kernel of the cup-product map [H ∗ (Σg )]⊗s → H ∗ (Σg ),

and satisfy γ1 · γ2 · α2 · β2 · · · αs · βs = 2ω ⊗ · · · ⊗ ω = 0. Therefore, 2s ≤ zcls (H ∗ (Σg )) ≤ TCs (Σg ). On the other hand, using the upper bound in Theorem 3.1, we get TCs (Σg ) ≤ 2s. Thus, TCs (Σg ) = 2s.

Motion planning in tori revisited

Jes´ us Gonz´ alez Departamento de Matem´ aticas, CINVESTAV del I.P.N., Apartado Postal 14-740, M´exico D.F., C.P. 07360, jesus@math.cinvestav.mx

B´ arbara Guti´errez Departamento de Matem´ aticas, CINVESTAV del I.P.N., Apartado Postal 14-740, M´exico D.F., C.P. 07360, bgutierrez@math.cinvestav.mx

Aldo Guzm´ an Departamento de Matem´ aticas, CINVESTAV del I.P.N., Apartado Postal 14-740, M´exico D.F., C.P. 07360, aldo@math.cinvestav.mx

Cristhian Hidber Departamento de Matem´ aticas, CINVESTAV del I.P.N., Apartado Postal 14-740, M´exico D.F., C.P. 07360, chidber@math.cinvestav.mx

Mar´ıa Mendoza Departamento de Matem´ aticas, CINVESTAV del I.P.N., Apartado Postal 14-740, M´exico D.F., C.P. 07360, marialuisa393@gmail.com

Christopher Roque Departamento de Matem´ aticas, CINVESTAV del I.P.N., Apartado Postal 14-740, M´exico D.F., C.P. 07360, croque@math.cinvestav.mx

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References [1] Bahri A.; Bendersky M.; Cohen F.; Gitler S., The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Adv. Math. 225:3 (2010), 1634–1668. [2] Basabe I.; Gonz´ alez J.; Rudyak J.; Tamaki D., Higher topological complexity and its symmetrization, Algebr. Geom. Topol. 14: 4 (2014), 2103–2124. [3] Cohen D.; Farber M., Topological complexity of collision-free motion planning on surfaces, Compos. Math. 147:2 (2011), 649–660. [4] Cohen D.; Pruidze G., Motion planning in tori, Bull. Lond. Math. Soc. 40:2 (2008), 249–262. [5] Farber M., Topological complexity of motion planning, Discrete Comput. Geom. 29:2 (2003), 211–221. [6] Farber M., Instabilities of robot motion, Topology Appl. 140:2-3 (2004), 245–266. [7] Farber M.; Grant M., Topological complexity of configuration spaces, Proc. Amer. Math. Soc. 137:5 (2009), 1841–1847.

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[8] Farber M.; Grant M.; Yuzvinsky S., Topological complexity of collision free motion planning algorithms in the presence of multiple moving obstacles, Topology and robotics, Contemp. Math. 438 (2007), 75–83. [9] Farber M.; Tabachnikov S.; Yuzvinsky S., Topological robotics: motion planning in projective spaces, Int. Math. Res. Not. 34 (2003), 1853–1870. [10] F´elix Y.; Tanr´e D., Rational homotopy of the polyhedral product functor, Proc. Amer. Math. Soc. 137:3 (2009), 891–898. [11] Gonz´ alez J.; Grant M., Sequential motion planning of noncolliding particles in Euclidean spaces, Accepted for publication in Proceedings of the American Mathematical Society. [12] Lupton G.; Scherer J., Topological complexity of H-spaces, Proc. Amer. Math. Soc. 141:5 (2013), 1827–1838. [13] Rudyak Y., On higher analogs of topological complexity, Topology and its Applications 57 (2010), 916–920 (erratum in Topology and its Applications 57 (2010), 1118. [14] Rudyak Y., On topological complexity of Eilenberg-MacLane spaces, http://arxiv.org/pdf/1302.1238v2.pdf. [15] Schwarz A., The genus of a fiber space, Amer. Math. Soc. Transl. Series 2 55 (1966), 49–140. [16] Yuzvinsky S., Topological complexity of generic hyperplane complements, Contemporary Math. 438 (2007), 115–119.

Morfismos, Vol. 19, No.1, 2015, pp. 19–38 Morfismos, Vol. 19, No.1, 2015, pp. 19–38

Toeplitz operators with piecewise ∗ quasicontinuous Toeplitz operators symbols with piecewise ∗ quasicontinuous Breitner Ocamposymbols Breitner Ocampo Abstract Abstract For a fixed subset of the unit circle ∂D, Λ := {λ1 , λ2 , . . . , λn }, we define the algebra P C of piecewise continuous functions in Λ. {λ Besides, ∂D \ ΛFor with one sided limits atunit eachcircle point∂D, λk Λ∈ := a fixed subset of the , λn }, 1 , λ2 , . . .we let QC stands the for the C ∗ -algebra quasicontinuous functions on in we define algebra P C of ofpiecewise continuous functions ∂D ∂D defined by D. in [5].at We QCBesides, as the we \ Λ with oneSarason sided limits eachdefine point then λk ∈PΛ. ∗ and QC. generated by PCC C ∗ -algebra let QC stands for the -algebra of quasicontinuous functions on 2 (D) standsbyforD.the Bergman of the unit then disk D, thatas is, the A ∂D defined Sarason in space [5]. We define P QC ∗ the C space of square integrable analytic -algebra generated by Pand C and QC. functions defined on D. , the algebra Toeplitz Our goal describe (D)tostands for Tthe Bergman space generated of the unit by disk D, that is, A2 is P QC operators whose symbols are certain extensions of functions the space of square integrable and analytic functions defined in on D. P QC acting A2 describe (D). Of course, function defined on ∂Dbycan be algebra generated Toeplitz Our goal on is to TP QC , athe extended to the disk symbols in many ways. The more naturalofextensions operators whose are certain extensions functions in are P the harmonic and ones. In the paper we describe QC acting on A2the (D).radial Of course, a function defined on ∂D the can be prove that this description not depend algebra TP QCtoand extended thewe disk in many ways. The moredoes natural extensions on the extension chosen. are the harmonic and the radial ones. In the paper we describe the algebra TP QC and we prove that this description does not depend 2010 Mathematics Subjectchosen. Classification: 32A36, 32A40, 32C15, 47B38, on the extension

47L80. 2010 Mathematics Classification: 32A36, 32A40, 32C15, 47B38, Toeplitz operator, Keywords and phrases:Subject Bergman spaces, C ∗ -algebras, 47L80. quasicontinuous symbols, piecewise continuous symbols. Keywords and phrases: Bergman spaces, C ∗ -algebras, Toeplitz operator, quasicontinuous symbols, piecewise continuous symbols.

1

Introduction

We1consider the C ∗ -algebra of quasicontinuous functions QC, which Introduction consists of all functions f : ∂D → C such that both f and its complex We consider the C ∗ -algebra of quasicontinuous functions QC, which ∗ This paper is part of the doctoral thesis of Breitner Ocampo under the supervision consists of all functions f : ∂D → C such that both f and its complex of Dr. Nikolai Vasilevski at the Mathematics Department of Cinvestav-IPN.

∗ This paper is part of the doctoral thesis of Breitner Ocampo under the supervision of Dr. Nikolai Vasilevski at the Mathematics Department of Cinvestav-IPN.

19

19

20

Breitner Ocampo

conjugate f¯ belong to H ∞ + C. Here H ∞ denotes the set (algebra) of boundary functions for bounded analytic functions on the unit disk D, and C stands for the algebra of continuous functions on ∂D. The space QC has two natural extensions to the disk, namely, the radial and the harmonic extension, we denote these extensions by QCR and QCH , respectively. We use A2 (D) to denote the Bergman space of L2 (D) which consists in all analytic functions. For A2 (D) ⊂ L2 (D), we denote by BD the Bergman projection BD : L2 (D) → A2 (D). Let K denote the ideal of compact operators acting on A2 (D). Recall that, for a bounded function f on D, the Toeplitz operator Tf acting on A2 (D) is defined by the formula Tf (g) = BD (f g). For a linear subespace A ⊂ L∞ (D) we denote by TA the (closed) operator algebra generated by Toeplitz operators with defining simbols in A. In this paper we describe the Calkin algebras TQCR /K and TQCH /K. We use the characterization of QC as the set of bounded functions with vanishing mean oscillation to prove that the Calkin algebras TQCR /K and TQCH /K are commutative, moreover TQCR ∼ = TQCH . For a finite set of points Λ := {λ1 , . . . , λn } of ∂D, we define the space of piecewise continuous functions P C := P CΛ as the algebra of continuous functions on ∂D \ Λ with one sided limits at each point λk ∈ Λ. We denote by P QC the C ∗ -algebra generated by both P C and QC. We use an extension of P QC to the disk and thus define the Toeplitz operator algebra TP QC ⊂ B(A2 (D)). There are several ways to extend the functions in P QC to D; two of them are: the radial extension, P QCR , and the harmonic extension, P QCH . The main goal of this paper is the description of the Calkin algebra TP QC /K, which is stated in Theorem 3.15. Finally, in Section 4, we prove that the result does not depend on the extension chosen for P QC, that is, TP QCR = TP QCH .

2

Preliminaries

First of all, we set some notation that will be used throughout the paper. Any mathematical symbol not described here will be used in its more common sense, · A stands for the norm in the space A. We denote by D the unit disk and by ∂D its boundary, the unit circle. The sets D and ∂D are endowed with the standard topology and with the Lebesgue measures dz = dxdy and dθ , where the point z = x + iy belongs to D and eiθ belongs to ∂D. All the functions in the paper are considered as

Toeplitz operators with piecewise quasicontinuous symbols

21

complex-valued. This section includes some basic facts about the space of Vanishing Mean Oscillation functions on ∂D, denoted here by V M O. The importance of this space lies in the fact that QC = V M O ∩ L∞ ( see [4]). For the convenience of the reader we recall the relevant material from [5] omitting proofs, thus making the exposition self contained. We define the following spaces of functions on ∂D: • L∞ := L∞ (∂D) = the algebra of bounded measurable functions f : ∂D → C, • H ∞ := H ∞ (∂D) = the algebra of radial limits of bounded analytic functions defined on D, • C := C(∂D) = the algebra continuous functions on ∂D.

Definition 2.1. [5, page 818] We define the C ∗ -algebra of quasicontinuous functions QC as the algebra of all bounded functions f on ∂D, such that, both f and its complex conjugate f¯ belong to H ∞ + C, that is; ∞ QC := (H ∞ + C) ∩ H + C .

Some of the statements below are formulated for segments in the real line, but they can also be formulated for arcs in ∂D. By an interval on R we always mean a finite interval. The length of the interval I will be denoted by |I|. For f ∈ L1 (I), the average of f over I is given by −1 f (t)dt. (1) I(f ) := |I| I

For a > 0, let Ma (f, I) :=

1 J⊂I,|J|<a |J| sup

J

|f (t) − J(f )|dt.

Note that 0 ≤ Ma (f, I) ≤ Mb (f, I) if a ≤ b, then let M0 (f, I) := lim Ma (f, I).

a→0

Definition 2.2. [5, page 81] A function f ∈ L1 (I) is of vanishing mean oscillation in the interval I ( or the arc I), if M0 (f, I) = 0. The set of all vanishing mean oscillation functions on I is denoted by V M O(I). In particular, if we replace I by ∂D in definitions above we get V M O := V M O(∂D).

22

Breitner Ocampo

A useful characterization of the space V M O is as follows: a function f belongs to V M O if and only if for any > 0 there exists δ > 0, depending on , such that −2 |J| |f (t) − f (s)|dsdt < , J

J

for every interval J ⊂ I with |J| < δ. Definition 2.3. [5, Page 818] Let f be an integrable function defined in an open interval containing the point Îť. We define the integral gap of f at Îť by Îť+δ Îť −1 −1 γΝ (f ) := lim sup δ f (t)dt − δ f (t)dt . δ→0 Îť

Îťâˆ’δ

Obviously, if f belongs to V M O(I), then γΝ (f ) = 0 for each interior point Ν of I. The most important use of Definition 2.3 is stablished in the following lemma: Lemma 2.4. [5, Lemma 2] Let I = (a, b) be an open interval, Ν a point of I, and f a function on I which belongs to both V M O((a, Ν)) and V M O((Ν, b)). If γΝ (f ) = 0, then f belongs to V M O(I).

We denote by M (QC) the space of all non-trivial multiplicative linear functionals on QC, endowed with the Gelfand topology. In the same way define M (C) and identify it with ∂D via the evaluation functionals. Since C is a subset of QC, every functional in M (QC) induces, by restriction, a functional in C. Here and subsequently, f0 denotes the function f0 (Îť) = Îť. The Stone-Weierstrass theorem implies that f0 and the function f (Îť) = 1 generate the C ∗ -algebra of all continuous functions on ∂D. Definition 2.5. [5, Page 822] For every Îť ∈ ∂D, we denote by MÎť (QC) the set of all functionals x in M (QC) such that x(f0 ) = Îť, that is MÎť (QC) := {x ∈ M (QC) : x(f0 ) = f0 (Îť) = Îť}. In other words, x belongs to MÎť (QC) if the restriction of x to the continuous functions is the evaluation functional at the point Îť. Definition 2.6. [5, Page 822] We let MÎť+ (QC) denote the set of x ∈ MÎť (QC) with the property that f (x) = 0 whenever f in QC is a function such that lim f (t) = 0. MÎťâˆ’ (QC) is defined in an analogous way. t→Ν+

Toeplitz operators with piecewise quasicontinuous symbols

23

Let f be bounded function on ∂D. The harmonic extension of f to the unit disk is denoted by fH and is given by the formula 1 (2) fH (z) := fH (r, θ) := Pr (θ − λ)f (λ)dλ, 2π ∂D

where Pr (θ) := Re

1 + reiθ 1 − reiθ

=

1 − r2 1 − 2rcos(θ) + r2

is the Poisson kernel for the unit disk. For every point z in D we define a functional in QC by the following rule: z(f ) = fH (z), so, we consider D as a subset of the dual space of QC. Under this identification we have that the weak-star closure of D contains M (QC) [5, Lemma 7]. Lemma 2.7. Let f be a function in QC which is continuous at the point λ0 . Then x(f ) = f (λ0 ) for every functional x in Mλ0 (QC). Proof. Consider the case where the function f is continuous at λ0 and such that f (λ0 ) = 0. Let x be a point in Mλ0 (QC). For > 0 there is δ0 > 0 such that |f (λ)| < for all λ in the arc Vλ0 = (λ0 − δ0 , λ0 + δ0 ). The values taken by the Poisson extension of f should be small if we evaluate points in D of an open disk with center at λ0 , i.e, there is a δ1 such that |fH (z)| < /2 if dist(z, λ0 ) < δ1 and z ∈ D. Using 1 = min{δ0 , δ1 , } we construct a neighbourhood Vx in QC ∗ with parameters f, f0 , 1 . By Lemma 7 in [5], there is a z in D such that z ∈ Vx , that is |fH (z) − x(f )| < 1 < and |f0 (z) − f0 (λ0 )| = |z − λ0 | < 1 < δ1 . This implies that dist(z, λ0 ) ≤ δ1 and then |fH (z)| < /2. Now we estimate x(f ), |x(f )| ≤ |x(f ) − fH (z)| + |fH (z)| < , consequently x(f ) = 0. In the general case, when f (λ0 ) = 0, we apply the previous argument to the function g = f − f (λ0 ). For g we obtain 0 = x(g) = x(f ) − f (λ0 ) and then x(f ) = f (λ0 ) for all x ∈ Mλ0 (QC). For z = 0 in D, we let Iz denote the closed arc of ∂D whose center is z/|z| and whose length is 2π(1 − |z|). For completeness, I0 = ∂D.

24

Breitner Ocampo

Lemma 2.8. [5, Lemma 5] For f in QC and any positive number , there is a positive number δ such that |fH (z) − Iz (f )| < whenever 1 − |z| < δ. The average of a function f over an arc I defines a linear functional on QC. Let us identify each arc I with the “averaging” functional in QC. The set of all these functionals is denoted by G. By Lemma 7 in [5] and Lemma 2.8 we come to the following lemma. Lemma 2.9. [5, Page 822] M (QC) is the set of points in the weak-star ∗ closure of G ( denoted here by G ) which does not belong to G.

For λ ∈ ∂D we denote by Gλ0 the set of all arcs I in G with center at λ. Let Mλ0 (QC) be the set of functionals in Mλ (QC) that lie in the weak-star closure of Gλ0 . By Lemma 2.8, the set Mλ0 (QC) coincides with the set of functionals in Mλ (QC) that lie in the weak-star closure of the radius of D terminating at λ. In [5], D. Sarason splits the space Mλ (QC) into three sets: Mλ+ (QC) \ Mλ0 (QC),

Mλ− (QC) \ Mλ0 (QC)

and

Mλ0 (QC).

These three sets are mutually disjoint due to the next lemma: Lemma 2.10. [5, Lemma 8] Mλ+ (QC) ∪ Mλ− (QC) = Mλ (QC) and Mλ+ (QC) ∩ Mλ− (QC) = Mλ0 (QC). The result in Lemma 2.10 allows us to draw the maximal ideal space M (QC). We consider the unit circle as the interval [0, 2π], where the points 0 and 2π represent the same point. At each point λ in [0, 2π] we draw a segment representing the fiber Mλ (QC). The segment Mλ (QC) is splitted into two parts, the upper part Mλ+ (QC) and the lower part Mλ− (QC). Their intersection is Mλ0 (QC), the central part of the fiber.

Mλ0 (QC)

Mλ+ (QC)

0

2π Mλ− (QC)

Mλ (QC)

Figure 1: The maximal ideal space of QC.

Toeplitz operators with piecewise quasicontinuous symbols

3

25

Toeplitz operators with piecewise quasicontinuous symbols on the Bergman space

This section deals with Toeplitz operators with symbols in certain extension of P QC acting on A2 (D). The C ∗ -algebra P QC is generated by both, the space P C of piecewise continuous functions, and QC, the space of quasicontinuous functions, both extended from ∂D to the unit disk D. The main result of this section (Theorem 3.15) describes the Calkin algebra TˆP QC := TP QC /K as the C ∗ -algebra of continuous sections over a bundle ξ constructed from the operator algebra TP QC . Definition 3.1. Let Λ := {λ1 , λ2 , . . . , λn } be a fixed set of n different points on ∂D. Define P C := P CΛ as the set of continuous functions on ∂D \ Λ with one sided limits at every point λk in Λ. For a function a in P C we set − a+ k := lim a(λ) and ak := lim a(λ), λ→λ+ k

λ→λ− k

following the standard positive orientation of ∂D. Definition 3.2. P QC is defined as the C ∗ -algebra generated by P C and QC. Our interest is to describe a certain Toeplitz operator algebra acting on the Bergman space A2 (D). For this we need to extend the functions in P QC to the whole disk. There are two most natural ways of such extensions • the harmonic extension gH , given by the Poisson formula 2, • the radial extension gR , defined by gR (r, θ) = g(θ). In this section we use the radial extension, however, we emphasize that the main result of this paper does not depend on the extensions mentioned above (Theorem 4.11). Recall that the Bergman space A2 (D) is the closed subspace of L2 (D) which consists of all functions analytic in D. Being closed, the space A2 (D) has the orthogonal projection BD : L2 (D) → A2 (D), called the Bergman projection. Let K denote the ideal of compact operators acting on A2 (D). Given a function g in L∞ (D), the Toeplitz operator Tg : A2 (D) → 2 A (D) with generating symbol g is defined by Tg (f ) = BD (gf ).

26

Breitner Ocampo

In [7], K. Zhu describes the largest C ∗ -algebra Q ⊂ L∞ (D) such that the map ψ : Q → B(A2 (D))/K f → Tf + K,

is a C ∗ -algebra homomorphism. This algebra is closely related to QC because both can be described using spaces of vanishing mean oscillation functions. Definition 3.3. [7, Page 633] Consider Γ := {f ∈ L∞ (D) : Tf Tg − Tf g ∈ K for all g ∈ L∞ (D)}. ¯ ∩ Γ. Let Q := Γ For z in D, we define Sz := {w ∈ D : |w| ≥ |z| and

| arg(z) − arg(w)| ≤ 1 − |z|},

The area of Sz , denoted by |Sz |, is π(1 − |z|)2 (1 + |z|). Definition 3.4. [7, Page 621] A function f in L1 (D) belongs to V M O∂ (D), the space of functions with vanishing mean oscillation near the boundary of D, if 1 1 f (w) − lim f (u)dA(u) dA(w) = 0. |Sz | Sz |z|→1− |Sz | Sz

Theorem 3.5. [7, Theorem 13] The algebra Q is the set of bounded functions with vanishing mean oscillation near the boundary, i.e., Q = V M O∂ (D) ∩ L∞ (D). For the proof we refer the reader to [7].

Lemma 3.6. Let f be a function in QC. Then, the function fR belongs to Q. Proof. According to Theorem 3.5 and Definition 3.4, we need to estimate 1 dA(w). f (w) − 1 (3) f (u)dA(u) |Sz | Sz |Sz | Sz

Using polar coordinates we get that this quantity is equal to

Toeplitz operators with piecewise quasicontinuous symbols

(4)

2 |Iz |2

Iz

Iz

27

|f (θ) − f (φ)|dA(θ)dA(φ).

If z is close to the boundary, then the measure of |Iz | is small. Hence, the expresion in (4) goes to zero because f is in QC. This implies that the expresion in (3) goes to zero if |z| goes to 1, thus fR is in Q as required. In Lemma 4.5 we prove that the harmonic extension fH also belongs to Q, but the tools needed for the proof of this fact are not stablished yet. From now on, and until further notice, we use only the radial extension of a function in P QC. To simplify the notation, we use P QC to denote functions defined on ∂D as well as radial extensions of such functions. Moreover g will denote both, the function on ∂D and its radial extension to D. By TP QC we denote the C ∗ -algebra generated by Toeplitz operators with symbols in P QC. We use TˆP QC to denote the Calkin algebra TP QC /K. The main goal of this paper is to describe the C ∗ -algebra TˆP QC . We use the Douglas-Varella Local Principle (DVLP for short) to describe the C ∗ -algebra TˆP QC . A complete description of this principle can be found, for example, in [6, Chapter 1]. Let A be a C ∗ -algebra with identity, Z be some of its central C ∗ subalgebras with the same identity, T be the compact of maximal ideals of Z. Furthermore, let Jt be the maximal ideal of Z corresponding to the point t ∈ T , and J(t) := Jt · A be the two sided closed ideal in the algebra A generated by Jt . We define Et := A/J(t) as the local algebra at the point t. [a]t stands for the class of the element a in the quotient algebra Et . Two elements a, b of A are say locally equivalents at the point t ∈ T if [a]t = [b]t in Et . Using the spaces Et E := t∈T

and T , there is a standard procedure to construct the C ∗ -bundle ξ = (p, E, T ), where p : E → T is a projection such that p|Et = t. This procedure gives to E a compatible topology such that the function a ˆ: T → E with a ˆ(t) = [a]t ∈ Et is continuous for each a in A.

28

Breitner Ocampo

A function γ : T → E is called a section of the C ∗ -bundle ξ, if p(γ(t)) = t. Let Γ(ξ) denote the C ∗ -algebra of all continuous sections defined on ξ. Theorem 3.7 (Douglas-Varela Local Principle). The C ∗ -algebra A is isomorphic and isometric to the C ∗ -algebra Γ(ξ), where ξ is the C ∗ bundle constructed from A and its central algebra Z. Lemma 3.6 and the results in [7] imply that the quotient TˆQC = TQC /K is a commutative C ∗ -subalgebra of TˆP QC . Thus we use TˆQC as the central algebra needed to apply the DVLP in the description of TˆP QC . The algebra TˆQC can be identified with QC TˆQC = {Tf + K|f ∈ QC}, hence we localize by points in M (QC). We first construct the system of ideals parametrized by points x in M (QC). Definition 3.8. For every point x ∈ M (QC), we define the maximal ideal of TˆQC , Jx := {f ∈ QC : f (x) = 0} = {Tf + K|f (x) = 0}. The ideal J(x) is defined as the set Jx · TP QC /K. We set the notation TˆP QC (x) := TˆP QC /J(x) for the local algebra at the point x. The class of the element Tf +K ∈ Tˆ in the quotient algebra Tˆ (x) shall be denoted by [Tˆf ]x , in order to simplify the notation we say “Tf is locally equivalent...” instead of “the class of [Tˆf ]x is loccaly equivalent...” Lemma 3.9. Let f be a function in QC and x a point of M (QC). The Toeplitz operator Tf is locally equivalent, at the point x, to the complex number f (x) (realized as the operator f (x)I). Proof. Let x be a point in M (QC) and f be a function in QC. The function f − f (x) belongs to J(x), thus, the operator Tf − Tf (x) = Tf −f (x) is zero in TˆP QC (x). This means that the operator Tf is locally equivalent to the operator Tf (x) = f (x)I and then, the operator Tf is locally equivalent to the complex number f (x). Lemma 3.10. Let x be a point of Mλ (QC) with λ ∈ / Λ and a be a function in P C. Then, the Toeplitz operator Ta , in the local algebra TˆP QC (x), is equivalent to the complex number a(λ) (realized as the operator a(λ)I).

Toeplitz operators with piecewise quasicontinuous symbols

29

The proof is very similar to the proof of Lemma 3.9 and is omitted. For the case when x ∈ MÎťk (QC), we use Lemma 2.10 to split the fiber MÎťk (QC) into three disjoints sets: MÎť+k (QC) \ MÎť0k (QC), MÎťâˆ’k (QC) \ MÎť0k (QC) and MÎť0k (QC). Lemma 3.11. Let x be a point of MÎť+k (QC) \ MÎť0k (QC) and a be a function in P C. Then, the Toeplitz operator Ta , in the local algebra TˆP QC (x), is equivalent to the complex number a+ k (realized as the oper+ ator ak I). Proof. Let a be a function for which a+ k = 0. If x belongs to MÎť+k (QC) \ MÎť0k (QC), then x belongs to MÎť+k (QC) and does not belong to MÎťâˆ’k (QC). This implies the existence of a function g in QC such that lim g(Îť) = 0

Îťâ†’Îťâˆ’ k

and g(x) = 1. The product ag is continuous at Îťk and ag(Îťk ) = 0. The difference Ta −Tag can be rewritten as T(1−g)a = T1−g Ta +K where K is a compact operator. Since the function 1 − g vanishes at x, T1−g belongs to Jx , and then Ta − Tag belongs to J(x). From this we conclude that the Toeplitz operator with symbol a is locally equivalent to the Toeplitz operator with symbol ag. At the same time, the Toeplitz operator Tag is locally equivalent to the complex number 0 = ag(Îťk ), hence, the operator Ta is locally equivalent to the complex number a+ k = 0. For the general case, if the function a in P C has limit a+ k = 0, we construct the function b(Îť) = a(Îť)−a+ . The function b has lateral limit k = 0, fulfilling the initial assumption of the proof. By the first part of b+ k + the proof, the Toeplitz operator Tb = Ta −ak I is locally equivalent to the complex number 0, thus the Toeplitz operator Ta is locally equivalent to the complex number a+ k. Similarly the following lemma holds: Lemma 3.12. Let x be a point of MÎťâˆ’k (QC) \ MÎť0k (QC) and a be a function in P C. Then, the Toeplitz operator Ta , in the local algebra TˆP QC (x), is equivalent to the complex number a− k (realized as the operI). ator a− k

30

Breitner Ocampo

Now we analize the case when x belongs to central part of the fiber MÎťk (QC), i.e, x ∈ MÎť0k (QC). For this case we use some results regarding Toeplitz operators with zero-order homogeneous symbols defined in the upper half plane Î . We consider A2 (Î ) as the Bergman space of Î , that is, the (closed) space of square integrable and analytic functions on Î . Let BÎ stands for the Bergman projection BÎ : L2 (Î ) → A2 (Î ). Denote by A∞ the C∗ -algebra of bounded mesureable homogeneous functions on Î of zero-order, or functions depending only in the polar coordinate θ. We introduce the Toeplitz operator algebra T (A∞ ) generated by all Toeplitz operators Ta : φ ∈ A2 (Î ) → BÎ (aφ) ∈ A2 (Î ) with defining symbols a(r, θ) = a(θ) ∈ A∞ . Theorem 3.13. [6, Theorem 7.2.1] For a = a(θ) ∈ A∞ , the Toeplitz operator Ta acting in A2 (Î ) is unitary equivalent to the multiplication operator Îła I acting on L2 (R). The function Îła (s) is given by Ď€ 2s a(θ)e−2sθ dθ. Îła (s) = 1 − e−2sĎ€ 0 + Let ∂D+ k denote the upper half of the circunference and Dk the upper half of the disk D both determined by the diameter passing through Îťk + − + and âˆ’Îťk . Denote by ∂D− k and Dk the complement of ∂Dk and Dk , respectively. Let H be a function in P C with lateral limits Hk+ and Hk− , we construct the function h in P C such that h = Hk+ in ∂D+ and h = Hk+ in ∂D− . The function H − h is continuous at Îťk and (H − h)(Îťk ) = 0. For any point x ∈ MÎť0k , the Toeplitz operator TH−h belongs to J(x) and thus TH and Th are locally equivalent at the point x. The previous paragraph implies that the C∗ -algebra generated by TH in TˆP QC (x) depends only on the values Hk+ and Hk− . To describe the local algebra at x, we need to analize the algebra generated by the Toeplitz operator which symbol h is constant on both ∂D+ and ∂D− . The radial extension of such function h is the function which is constant in D+ with value A and constant in D− with value B for some complex constants A and B. Let φ be a M¨ obius transformation which sends the upper half plane to the unit disk and such that: φ(0) = Îťk , φ(i) = 0 and φ(∞) = âˆ’Îťk .

Toeplitz operators with piecewise quasicontinuous symbols

31

Using the function φ we construct a unitary transformation W which sends L2 (D) onto L2 (Π). Under the unitary transformation W , the Toeplitz operator with symbol h, acting on A2 (D), is unitary equivalent to the Toeplitz operator Th(φ(w)) acting on A2 (Π). The corresponding symbol h(φ(w)) is a homogeneous function of zero-order. By Theorem 3.13, the Toeplitz operator with symbol h(φ(w)), acting on A2 (Π), is unitary equivalent to the multiplication operator by the function γh(φ(w)) , acting on L2 (R). Following the unitary equivalences we deduce that Th is unitary equivalent to γh(φ(w)) . ¯= By Corollary 7.2.2 in [6], the function γh(φ) (s) is continuous in R R ∪ {−∞, +∞}, the two point compactification of R; furthermore, for 1 1−e−sπ The the function h = χD+ we have γh(φ) (s) = 1−e −2sπ = 1+e−2sπ . k function γh(φ) (s) and the identity function 1 generate the algebra of ¯ [6, Corollary 7.2.6]. continuous functions on R Recall that all piecewise constant functions are generated by linear combinations of the identity and the function χD+ . Thus, using the k change of variables 1 , t= 1 + e−2sπ ¯ we conclude that the which is a homeomorphism between [0, 1] and R, ˆ local algebra TP QC (x) is isomorphic to C[0, 1] for every x ∈ Mλ0k (QC); further, such isomorphism, denoted here by ψ, acts on the generator Tχ + as follows: D

k

Tχ

D+ k

→ t.

This implies that the Toeplitz operator with symbol a in P C is sent to + C([0, 1]), via ψ, to the function a− k (1 − t) + ak t. Thus we come to the following lemma. Lemma 3.14. If x belongs to Mλ0k (QC), then the local algebra generated by the Toeplitz operators with symbols in P QC is isometric and isomorphic to the algebra of all continuous functions in [0, 1]. With the set M (QC), we construct the C∗ -bundle ξPQC := (p, E, M (QC)). We use the description of the local algebras given by Lemmas 3.10, 3.11, 3.12 and 3.14 to construct the bundle E := Ex x∈M (QC)

32

Breitner Ocampo

where • Ex = C, if x ∈ Mλ (QC) with λ ∈ / Λ, • Ex = C, if x ∈ Mλ+k (QC) \ Mλ0k (QC), λk ∈ Λ, • Ex = C, if x ∈ Mλ−k (QC) \ Mλ0k (QC), λk ∈ Λ, • Ex = C([0, 1]), if x ∈ Mλ0k (QC), λk ∈ Λ. The function p is the natural projection from E to M (QC). Let Γ(ξP QC ) denote the algebra of all continuous sections of the bundle ξP QC . Applying the DVLP (Theorem 3.7) we get the following theorem: Theorem 3.15. The C∗ algebra TˆP QC is isometric and isomorphic to the C∗ -algebra of continuous sections over the C∗ -bundle ξPQC . As a corollary of Theorem 3.15, the algebra TˆP QC is commutative, thus there exists a compact space X = M (TˆP QC ), such that TˆP QC ∼ = C(X) = C(M (TˆP QC )). The compact space M (TˆP QC ) can be constructed using the irreducible representations of TˆP QC . ˆ be the set ∂D cut by the points λk of Λ. The pair of points Let ∂D ˆ which correspond to the point λk will be denoted by λ+ and λ− , of ∂D k k following the positive orientation of ∂D. Let I n := ni=1 [0, 1]k be the disjoint union of n copies of the interval [0, 1]. ˆ and In with the point identification Denote by Σ the union of ∂D λ− k ≡ 0k

λ+ k ≡ 1k ,

where 0k and 1k are the boundary points of [0, 1]k , k = 1, . . . , n. Let M (TˆP QC ) := Mλ (TP QC ) where each fiber corresponds to λ∈Σ

ˆ Mλ (TˆP QC ) :=Mλ (QC) if λ ∈ ∂ D, Mλ+ (TˆP QC ) := Mλ+k (QC) \ Mλ−k (QC) ∪ Mλ0k (QC), k Mλ− (TˆP QC ) := Mλ−k (QC) \ Mλ+k (QC) ∪ Mλ0k (QC), k

Mt (TˆP QC ) :=Mλ0k (QC) if t ∈ (0, 1)k ,

λk ∈ /Λ

λk ∈ Λ, λk ∈ Λ, k = 1, . . . , n.

With the help of Figure 1, we draw the maximal ideal space for TˆP QC . The idea is to duplicate the set Mλ0 (QC) and then connect this two copies by the interval [0, 1].

Toeplitz operators with piecewise quasicontinuous symbols

33

Mλ− (T P QC ) i

Mt (T P QC )

Mλ−i (QC)\Mλ+i (QC)

Mλ0 (QC)

Mλ+i (QC)\Mλ−i (QC)

Mλ (T P QC ) = Mλ (QC)

Mλ+ (T P QC ) i

Figure 2: The maximal ideal space of T P QC .

We use the topology of M (QC) in order to describe the topology of M (TˆP QC ). We only describe the topology of the fibers Mλ± (TˆP QC ) k and Mt (TˆP QC ), since the topology on the other fibers corresponds to the topology of Mλ (QC). For x in M (QC), let Ω(x) denote the family of open neighbourhoods of x. For x ∈ Mλ (QC) and N in Ω(x), let Nλ = N ∩ Mλ (QC), and let Nλ+ and Nλ− denote the sets of points in − N that lie above the semicircles ∂D+ k and ∂Dk , respectively. Consider the fiber Mλ+ (TˆP QC ). The sets N in Ω(x) satisfying N = k

Nλk ∪Nλ+ form neighbourhoods of x ∈ Mλ+k (QC)\Mλ−k (QC). Let Ω+ (x) k be the set of neighbourhoods N in Ω(x) satisfying N = Nλ ∪ Nλ+ . The sets (Nλk × (1 − , 1]) ∪ Nλ+ k

N ∈ Ω+ (x),

and

0 < < 1,

form open neighbourhoods of points x in Mλ0k (QC). The open neighbourhoods for points in the fiber Mλ− (TˆP QC ) are k constructed analogously. The sets N in Ω(x) satisfying N = Nλk ∪ Nλ− form neighbourhoods k

of x ∈ Mλ−k (QC) \ Mλ+k (QC). The sets (Nλk × [0, )) ∪ Nλ− k

N ∈ Ω− (x),

and

0 < < 1,

form open neighbourhoods of points x in Mλ0k (QC). Each set Mλ0k (QC) × (0, 1) is open in M (TˆP QC ) and carries the product topology.

34

Breitner Ocampo

Theorem 3.16. Let X := M (TˆP QC ) as described above. The algebra TˆP QC is isomorphic to the algebra of continuous functions over X, the isomorphism acts on the generators in the following way: • For generators which symbols is   a(λ),    a+ , k Φ(Tˆa )(x) = −  a  k,    − ak (1 − t) + a+ k t,

a function a in P C if x ∈ Mλ (TˆP QC ) with λ = λk ; if x ∈ Mλ+ (TˆP QC ); k if x ∈ M − (TˆP QC ); λk

if x ∈ Mt (TˆP QC ).

• For generators which symbols are functions f in QC, Φ(Tˆf )(x) = f (x).

4

Independence of the result on the extension chosen

In this section we prove that the description of the algebra TˆP QC does not depend of the extension chosen for functions in P QC. Recall that P QC is the algebra generated by P C and QC. This algebra is defined on ∂D and then extended to the whole disk by two different ways: • the harmonic extension gH given by the Poisson formula 2, • the radial extension gR , defined by gR (r, θ) = g(θ). Let a be a function in P C. At the point x ∈ Mλ (QC), for λ ∈ / Λ; the Toeplitz operator TaR is locally equivalent to the complex number a(λ). The same still true if we use the harmonic extension aH . For points x in Mλ+k (QC) \ Mλ0k (QC) (respectively Mλ−k (QC) \ Mλ0k (QC)), the Toeplitz operators TaR and TaH are equivalent to the number a+ k (Respectively a− ), and then, the local algebras are the same. k ˆ be a Now, we analize the case when x belongs to Mλ0k (QC). Let a + function in P C, we construct a function a such that a = a ˆk in ∂D+ k − − and a = a ˆk in ∂Dk . The Toeplitz operator with symbol a ˆH is locally equivalent to TaH . As in Section 2, we use a M¨ obius transformation φ to generate a 2 unitary operator between L (D) and L2 (Π). For the function a in P C described earlier, the function aH (φ(z)) is harmonic in Π and corresponds to the harmonic extension of a(φ(t)).

Toeplitz operators with piecewise quasicontinuous symbols

35

+ + θ − The harmonic extension of a(φ(t)) is aÎ H := Ď€ (ak −ak )−ak , which is a zero-order homogeneous function on Î . By Theorem 3.13, the Toeplitz operator TaÎ is unitary equivalent to the multiplication operator ÎłaÎ . H H The function ÎłaÎ is given by H

ÎłaÎ = A H

1 1 − 2sĎ€ e−2sĎ€ − 1

+ B,

for suitable complex constants A and B. Corollary 7.2.7 of [6] shows that the algebra generated by ÎłaÎ and the identity is the algebra of H ÂŻ continuous functions on R. Following the unitary equivalences from TaH to ÎłaÎ and making H a change of variables, we have that the algebra generated by TaH is isomorphic to the algebra of continuous functions over the segment [0, 1]. We already know, from Theorem 3.14, that the Toeplitz operator with symbol aR generates the algebra of continuous functions over [0, 1] as well, so, locally, the algebras genereated by TaH and TaR are the same. We have thus proved Theorem 4.1. Consider the algebra P C defined on ∂D and its extensions P CR and P CH . The local algebras TˆP CR (x) and TˆP CH (x) are the same. To show the same theorem for functions f in QC we need to stablish some definitions related to the space Q in Definition 3.3. Further information on the theorems and definitions below can be found in [7]. Definition 4.2. For a function g ∈ L∞ (D) we define its Berezin transform gËœ by the formula gËœ(z) :=

D

g(w)

1 − |w|2 dA(w). (1 − z w) ÂŻ 2

g ∞ ≤ g ∞ . Note that g˜ belongs to L∞ (D) and ˜ Definition 4.3. Define B as the set of bounded functions on D such that its Berezin transform goes to zero as z approaches to the boundary of D, that is, B := {f ∈ L∞ (D) : lim f˜(z) = 0}. |z|→1

36

Breitner Ocampo

In [1], S. Axler and D. Zheng proved that a Toeplitz operator Tg , with bounded symbol g, is compact if and only if g is in B. The next lemma is due to D. Sarason and is a combination of some results in [5]. Theorem 4.4. The set Q in Definition 3.3 is described as Q = {f ∈ L∞ (D) :

|2 (z) − |f˜(z)|2 = 0}. lim |f

|z|→1

The set B ∊ Q is an ideal of Q and, for f ∈ Q, the Toeplitz operator Tf is compact if and only if f belongs to B ∊ Q. Lemma 4.5. For a function f in QC, the function fH belongs to Q. Proof. For this proof we use two facts: 1. The Berezin transform of a harmonic function is the function itself, in our case, f˜H = fH . 2. By [3], the harmonic extension is asymptotically multiplicative in QC, that is lim |f |2H (z) − |fH (z)|2 = 0. |z|→1−

Now we proceed with the proof: 2 2 2 2 2 (z) − |f | (z)| ≤ |f | (z) − |f | (z) |f H + |f |H (z) − |fH (z)|2 H H H 2 2 ≤ |fH | (z) − |f |H (z) + |f |2H (z) − |fH (z)|2 ,

the last two sumands goes to zero as z approaches to the boundary ∂D; the later because of item 2, and the former is due to items 2 and 1. Finally, using Theorem 4.4, we have that fH is in Q.

Definition 4.6 (page 626, [7]). For each point z in D we define 1 − |z| . Sz := w ∈ D : |w| ≼ |z| and | arg(z) − arg(w)| ≤ 2 Definition 4.7 (page 627, [7]). For a function f in L∞ (D) define 1 ˆ f (z) := f (w)dA(w). |Sz | Sz

Toeplitz operators with piecewise quasicontinuous symbols

37

Definition 4.8 (page 626, [7]). Let f be in L∞ (D, dA). We say f is in ESV (D) if and only if for any > 0, and σ ∈ (0, 1), there exists δ0 > 0 such that |f (z)−f (w)| < whenever w ∈ Sz and |z|, |w| ∈ [1−δ, 1−δσ], with δ < δ0 . The notation ESV (D) means eventually slowly varying and was introduced by C. Berger and L. Coburn in [2]. Theorem 4.9. [7, Theorem 5] Q = ESV + Q ∩ B. A decomposition is given by f = fˆ + (f − fˆ). Moreover ESV (D) ∩ B = {f ∈ L∞ (D) | f (z) → 0 as |z| → 1− }. We calculate fˆR and get fˆR (z) = Iz (f ). Then, Theorem 4.9 gives us the decomposition fR (z) = Iz (f ) + (fR (z) − Iz (f )), where Iz (f ) belongs to ESV (D) and fR (z) − Iz (f ) belongs to Q ∩ B. Lemma 4.10. Consider the function f in QC. The Toeplitz operator with symbol fR − fH is compact. Proof. We write fR (z) − fH (z) = (Iz (f ) − fH (z)) + (fR (z) − Iz (f )). The first summand goes to zero as |z| goes to 1 by Theorem 2.8. Then by Theorem 4.9, the function Iz (f ) − fH (z) belongs to ESV (D) ∩ B. By the decomposition of Q as ESV (D)+Q∩B we have that (fR (z)−Iz (f )) belongs to Q ∩ B. In summary, the function fR (z) − fH (z) belongs to Q ∩ B and then the Toeplitz operator with symbol fR − fH is compact. Now we stablish the main result of this section: the algebra described in Theorem 3.15 does not depend on the extension chosen for the symbols in P QC. Theorem 4.11. Let P QCR and P QCH denote, respectively, the radial and the harmonic extension to the disk of functions in P QC. Then, the Calkin algebras TP QCR /K and TP QCH /K are the same. Proof. The proof follows from Theorem 4.1 and Lemma 4.10. Ocampo, B. Departmento de Matem´ aticas, Centro de Investigaci´ on y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07360 M´exico, D.F. bocampo@math.cinvestav.mx

38

Breitner Ocampo

References [1] Axler S.; Zheng D., Compact operators via the Berezin transform, Indiana Univ. Math. J. 47:2 (1998), 387–400. [2] Berger C.A.; Coburn L. A., Toeplitz operators and quantum mechanics, J. Funct. Anal. 68:3 (1986), 273–299. [3] Sarason D., Algebras of functions on the unit circle, Bull. Amer. Math. Soc. 79 (1973), 286–299. [4] Sarason D., Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. [5] Sarason D., Toeplitz operators with piecewise quasicontinuous symbols, Indiana Univ. Math. J. 26:5 (1977), 817–838. [6] Vasilevski N. L., Commutative Algebras of Toeplitz Operators on the Bergman Space, vol. 185 of Operator Theory: Advances and Applications, Birkh¨ auser Verlag, Basel, 2008. [7] Zhu K. H., VMO, ESV, and Toeplitz operators on the Bergman space, Trans. Amer. Math. Soc. 302:2 (1987), 617–646.

Morfismos se imprime en el taller de reproducci´ on del Departamento de Matem´ aticas del Cinvestav, localizado en Avenida Instituto Polit´ecnico Nacional 2508, Colonia San Pedro Zacatenco, C.P. 07360, M´exico, D.F. Este n´ umero se termin´ o de imprimir en el mes de julio de 2015. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm. consta de 50 ejemplares con pasta tintoreto color verde.

Apoyo t´ecnico: Omar Hern´ andez Orozco.

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