Determinants in K-theory and operator algebras by Joseph Migler B.S., University of California, Santa Barbara, 2010 M.A., University of Colorado, Boulder, 2012 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics 2015 This thesis entitled: Determinants in K-theory and operator algebras written by Joseph Migler has been approved for the Department of Mathematics Professor Alexander Gorokhovsky Professor Carla Farsi Professor Judith Packer Professor Arlan Ramsay Professor Bahram Rangipour Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Migler, Joseph (Ph.D., Mathematics) Determinants in K-theory and operator algebras Thesis directed by Professor Alexander Gorokhovsky A determinant in algebraic K-theory is associated to any two Fredholm operators which commute modulo the trace class. This invariant is defined in terms of the Fredholm determinant, which itself extends the usual notion of matrix determinant. On the other hand, one may consider a homologically defined invariant known as joint torsion. This thesis answers in the affirmative a conjecture of R. Carey and J. Pincus, namely that these two invariants agree. The strategy is to analyze how joint torsion transforms under an action by certain groups of pairs of invertible operators. This allows one to reduce the calculation to the finite dimensional setting, where joint torsion is shown to be trivial. The equality implies that joint torsion has continuity properties, satisfies the expected Steinberg relations, and depends only on the images of the operators modulo trace class. Moreover, we show that the determinant invariant of two commuting operators can be computed in terms of finite dimensional data. The second main goal of this thesis is to investigate how joint torsion behaves under the functional calculus. We study the extent to which the functional calculus commutes, modulo operator ideals, with projections in a finitely summable Fredholm module. As an application, we recover in particular some results of R. Carey and J. Pincus on determinants of Toeplitz operators and Tate tame symbols. In addition, we obtain variational formulas and explicit integral formulas for joint torsion. iv Acknowledgements I am indebted to Alexander Gorokhovsky, whom it is my honor to have as an advisor. His endless patience and mathematical insight are a source of inspiration for me. I am grateful for his guidance, for countless enlightening conversations, and for his generosity in sharing both his time and knowledge. His help was never more than a hallway away. I wish to thank Carla Farsi, Judith Packer, Arlan Ramsay, and Bahram Rangipour for serving on my dissertation committee. I would also like to thank Carla Farsi for her encouragement and guidance during several classes I taught as a graduate student. I wish to thank Judith Packer for organizing many fascinating seminars which I had the pleasure of attending. I would like to thank Arlan Ramsay for his careful reading of this dissertation. I wish to express my gratitude to Bahram Rangipour for generously sharing his time with me during his stay in Boulder. I am grateful for the help of many mathematicians, in particular Richard Carey, Guillermo Corti~nas,Ra´ulCurto, J¨orgEschmeier, Karl Gustafson, Nigel Higson, Matthias Lesch, Jens Kaad, Jerry Kaminker, Ryszard Nest, Joel Pincus, Rapha¨elPonge, Dan Voiculescu, and Mariusz Wodz- icki. Their comments on the subject of this thesis and related topics have greatly enhanced my understanding. Finally I wish to thank my family { this degree is as much theirs as it is mine. I would like to thank my parents for their support at every stage of my education. They taught me integrity and most everything I know today. I am especially grateful for their encouragement during my years of studying mathematics. I would like to thank my sisters and their families for the joy they have shared with me and for their help throughout my life. Contents Chapter 1 Introduction 1 2 Preliminaries 8 2.1 Fredholm theory . 8 2.1.1 Schatten classes . 8 2.1.2 Fredholm determinant . 10 2.1.3 Fredholm index . 13 2.2 Algebraic K-theory . 14 2.2.1 Algebraic K0 .................................... 14 2.2.2 Algebraic K1 .................................... 16 2.2.3 Algebraic K2 .................................... 17 2.2.4 The determinant invariant . 18 2.3 Toeplitz operators . 21 2.3.1 Index and spectral theory . 22 2.3.2 Commutators of Toeplitz operators . 22 3 Joint torsion 24 3.1 Background: The work of Carey and Pincus . 24 3.2 Commuting operators . 27 3.2.1 Algebraic torsion . 27 vi 3.2.2 Joint torsion of commuting operators . 28 3.2.3 The case of two commuting operators . 29 3.3 Almost commuting operators . 31 3.3.1 Perturbation vectors . 32 3.3.2 Joint torsion of two almost commuting operators . 35 4 Equality 36 4.1 The finite dimensional case . 36 4.1.1 Torsion of a double complex . 37 4.1.2 Joint torsion in finite dimensions . 39 4.2 Factorization results . 44 4.2.1 Perturbation vectors . 44 4.2.2 Joint torsion . 47 4.3 Main results . 51 4.3.1 Group actions on commuting squares . 51 4.3.2 A proof of equality . 52 4.4 Applications . 53 4.4.1 Properties of the determinant invariant . 53 4.4.2 Properties of joint torsion . 54 4.4.3 Joint torsion and Hilbert-Schmidt operators . 59 5 Explicit formulas for joint torsion 62 5.1 Transformation rules for joint torsion . 62 5.1.1 Commutators . 62 5.1.2 Perturbations . 64 5.1.3 Joint torsion . 66 5.1.4 Positivity . 68 5.2 Fredholm modules . 71 vii 5.3 Toeplitz operators and tame symbols . 76 5.3.1 H1 symbols . 76 5.3.2 L1 symbols . 79 5.3.3 An integral formula . 82 Bibliography 85 Appendix A The existence of perturbations 89 Chapter 1 Introduction Let A and B be two invertible operators on a complex separable Hilbert space H that com- mute modulo the trace class L1(H), which consists of compact operators with summable singular values. The multiplicative commutator ABA−1B−1 is an invertible determinant class operator, that is it differs from the identity by a trace class operator, and therefore has a nonzero Fredholm determinant. The assignment (A; B) 7! det(ABA−1B−1) is bimultiplicative and skew-symmetric. Moreover, det(ABA−1B−1) = det(A~B~A~−1B~−1) for any invertible trace class perturbations A~ and B~ of A and B, respectively. L. Brown observed in [8] that this is a special case of a more general phenomenon. Indeed, any two bounded Fredholm operators A and B that commute modulo trace class have invertible commuting images a and b, respectively, in the quotient B=L1. Here B = B(H) is the algebra of bounded linear operators on H, and L1 = L1(H) is the ideal of trace class operators. As we shall see in Section 2.2, these images determine a canonical element fa; bg, known as a Steinberg 1 symbol, in the second algebraic K-group K2(B=L ). Consequently, there is a determinant invariant d(a; b) = det @fa; bg 2 C×. Brown showed that when A and B are invertible, this invariant recovers the classical determinant of a multiplicative commutator, that is, d(a; b) = det(ABA−1B−1). This immediately yields the remarkable fact that the determinant of the multiplicative commutator depends only on the Steinberg symbol in K-theory. See also the paper of J. W. Helton and R. Howe [32] for work in this area around the same time. As an example, consider two non-vanishing smooth functions f and g on the unit circle. Then 2 one may form the Toeplitz operators Tf and Tg, which are compressions of multiplication operators by f and g on L2(S1) to the Hardy space H2(S1) (Section 2.3). Since f and g are non-vanishing and continuous, it turns out that Tf and Tg are Fredholm. Moreover, since f and g are smooth, we shall see that Tf and Tg commute with each other modulo trace class. In [15] R. Carey and J. Pincus obtain an integral formula for the determinant invariant of Tf and Tg: Z Z 1 1 1 d(Tf + L ;Tg + L ) = exp log f d(log g) − log g(p) d(log f) : (1.1) 2πi S1 S1 The integrals are taken counterclockwise starting at any point p 2 S1. If h(eiθ) = jh(eiθ)jeiφ(θ) for a continuous real-valued function φ, then we take log h(eiθ) = log jhj + iφ(θ). Any other choice of log h will differ from this one by a multiple of 2πi and hence will leave the quantity in the formula unaffected. Now let A and B be commuting operators with images a and b in B=L1. In [13] Carey and Pincus expressed the determinant invariant d(a − z1; b − z2) in terms of multiplicative Lefschetz numbers of the form det(B − z2)j det(A − z1)j ker(A−z1) coker (B−z2) (1.2) det(B − z2)jcoker (A−z1) det(A − z1)jker(B−z2) whenever a − z1 and a − z2 are invertible and (z1; z2) 2= σT (A; B), the Taylor joint spectrum of A and B. However, the determinant invariant is defined even if (z1; z2) 2 σT (A; B). Thus, Carey and Pincus introduce an invariant τ(A; B), known as joint torsion, for any two commuting Fredholm operators A and B.