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 H∞ symbols ...... 76
5.3.2 L∞ 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 {a, b}, 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 ∂{a, b} ∈ 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 ∈ S1. If h(eiθ) = |h(eiθ)|eiφ(θ) for a continuous real-valued function φ, then we take log h(eiθ) = log |h| + 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)| det(A − z1)| ker(A−z1) coker (B−z2) (1.2) det(B − z2)|coker (A−z1) det(A − z1)|ker(B−z2) whenever a − z1 and a − z2 are invertible and (z1, z2) ∈/ σT (A, B), the Taylor joint spectrum of A and B. However, the determinant invariant is defined even if (z1, z2) ∈ 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. This generalizes the multiplicative Lefschetz number in (1.2). By replacing A by exp(A) and B by I, joint torsion recovers the Fredholm index of A, so the subject of this thesis may be seen as a type of multiplicative index theory.
In order to define joint torsion, Carey and Pincus use the notion of algebraic torsion from homological algebra. For any finite length exact sequence (V•, d•) of finite dimensional vector spaces, the algebraic torsion τ(V•, d•) is a canonical generator of the determinant line
∗ dim Vn dim Vn−1 det V• = Λ Vn ⊗ Λ Vn−1 ⊗ · · ·
Section 3.2.1 reviews the details of this construction. For example, the torsion of an automorphism of a finite dimensional vector space is its determinant. As another example, J.-M. Bismut, H. Gillet, 3 and C. Soul´ehave shown in [6] that the Ray-Singer analytic torsion can be calculated as the norm of τ(V•, d•).
J. Kaad has generalized the notion of joint torsion to n ≥ 2 commuting operators [37].
Moreover, he shows that joint torsion is multiplicative, satisfies cocycle identities, and is trivial under appropriate Fredholm assumptions. He has also investigated the relationship between joint torsion on the one hand, and determinant functors and K-theory of triangulated categories on the other. In the case of n = 2 commuting operators, he has shown in [36] that the determinant invariant coincides with the Connes-Karoubi multiplicative character [20]. Furthermore, Kaad has constructed a product in relative K-theory and investigated the relative Chern character with values in continuous cyclic homology [35]. He uses these results to calculate the multiplicative character applied to Loday products of exponentials. See also Section 3.2 below for a discussion of this commuting case.
Carey and Pincus extended their definition of joint torsion, in a different direction, to two almost commuting Fredholm operators [16]. Thus, let A and B be Fredholm operators with trace class commutator, and moreover, assume the existence of operators C and D such that AB = CD,
A − D ∈ L1, and B − C ∈ L1. They then proceed as before, defining τ(A, B, C, D) in terms of short exact sequences of Koszul complexes. However, the result is no longer a scalar, but rather an element of a certain determinant line. To obtain a scalar, Carey and Pincus associate a perturbation
0 0 1 vector σA,A0 to each pair of Fredholm operators A and A with A − A ∈ L . See Section 3.3 for details on the constructions. More recently, J. Kaad and R. Nest have generalized the notion of perturbation vectors to finite rank perturbations of Fredholm complexes [39].
Perturbation vectors can be seen as a generalization to the singular setting of the classical perturbation determinant det(A−1A0). Carey and Pincus have shown in [16] that perturbation vectors form a non-vanishing section of a Quillen determinant line bundle [48]. Moreover, they have applied joint torsion to Toeplitz operators, especially problems where standard techniques only apply to symbols with zero winding number. They prove Szeg˝o-type limit theorems on the asymptotic behavior of determinants of Toeplitz operators whose symbols have nonzero winding 4 number [16]. See Section 3.1 for a discussion of this and other related work by Carey and Pincus.
∞ 1 In the case when f, g ∈ H (S ), the joint torsion τ(Tf ,Tg) is the product of tame symbols [15], and can be expressed in terms of Deligne cohomology [23]. In particular, the determinant invariant
1 1 d(Tf + L ,Tg + L ) is equal to the joint torsion τ(Tf ,Tg) when f and g are smooth functions in
H∞(S1). More generally, Carey and Pincus state in [16, Section 8, p. 345]:
The existence of the identification map (in the existence theorem for the pertur-
bation vector) has uncovered a basic problem – which deserves to be stated as a
question or perhaps as a conjecture:
Let a, b be commuting units in B(H)/L1(H). Let A, B, C, D be elements in B(H)
so that AB = CD and let A, D denote lifts of a and B,C denote lifts of b. In what
generality is it true that det ∂{a, b} = τ(A, B, C, D)?
This is quickly seen to be true for invertible operators A, B, C, and D [16, Section 8]. Carey and Pincus proved in [13] that this is also true for commuting Fredholm operators A and B with acyclic Koszul complex K•(A, B). We will see later that this follows from (1.2). More generally, they have shown in unpublished work that d(a, b) = τ(A, B, B, A) for any commuting Fredholm operators A and B (see [15, Theorem 2]). One of the main results of this thesis is Theorem 4.3.3, which answers the above question in full generality.
Let us take a moment to outline the strategy, which is carried out in Chapter 4. Since the determinant invariant is trivial in finite dimensions, the first order of business is to show that this true for joint torsion as well. This is done in Section 4.1, closely following the work of Kaad [37].
Our main tool is the double complex (4.4), which is composed of the homology spaces of modified
Koszul complexes. The resulting vertical and horizontal torsion vectors agree, up the sign of a permutation which appears in the definition of joint torsion. By picking generators carefully, we will see that these vectors comprise the joint torsion, which is consequently trivial.
Then in Section 4.2 we consider multiplicative properties of perturbation vectors and joint torsion with respect to invertible operators. This will allow us to investigate the transformation 5 of joint torsion under certain group actions in Section 4.3. More specifically, consider a quadruple
(A, B, C, D) where A, B, C, and D are Fredholm operators such that
A − D ∈ L1,B − C ∈ L1, and AB = CD.
Let U and V be invertible operators such that U − V ∈ L1 and the commutator [U, B] ∈ L1 as well. The set of all such pairs (U, V ) forms a group which acts on quadruples by
(A, B, C, D) • (U, V ) = (U −1A, B, U −1CU, V −1D).
We will see that joint torsion transforms according to the rule
τ((A, B, C, D) • (U, V )) = d(U −1 + L1,B + L1) · τ(A, B, C, D).
A similar formula holds for pairs of invertibles that commute with the first operator A modulo L1.
These group actions will allow us to reduce the general case to operators A, B, C, D in the coset of the identity modulo finite rank operators. The proof of Theorem 4.3.3 then follows quickly from the finite dimensional case.
In Section 4.4 we record a number of consequences of this equality. On the one hand, joint torsion is a finite dimensional object, at least for commuting operators. The determinant invariant on the other hand is defined in terms of the infinite Fredholm determinant. Hence it is somewhat surprising that the determinant invariant turns out to be expressible in terms of finite dimensional data. Moreover, we will see that joint torsion enjoys continuity properties and satisfies the usual
Sternberg relations (e.g. it is multiplicative and skew-symmetric). Finally, we will use these results to investigate the behavior of joint torsion with respect to Hilbert-Schmidt operators.
The second main goal of this thesis is to study the transformation of joint torsion under the functional calculus. J. Kaad and R. Nest have investigated local indices of commuting tuples of operators under the holomorphic functional calculus [38], and they obtain a global index theorem originally due to J. Eschmeier and M. Putinar [29]. In Section 5.1, we establish a multiplicative analogue of such transformation rules: the joint torsion τ(f(A),B) is given by
Y τ(A − λ, B)ordλ(f) · τ(q(A),B). {λ∈σ(A) | f(λ)=0} 6
Here q(A) is an invertible operator, so the second factor may be viewed as a type of multiplicative
Lefschetz number. In addition, we investigate variational formulas for joint torsion (Corollaries
4.4.5, 4.4.9, and 5.1.15).
In [26], T. Ehrhardt generalizes the Helton-Howe-Pincus formula on determinants of expo- nentials (Proposition 2.1.13) by showing that
eAeB − eA+B ∈ L1 (1.3) whenever [A, B] ∈ L1, and moreover,