Introduction to the Baum-Connes conjecture Nigel Higson, John Roe PSU NCGOA07 Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 1 / 15
History of the BC conjecture Lecture 1: Elliptic operators and the BC Conjecture Throughout, G will be a group (perhaps with a locally compact topology). Definition A unitary representation of G is a (strongly) continuous homomorphism ρ: G U(H) from G to the group of unitary operators on a Hilbert space. There is a natural notion of direct sum of representations. A representation is irreducible if it cannot be nontrivially decomposed as a direct sum. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 2 / 15
An important example History of the BC conjecture Let G be the group Z (with the discrete topology). Then every complex number u with u = 1 corresponds to a 1-dimensional irreducible representation of G, on which n Z acts by multiplication by u n. Moreover, these are all the irreducible representations of G. Thus the space of irreducible (unitary) representations is naturally identified with the topological space S 1 (the unit circle). The goal of the Baum-Connes conjecture is to understand irreducible (unitary) representations topologically. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 3 / 15
Important clarification History of the BC conjecture In the case G = Z every unitary representation can be uniquely decomposed into a direct sum of irreducibles. Moreover the space of irreducibles carries a natural Hausdorff topology and can be studied as a conventional ( commutative ) object. These are hallmarks of the Type I situation. Outside Type I We do not have uniqueness of decompositions into irreducibles. The natural topology on the space of irreducibles is not Hausdorff, or even T 1. We avoid these difficulties by replacing the space of irreducibles by a C -algebra - a noncommutative space. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 4 / 15
History of Baum-Connes History of the BC conjecture Baum-Connes (Kingston Conference 1982) foliations and Lie groups Kasparov s Conspectus 1983 Baum-Connes-Higson 1994 Key idea: the relationship between topology and representation theory is mediated by elliptic operators. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 5 / 15
Unbounded operators Unbounded self-adjoint operators A bounded operator T on a Hilbert space H is self-adjoint if Tx, y = x, Ty x, y H. This can be generalized to unbounded, densely defined operators provided that careful attention is paid to domain questions. An unbounded, self-adjoint operator T gives rise to a functional calculus homomorphism φ T : f f (T ), C 0 (R) B(H), and the homomorphism completely determines the operator. We could say that an unbounded operator is a functional calculus homomorphism. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 6 / 15
Unbounded operators We shall be interested in graded Hilbert spaces H = H + H. In this case an odd unbounded operator is identified with a grading-preserving functional calculus homomorphism φ T : f f (T ), S B(H) where S denotes the algebra C 0 (R) graded by even and odd functions. Definition The unbounded self-adjoint operator T has compact resolvent if f (T ) is a compact operator for all f C 0 (R). (Equivalent spectral definition does not generalize.) Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 7 / 15
Example Unbounded operators Consider the space H = L 2 (S 1 ) of functions on the circle (f (θ) = f (θ + 2π)). The operator D = id/dθ is self-adjoint on H (integration by parts). In fact D is diagonal with respect to the basis {e inθ } of H, and the eigenvalues are {n}. Conclusion: D has compact resolvent. Note that the functions e inθ are the characters of the irreducible representations of Z. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 8 / 15
Elliptic operators Let M be a smooth manifold, D a first order (for simplicity) differential operator acting on sections of a vector bundle S over M. Definition The symbol of D is the bundle map σ D : T M End(S) that satisfies the Liebniz rule D(fs) = fds iσ D (df ) s f C (M), s C (S). The symbol is an algebraic representation of the highest order part of D. Definition D is elliptic if σ D (ξ) is invertible for all nonzero ξ T M. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 9 / 15
Elliptic operators Theorem A (self-adjoint) elliptic operator on a compact manifold has compact resolvent. Proof. Think first about constant coefficient operators (on the torus). Quantify the compact resolvent property in terms of an estimate on Sobolev norms. Use an approximation argument to show that the same estimate holds for variable-coefficient operators on small patches of a general manifold. Glue the patches together using a partition of unity. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 10 / 15
Clifford modules Elliptic operators Let V be a (real) vector space equipped with a symmetric bilinear form (, ). Definition A vector space W is a Clifford module over V if there is given a linear map c : V End(W ) such that c(u)c(v) + c(v)c(u) = 2(u, v)i u, v V. Consider the example V = inner product space of dimension 2k. Then there is a fundamental Clifford module of dimension 2 k, and any Clifford module is the direct sum of a number of copies of this fundamental one. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 11 / 15
Dirac operators Elliptic operators Let M be a Riemannian manifold. Suppose that S is a Clifford bundle over M that is, a vector bundle whose fiber S x at each point x is a Clifford module over the inner product space T x M. We can equip S with a (compatible) connection. Then the Dirac operator of S is the (first order differential) operator D : C (S) C (S) defined by C (S) C (T M S) g C (TM S) c C (S) Lemma The Dirac operator is self-adjoint and elliptic. Indeed, the symbol is σ D = c : T (M) End(S), which is (up to a scalar) its own inverse for nonzero ξ. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 12 / 15
Examples Elliptic operators The de Rham operator d + d on an oriented Riemannian manifold. (Its square is the classical Laplacian.) The Dolbeault operator + on a Kähler manifold. The Dirac operator of a Spin c manifold see below. Definition An (even-dimensional) Riemannian manifold is Spin c if it carries a Clifford bundle whose fiber at each point is a copy of the fundamental Clifford module. This is a generalized orientability condition. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 13 / 15
The index K -theory and the index The examples of Dirac operators discussed above (on even-dimensional manifolds) are all graded, i.e. S = S + S is a graded bundle and D is an odd operator with respect to the grading. If the underlying manifold is also compact, then the kernel Ker D is a graded (finite-dimensional) vector space and one defines Index(D) = dim(ker D) + dim(ker D). This is a topological invariant of the operator D. Recall that a (graded) operator with compact resolvent (such as D) gives rise to a (graded) homomorphism S K, f f (D). Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 14 / 15
K -theory and the index Lemma The collection of homotopy classes of graded homomorphisms S K is naturally parameterized by Z. This parameterization sends an operator D to its index. This is a (very) special case of Higson s spectral picture of K -theory. Generalizations: Hilbert modules on right, giving K (A). Algebra on left, giving morphisms on K -theory. Replace morphism by asymptotic morphism. Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 15 / 15