Handout for the workshop on the Baum-Connes conjecture Useful facts to keep in mind

speakers: Paul Baum, Pierre Julg, and Guoliang Yu

May 22, 2002

1 C^{∗}-algebras

• A C^{∗}-algebra A is

– a ℂ-algebra

– together with an involution ∗: A → A such that (λab)^{∗} = \overline{λ}b^{∗}a^{∗} for all λ ∈ ℂ, a, b ∈ A.

– It is equipped with an algebra norm, i.e. |ab| ≤ |a| |b|.

– A is complete with this norm.

– The C^{∗}-identity |a^{∗}a| = |a|^{2} is satisfied ∀a ∈ A

A C^{∗}-algebra homomorphism is a continuous map which respects all the algebraic structure.

• Examples of C^{∗}-algebras:

– ℂ

– every closed ∗-subalgebra of the bounded operators B(H) on a Hilbert space H. By the GNS-construction, every C^{∗}-algebra is isomorphic to such an algebra.

– If X is a (locally compact Hausdorff) space, the algebra C_{0}(X) of complex valued continuous functions on X, which vanish at infinity, is a commutative C^{∗}-algebra. Every commutative C^{∗}-algebra is isomorphic to such an algebra.

– If X is a measure space, L^{∞}(X) with its norm is another example of a commutative C^{∗}-algebra. (This is usually very different from C(X)!).

– Given an arbitrary C^{∗}-algebra A and X as before, C_{0}(X; A) is the algebra of continuous functions on X with values in A, which vanish at infinity. The algebraic operations are defined pointwise, the norm is the sup-norm.

– Given a Hilbert space H, K(H) is the algebra of compact operators k: H → H, i.e. the closure of the algebra of all operators of finite rank (i.e. with finite dimensional image). This is a two-sided ideal inside B(H).

– Given an arbitrary C^{∗}-algebra A, the algebra of n-by-n-matrices M_{n}(A) also is a C^{∗}-algebra.

• Short exact sequences of C^{∗}-algebras:

0 → I → A → B → 0,

consist of C^{∗}-algebras together with C^{∗}-algebra homomorphism, such that kernel of outgoing and image of incoming map coincide at each algebra in question. In such a diagram, I is a two-sided closed ideal of A, and B = A/I is the quotient.

Example: 0 → K(H) → B(H) → C(H) → 0, where by definition C(H) = B(H)/K(H) is the Calkin algebra.

• a ∈ A is called

– self adjoint, if a = a^{∗}

– unitary, if a^{∗} = a^{−1}

– a projection, if a = a^{2} = a^{∗}.

• For self-adjoint and unitary element a of a C^{∗}-algebra A, functional calculus is defined, i.e. a C^{∗}-homomorphism

C_{0}(ℂ) → A: f ↦ f(a)

with id(a) = a. If we replace for the domain of f ℂ by the spectrum of a (a compact subset of ℂ), this map becomes a C^{∗}-algebra embedding.

• Given a discrete group Γ, the reduced group C^{∗}-algebra C_{r}^{∗}Γ is defined as the norm closure of the (algebraic) group ring ℂΓ inside the bounded operators on l^{2}(Γ), where l^{2}(Γ) is the Hilbert spaces of L^{2}-functions on Γ, which we equip with the discrete measure (every element has volume one). Observe that this is a Haar measure on Γ.

• In a similar way, we can form the C^{∗}-algebra of a topological group G (e.g. a Lie group) as completion of the convolution algebra C_{comp}(G) inside B(L^{2}(G)). Attention: C_{comp}(G) consists of functions on G with compact support, but the product is not pointwise, it is given by convolution (again using a fixed Haar measure).

• One can form (completed) tensor products of C^{∗}-algebras. Of particular importance is

– A ⊗ M_{n}(ℂ) = M_{n}(A)

– A ⊗ K(l^{2}(ℤ)), a limit of M_{n}(A) for n → ∞.

– C_{0}(X) ⊗ A = C_{0}(X; A).

2 K-theory

• For a C^{∗}-algebra A, we define the abelian groups

– K_{0}(A), consisting of (formal differences of) (equivalence classes of) projections in A, M_{n}(A) (and A ⊗ K). One of the (several different) ways to define equivalence is as: homotopic through the spaces of projections.

– K_{1}(A) consists of unitary elements of A, M_{n}(A) (and A ⊗ K), again module equivalence given by homotopies through the space of unitaries.

• Both K_{0} and K_{1} are functors from the category of C^{∗}-algebras to the category of abelian groups.

• Examples:

A  ℂ K B C C_{r}^{∗}G for G compact C_{0}(X) C_{r}^{∗}G
K_{0}(A) ℤ ℤ 0 0 Rep(G)  K^{0}(X) ??
K_{1}(A) 0 0 0 ℤ 0  K^{1}(X) ??

K^{0}(X) is the topological K-theory of X, defined as (equivalence classes of formal differences) of vector bundles over X.

• Morita-Equivalence:

K_{∗}(A) = K_{∗}(M_{n}(A)) = K_{∗}(A ⊗ K).

• Bott periodicity and six term exact sequence (some people call this excision): Given a short exact sequence 0 → I → A → B → 0, we get an induced exact sequence

K_{0}(I) ⟶ K_{0}(A) ⟶ K_{0}(B)
↑  ↓
K_{1}(B) ⟵ K_{1}(A) ⟵ K_{1}(I).

The map δ: K_{1}(B) → K_{0}(I) is the index map, the map K_{0}(B) → K_{1}(I) involves in addition Bott periodicity.

• Example: for 0 → K → B → C → 0, we get the K-theory exact sequence

ℤ ⟶ 0 ⟶ 0
↑  ↓
ℤ ⟵ 0 ⟵ 0

• Homotopy invariance: two C^{∗}-homomophism f_{0}, f_{1} : A → B are called homotopic, if they can be joint by a (pointwise continuous) path f_{t}: A → B of C^{∗}-homomorphism. In this situation, f_{0} and f_{1} induce identical maps on K_{∗}.

• Every trace on a C^{∗}-algebra A gives rise to a homomorphism K_{0}(A) → ℂ. The usual operator trace is defined on all of K, it gives rise to the isomorphism

tr: K_{0}(K) → ℤ.

For a projection p ∈ K, tr(p) is the dimension of the image of p.

3 Index and differential operators

• Every element of K_{1}(C) can be represented by a unitary in C. By definition, C = B/K, i.e. u ∈ C comes from an element U of B. Being unitary in C means that U^{∗}U = 1 − k_{0} and UU^{∗} = 1 − k_{1} for suitable k_{0}, k_{1} ∈ K. By Atkinsons theorem, such a bounded operator U: H → H is Fredholm, i.e. has finite dimension kernel and cokernel (and closed image). Therefore, its (Fredholm) index

ind(U) := dim(ker(U)) − dim(coker(U)) = dim(ker(U)) − dim(ker(U^{∗}))

is defined. Using the index map δ: K_{1}(C) → K_{0}(K), the following holds:

tr δ([u]) = ind(U).

This gives an isomorphism K_{1}(C) ≅ ℤ.

• In the following, we will encounter unbounded operators on a Hilbert space H. Such operators are only defined on a dense subspace of H. The adjoint of an unbounded operator is another unbounded operator. If an unbounded operator A is self adjoint, the functional calculus for A is defined, i.e. there is a ∗-homomorphism

L^{∞}(ℝ) → B(H): f ↦ f(A),

with the property that if f(t) = tg(t), the f(A) = Ag(A), where on the right hand side we compose the operators A and g(A). If we replace ℝ by the spectrum of A, this becomes a C^{∗}-algebra embedding.

• A generalized Dirac operator D on a (Riemannian) manifold M is a certain type of first order differential operator on M (defining a map between smooth section of two (Hermitian) vector bundles E and F over M). The following properties are important:

– D is a self adjoint unbounded operator on L^{2}(E), the Hilbert space of L^{2}-section.

– In particular, functions like ∗ D(1 + D^{2})^{−1/2}

∗ e^{itD}; t ∈ ℝ

∗ ^{−tD^{2}}; t ≥ 0 are defined. The first one is of the type χ(D) with χ :: ℝ → [−1, 1] even, χ(t) ⟶^{t→±∞} ±1. Such a χ is called a chopping function.

– D satisfies elliptic regularity. Classically: if Df is smooth (on some open subset V of M, then f is smooth (on V). This can be refraised in terms of operators: if h ∈ C_{0}(ℝ), and f, g ∈ C_{0}(M), then fh(D)g is compact. Here f and g denote the operators on L^{2}(E) which are given by pointwise multiplication. If M is compact, this means that h(D) itself is compact.

– D has finite propagation: gDf = 0 if the support of g and f are disjoint (this is simply true because D is a differential operator). It translates to corresponding, but weaker statements for functions like h(D).

• Example: on a Riemannian manifold M, let d be the exterior differential on the direct sum Ω^{∗}(M) of all spaces of differential forms, and δ = d^{∗} its adjoint. Then d + δ is a generalized Dirac operator.

• Grading Usually (on even dimensional manifolds) a generalized Dirac operator comes with a grading E = E_{+} ⊕ E_{−}, and

D = (\begin{matrix}0 & D_{−} \\ D_{+} & 0: L^{2}(E_{+}) ⊕ L^{2}(E_{−}) → L^{2}(E_{+}) ⊕ L^{2}(E_{−}). \end{matrix})

. If χ as above is an odd function, then

χ(D) = (\begin{matrix}0 & χ(D_{−}) \\ χ(D_{+}) & 0: L^{2}(E_{+}) ⊕ L^{2}(E_{−}) → L^{2}(E_{+}) ⊕ L^{2}(E_{−}). \end{matrix})

.

• Example of a grading: on Ω^{∗}(M) we can define a grading Ω^{ev}(M) ⊕ Ω^{odd}(M) by distinguishing between differential forms of even and odd degree. Then d + δ decomposes as described.

• Using a unitary isomorphism U: L^{2}(E_{−}) → L^{2}(E_{+}) (which usually exists, we assume this now) and a chopping function, we obtain an operator T := Uχ(D_{+}): L^{2}(E_{+}) → L^{2}(E_{+}).

If M is a compact manifold, then T^{∗}T − 1 and T T^{∗} − 1 are compact. Therefore T defines a class in K_{1}(C). We define the index of D to be δ([T]) ∈ K_{0}(K).

In this case, identifying K_{0}(K) with ℤ gives a number, which is indeed the difference of the dimension of kernel and cokernel of D, i.e. the Fredholm index of D. Using appropriate C^{∗}-algebras instead of K, B, and C, one can define indices of Dirac operators for non-compact manifolds as elements in K-groups of certain C^{∗}-algebras. Moreover, one can refine the analysis, if additional structure is given, e.g. if there is symmetry (a group acting on M such that D is equivariant), to obtain indices in more interesting K-groups like K_{∗}(C_{r}^{∗}Γ).

• No matter how and where the index lives, if it is non-zero, then zero does belong to the spectrum of D, i.e. D can not have a bounded inverse. This is important for applications to geometry, where D often has a bounded inverse if some particular geometric condition is satisfied.

4 K-homology

• For every C^{∗}-algebra A on can also define K-homology groups K^{0}(A) and K^{1}(A). This gives a contravariant functor (i.e. arrows are reversed) from the category of C^{∗}-algebras to the category of abelian groups. Properties very similar to K-theory are satisfied.

• Most important is, that there is a pairing

K^{∗}(A) ⊗ K_{∗}(A) → ℤ.

• For us most important is the case A = C(X). In this case, every Dirac type differential operator D produces an element [D] ∈ K^{∗}(C(X)). (If X is a manifold, one will use operators defined over X, in general, every (proper) map M → X can be used to push K-homology classes given by such operators from K^{∗}(C(M)) to K^{∗}(C(X)) (observe that we get a C^{∗}-algebra map C(X) → C(M)).

• In the case A = C(X), the pairing

K^{∗}(C(X)) ⊗ K_{∗}(C(X)) → ℂ

is given as follows: if a K-homology class is represented by a Dirac type operator D, and a K-theory class by a vector bundle E, then

[D] ⊗ [E] ↦ ind(D_{E}).