Spectral Theory for Hilbert Space Operators

Index Let B be a unital Banach algebra and let G denote the group of invertible elements in B. Let G0 be the connected component of G containing the identity of G. Note that any connected component of G is open because all connected neighborhoods of its element are contained in it. It follows immediately that any connected component of G is closed in G, as well. In particular, G0 is open as well as closed subset of G. Example 1.3.9 Let X be a compact Hausdorff space. Consider the Banach algebra C(X) of continuous functions from X into C. Note that the group G of invertible elements in C(X) is given by G = { f ∈C(X) : f is nowhere vanishing on X}. Moreover, 1 is the identity for C(X). Let f ∈ C(X) belong to the connected component G0 of G containing 1. Since G0 is locally path-connected (G0 is open and C(X) is a vector space, therefore locally path-connected), there exists a collection { fλ}λ∈[0,1] of continuous functions in C(X) such that f0 = 1 and f1 = f . Define Γ : X × [0,1]→G by Γ(x,λ) = fλ(x), and note that Γ(x,0) = 1 and Γ(x,1) = f (x) for every x ∈ X. In other words, G0 consists of precisely those elements in G that are homotopic to 1. Recall that Fn(H) is the set of Fredholm operators in L(H) of index n. Lemma 1.3.4 Let G (H) be the group of invertible elements in Q(H). If G0(H) denotes the connected component of G (H) containing the identity element in G(H), then G (H) = {π(T ) : T ∈ F (H)} and G0(H) = {π(T ) : T ∈ F0(H)}, where π :L(H)→Q(H) is the Calkin map. Proof Since Fn(H) is a connected component of F (H) (see Corollary 1.3.3) and since π is an open map, G (H) is the union of clopen sets {π(T ) : T ∈ F0(H)} and {π(T ) : T ∈ F (H)\F0(H)}. Moreover, this union is disjoint. Indeed, if S n ∈ Fn(H) and S m ∈ Fm(H), then π(S n) = π(S m) implies S n − S m is a compact operator. Since the Fredholm index is invariant under compact perturbations (Theorem 1.3.1), we have m = n in this case. Let B be a unital Banach algebra and let G denote the group of invertible elements in B. Let G0 be the connected component of G containing the identity of G. For f ,g ∈G0, we record the following observations: • Since Lg : G0→ gG0 given by Lg( f ) = g f is surjective and continuous, every coset of G0 is connected. • Since f g and f belongs to the connected subset fG0 of B, fG0 ∪G0 is connected and non-empty. In particular, fG0∪G0 ⊆G0, and hence, G0 is a subgroup. • For any h ∈G, hG0h−1 is a connected subset containing identity, and hence, hG0h−1 = G0. That is, G0 is normal. i i “BDFT ̇Book” — 2021/5/18 — 12:14 — page 31 — #43 i i i i i i Spectral Theory for Hilbert Space Operators 31 • The cosets of G0 are connected components of G. Since G0 is a clopen subset of G, the quotient G/G0 is a discrete group. The previous discussion forms the basis of the following definition. Definition 1.3.10 Let B be a unital Banach algebra. Let G denote the group of invertible elements in B and let G0 denote the connected component of G containing the identity of G. The abstract index group for B is the quotient group G/G0. In what follows, we will be particularly interested in the abstract index group for the Banach algebra C(X). Example 1.3.11 (Example 1.3.9 continued) Recall that the group G of invertible elements in C(X) is the set of nowhere vanishing continuous functions in C(X). It turns out that the connected component G0 of C(X) containing the identity element 1 in C(X) is the group exp(C(X)) of functions of the form e f for some f ∈ C(X). To see this, note that if f lies in an open unit ball in C(X) around 1, then f = eg, where g ∈ B is given by