Partially positive square root and Kuiper type theorem for operators on C*-Hilbert modules

E. V. Troitsky

Abstract

Let us denote by End_{A}l_{2}(A) the Banach algebra of all bounded A-morphisms of Hilbert A-module l_{2}(A), and by End_{A}^{∗}l_{2}(A) the C^{∗}-algebra of operators, admitting adjoint. Let GL and GL^{∗} denote the corresponding groups of invertible operators. In the present paper we prove that GL is norm contractible for A being σ-unital. This completes the research of a number of authors during approximately 20 years.

1 Introduction

Let us denote by End_{A}l_{2}(A) the Banach algebra of all bounded A-homomorphisms of Hilbert A-module l_{2}(A), and by End_{A}^{∗}l_{2}(A) the C^{∗}-algebra of operators, admitting adjoint. Let GL and GL^{∗} denote the corresponding groups of invertible operators. The question about the contractibility of general linear groups is very important for K–theory to construct classifying spaces in terms of Fredholm operators. To this question a series of papers was devoted: [13, 5, 20, 14]. The author used these results to construct [19] the classifying spaces of K-theory K^{p,q}(X; A) which arises in analytical approach to the Novikov Conjecture on higher signatures. In paper [3] J. Cuntz and N. Higson proved the contractibility of GL^{∗} for A with strictly positive element (or, equivalent, with countable approximate unit = σ-unital). In the paper [21] (see also [23]) a simple proof of the theorem of Cuntz and Higson was given and the contractibility of GL was proved in some special cases, in particular, for A, being a subalgebra of algebra of compact operators in separable Hilbert space, and for A = C_{0}(M), where M is a finite-dimensional manifold. The further results based on this approach were obtained by my students in [18] and [2].

In the present paper we obtain a sort of partially positive polar decomposition for GL . With its help we prove the following statement.

Theorem 1 The group GL is norm contractible for any σ-unital algebra A.

Acknowledgment. The present results were obtained due to the creative and helpful atmosphere of Max-Planck-Institut für Mathematik (Bonn) while the author was visiting there in the Fall 2000.

The author is grateful to L. G. Brown, J. Cuntz, M. Frank, V. M. Manuilov and A. S. Mishchenko for helpful discussions at different stages of the present research.

The preliminary research was partially supported by the RFBR (Grant 99-01-01202) and by the President of RF (Grant 00-15-99263).

2 Preliminary notes

It is known, that the set of invertible operators in a Banach space is open with the respect to the topology of a norm, while the set of bounded A-homomorphisms is closed in the set of all endomorphisms. Thus, GL is an open set in a Banach space. According to the Milnor theorem [12] such sets have the homotopy type of CW -complexes, and, therefore, by the theorem of Whitehead, strong and weak homotopy triviality are equivalent for them. Together with the contractibility of GL^{∗} [3] this implies the following statement.

Lemma 2 To prove the contractibility of GL it is sufficient to verify the following. Let f : S → GL be a continuous map of a sphere of arbitrary dimension. Then f is homotopic to a map valued in GL^{∗}. □

Lemma 3 (a variant of the Atiyah theorem about small balls) [21, Lemma 2.2] Let f : S → G be a continuous map of a sphere of arbitrary finite dimension. Then f is homotopic to a map f′ such that f′(S) is a finite polyhedron in End_{A}l_{2}(A), laying in GL together with the homotopy.

One more step from the original work of Kuiper [7] is universal. Let us denote orthogonal (A-Hilbert) sum by ⊕ and Banach module one by ˜⊕.

Lemma 4 [21, Lemma 2.4] Subset V ⊂ GL , defined as

V = {g ∈ GL | g|_{H′} = Id_{H′}, g(H_{1}) = H_{1}},

where

l_{2}(A) = H′ ⊕ H_{1}, H′ ≅ H_{1} ≅ l_{2}(A),

is contractible in GL to 1 ∈ GL .

Lemma 5 [21, Lemma 2.5] Subset W ⊂ GL , defined as

W = {g ∈ GL | g|_{H′} = Id_{H′} },

where

l_{2}(A) = H′ ⊕ H_{1}, H′ ≅ H_{1} ≅ l_{2}(A),

is contractible inside GL to

V = {g ∈ GL | g|_{H′} = Id_{H′}, g(H_{1}) = H_{1}}.

3 Technical lemmas

Suppose, K_{A} is the C^{∗}-algebra of A-compact operators on l_{2}(A), LM(K_{A}) ≅ End_{A}l_{2}(A) is the algebra of the left multipliers, M(K_{A}) ≅ End_{A}^{∗}l_{2}(A) is the C^{∗}-algebra of multipliers and QM(K_{A}) ≅ Hom_{A}(l_{2}(A), l_{2}(A)′) is the space of quasi-multipliers (see [1, 4, 6, 9, 15, 16, 17, 22, 23]).

Suppose, α is a strictly positive element in a σ-unital algebra A, α_{i} := φ_{i}(α) is a countable approximate unit, where φ_{i} is defined by the formula

φ_{i}(t) := \{ \begin{matrix}0, & if t ∈ [0, \frac{1}{i}], \\ 1, & if t ∈ [\frac{1}{i−1}, ∞), \\ linear, & if t ∈ [\frac{1}{i}, \frac{1}{i−1}], \end{matrix}  (1)

Suppose,

β_{i} := ψ_{i}(α),  ψ_{i}(t) := \{ \begin{matrix}0, & if t = 0, \\ 1, & if t ∈ [\frac{1}{i}, ∞), \\ linear, & if t ∈ [0, \frac{1}{i}], \end{matrix}  (2)

Then β_{i} is a strictly positive element in A and

α_{i}β_{i} = α_{i}α_{i+k} = α_{i},  k = 1, 2, …  (3)

Put ω_{i} := (α_{i} − α_{i−1})^{1/2} for i ≥ 3 and ω_{2} = α_{2}^{1/2}, so

ω_{j}α_{i} = α_{i}ω_{j} = 0, j = i + 2, i + 3, …,  ω_{j}α_{i} = α_{i}ω_{j} = ω_{j}, j = 1, …, i − 1. (4)

Since there is no unit in A, the notion of “standard base” {e_{i}} of module l_{2}(A) makes no sense. Nevertheless, the elements e_{i}γ are well defined for any γ ∈ A, namely,

e_{i}γ := (0, …, 0, γ, 0, …), γ at i-th place.

Let us denote the correspondent orthoprojections on these one-dimensional submodules A_{i} by q_{i}, and p_{i} := q_{1} + … + q_{i}, L_{i} := Im p_{i}.

We say that a linear mapping F is topologically injective, if there is a constant c_{0} > 0 such that ∥F x∥ ≥ c_{0}∥x∥.

Lemma 6 [21] The injection J : l_{2}(A) → l_{2}(A) under the formula

(a_{1}, a_{2}, …) ↦ ∑_{j} ∑_{i} v_{ij}a_{j},  ⟨v_{ij}, v_{ij}⟩ = ω_{i}^{2},  v_{ij} ∈ M_{k(i,j)},

l_{2}(A) = M_{1} ⊕ M_{2} ⊕ …, M_{r} = { (0, …, 0, a_{s(r)}, …, a_{s(r+1)−1}, 0, …) },
{k(1, 1); k(1, 2), k(2, 1); k(1, 3), k(2, 2), k(3, 1); …} = {1, 2, …},

leaves the inner product unchanged and admits an adjoint. In particular, the image is defined by a selfadjoint projection of the form JJ^{∗}.

We have the following evident lemma.

Lemma 7 Let U ∈ GL^{∗} be a unitary operator. For any x ∈ l_{2}(A) and any ε > 0 there exists a number k_{0}, such that for any γ ∈ A, ∥γ∥ ≤ 1, and any k > k_{0}

∥⟨Ue_{k}γ, x⟩∥ < ε.

Let us denote as obvious Re a := \frac{a+a∗}{2} ∈ A for a ∈ A.

Lemma 8 Suppose, a, b, c, d ∈ A. Then

Re (a^{∗}cb + b^{∗}da) ≥ −(∥c∥ + ∥d∥)(a^{∗}a + b^{∗}b).

Proof:

Re (a^{∗}cb + b^{∗}da) = a^{∗} \frac{c + d^{∗}}{2} b + b^{∗} \frac{c^{∗} + d}{2}a =

= (∥c^{∗} + d∥^{1/2} a + \frac{c + d^{∗}}{2∥c^{∗} + d∥^{1/2}} b)^{∗} (∥c^{∗} + d∥^{1/2} a + \frac{c + d^{∗}}{2∥c^{∗} + d∥^{1/2}} b) −

−∥c^{∗} + d∥ a^{∗}a − b^{∗} \frac{(c + d^{∗})^{∗}(c + d^{∗})}{4∥c^{∗} + d∥} b ≥

≥ −(∥c∥ + ∥d∥) (a^{∗}a − b^{∗}b).

□

Lemma 9 Let x and y be elements of l_{2}(A). Then

Re ⟨x, y⟩ ≤ \frac{1}{2} (⟨x, x⟩ + ⟨y, y⟩).

Proof: Passing to coordinates x = (x_{1}, x_{2}, …), y = (y_{1}, y_{2}, …), it is sufficient to verify

Re (a^{∗}b) ≤ \frac{1}{2}(a^{∗}a + b^{∗}b), a, b ∈ A.

Since 0 ≤ (a − b)^{∗}(a − b) = a^{∗}a + b^{∗}b − 2Re (a^{∗}b), we are done. □ Lemma 10 Let S, T : l_{2}(A) → l_{2}(A) be bounded A-operators. Suppose,

c ⟨x, x⟩ < Re ⟨Sx, T x⟩

for some x ∈ l_{2}(A). Then

\frac{c^{2}}{∥T ∥^{2}} ⟨x, x⟩ < ⟨Sx, Sx⟩.

Proof: By Lemma 9

c⟨x, x⟩ < Re ⟨Sx, T x⟩ = Re ⟨\frac{∥T ∥}{\sqrt{c}}Sx, \frac{\sqrt{c}}{∥T ∥}T x⟩ ≤
≤ \frac{1}{2} (\frac{∥T ∥^{2}}{c} ⟨Sx, Sx⟩ + c⟨x, x⟩) ,

\frac{c^{2}}{∥T ∥^{2}} ⟨x, x⟩ < ⟨Sx, Sx⟩.

□

Lemma 11 (see, e.g. [10]) Let L be a closed subspace of a Banach space X, which is not equal to X. Then for any ε > 0 there exists y ∈ X such that ∥y∥ = 1 and ∥x − y∥ > 1 − ε for any x ∈ L.

Lemma 12 Suppose, l_{2}(A) = H_{0} ⊕ H_{1}, V ∈ GL^{∗}, V (H_{0})⊥V (H_{1}), F ∈ GL , and for any x ∈ H_{0} one has

Re ⟨V x, F x⟩ ≥ c · ⟨x, x⟩.

Then the linear homotopy

V (x, t) := \{ \begin{matrix}(1 − t)F (x) + tV (x) & if x ∈ H_{0}, \\ V (x) & if x ∈ H_{1}, \end{matrix}

t ∈ [0, 1], is in GL .

Proof: Let p be the orthogonal projection on V (H_{0}). We have to verify that

p|_{((1−t)F +tV)(H_{0})} : ((1 − t)F + tV)(H_{0}) → V (H_{0})

is an isomorphism. This will complete the proof.

Let us start from the topological injectivity. For x ∈ H_{0} we have

c⟨x, x⟩ < Re ⟨F x, V x⟩ = (1 − t)Re ⟨V (x, t), V x⟩ − t⟨V x, V x⟩,

(c + \frac{t}{∥V^{−1}∥^{2}} ) ⟨x, x⟩ < (1 − t)Re ⟨V (x, t), V x⟩,

c⟨x, x⟩ < Re ⟨V (x, t), V x⟩.

Let us remark that

⟨V (x, t), V x⟩ = ⟨pV (x, t), V x⟩.

By Lemma 10

\frac{c^{2}}{∥V ∥^{2}} ⟨x, x⟩ < ⟨pV (x, t), pV (x, t)⟩.

Topological injectivity is verified.

Hence, the image is a closed subspace of V (H_{0}). Suppose, it is not equal to V (H_{0}). Then we can apply Lemma 11 and find an element y = V z ∈ V (H_{0}) such that ∥y∥ = 1 and ∥x − y∥ > 1 − ε for any x from our closed subspace. In particular, we can take ε such that

(1 − ε)^{2} > 1 − \frac{c^{2} ∥V^{−1}∥^{2}}{(∥F ∥ + ∥V ∥)^{2}} ,  (5)

and x = τpV (z, t) with

τ = \frac{c}{(∥F ∥ + ∥V ∥)^{2}} .  (6)

We have

⟨x − y, x − y⟩ ≤ τ^{2}(∥F ∥ + ∥V ∥)^{2}⟨z, z⟩ − 2cτ⟨z, z⟩ + ⟨y, y⟩ =
= τ^{2}(∥F ∥ + ∥V ∥)^{2}⟨V^{−1}y, V^{−1}y⟩ − 2cτ⟨V^{−1}y, V^{−1}y⟩ + ⟨y, y⟩ ≤
≤ [ τ^{2}((∥F ∥ + ∥V ∥) ∥V^{−1}∥)^{2} − 2cτ∥V^{−1}∥^{2} + 1] ⟨y, y⟩ =
= [ (τ(∥F ∥ + ∥V ∥)^{2} − 2c)τ∥V^{−1}∥^{2} + 1] ⟨y, y⟩ =
= [1 − \frac{c^{2} ∥V^{−1}∥^{2}}{(∥F ∥ + ∥V ∥)^{2}} ] ⟨y, y⟩.

Hence,

1 − \frac{c^{2} ∥V^{−1}∥^{2}}{(∥F ∥ + ∥V ∥)^{2}} < (1 − ε)^{2} < ∥x − y∥^{2} ≤ 1 − \frac{c^{2} ∥V^{−1}∥^{2}}{(∥F ∥ + ∥V ∥)^{2}} ,

and we obtain a contradiction. □

Lemma 13 Let p and q be two orthogonal projections in some Hilbert module M. Suppose, g := q|_{Im p} : Im p → Im q is an isomorphism. Then ig^{−1}q is adjointable where i : Im p ⊂ M is the inclusion.

Proof: This follows immediately from the self-adjointness of p, i, and q, and the equality g = q ∘ i. □

Lemma 14 Suppose, H_{0} ⊕ H_{1} = l_{2}(A), H_{0} ≅ H_{1} ≅ l_{2}(A), F ∈ GL and GL ∋ F̃ =

\{ \begin{matrix}F & on H_{0}, \\ F′ & on H_{1}, \end{matrix} are continuous families. Then they are homotopic each other.

Proof: Let us consider the continuous family G = \{ \begin{matrix}Id & on F (H_{0}) \\ F′F^{−1} & on F (H_{1}) \end{matrix} in GL , so

F̃ = GF. Then there is a linear homotopy G ∼ G̃ in GL , where

G̃ = \{ \begin{matrix}Id & on F (H_{0}), \\ p_{F} F′F^{−1} & on F (H_{1}), \end{matrix}

and p_{F} is the projection onto F (H_{1}) along F (H_{0}) (cf. [21, Lemma 2.5] and Lemma 5). One can complete the proof in the same way as in [21, Lemma 2.5] by verifying G̃ ∼ Id .

□

Lemma 15 Let F ∈ K ⊂ GL , where K is a finite polyhedron. Suppose, l_{2}(A) = H′ ⊕H′_{0}, l_{2}(A) ≅ H′ ≅ H′_{0}, and there exists a continuous mapping F ↦ D_{F}, where

1) D_{F} : H′_{0} → l_{2}(A) is an adjointable injection;

2) (Im D_{F})^{⊥} ≅ l_{2}(A).

Then there exists a continuous mapping F ↦ V_{F} ∈ GL such that

(i) V_{F} |_{H′_{0}} = D_{F};

(ii) V_{F} (H′) = (D_{F} (H′_{0}))^{⊥};

(iii) V_{F} ∈ GL^{∗}, i.e. admits an adjoint.

Proof: Denote by P_{F} the orthoprojection on Im D_{F}. Let us choose a sufficiently small subdivision of K such that P_{F_{2}}|_{Im D_{F_{1}}} : Im D_{F_{1}} → Im D_{F_{2}} is an isomorphism for any F_{1} and F_{2} from one simplex. Then the same is true for 1 − P_{F}. Denote by

γ_{F_{1},F_{2}} := (1 − P_{F_{2}})|_{Im(1−P_{F_{1}})} : Im(1 − P_{F_{1}}) ≅ Im(1 − P_{F_{2}})

the corresponding isomorphism.

Let us define the operators V_{F} by the induction over the dimension of skeletons. Each time we will define V_{F} by (i) and (ii) and in such a way that V′_{F} := V_{F} |_{H′} : H′ → (D_{F} (H′_{0}))^{⊥} = (1 − P_{F})(l_{2}(A)) is adjointable. In fact, we only need to define by induction this family V′_{F}. It is well known that two Banach-isomorphic countably generated Hilbert A-modules are unitary isomorphic (see, e.g. [4] or [11]). Hence we can define F ↦ V′_{F} over 0-skeleton. Suppose, we have defined the mapping over the boundary ∂σ of an n-simplex σ with vertices F_{0}, …, F_{n} and we would like to extend it up to the whole simplex. Let us consider a continuous mapping

μ : F ↦ (1 − P_{F_{0}})V′_{F} = γ(F, F_{0})V′_{F} , F ∈ ∂σ,

valued in the space GL^{∗}(H′, (1 − P_{F_{0}})(l_{2}(A))) of adjointable invertible operators H′ → (1 − P_{F_{0}})(l_{2}(A)). The last space is contractible by Cuntz and Higson theorem [3] (see also [21]). Hence we have an extension μ̃ : σ → GL^{∗}(H′, (1 − P_{F_{0}})(l_{2}(A))) of μ. Let us define V′_{F} for F ∈ σ by the formula

V′_{F} = (γ(F, F_{0}))^{−1}μ̃(F).

By Lemma 13 this operator is adjointable. This gives the step of induction. □

4 Proof of the main theorem

Let us choose a small subdivision of the polyhedron K, s.t.

diam(any simplex) < δ = \frac{1}{4 C^{3}} ,  (7)

where

∥F ∥ < C, ∥F^{−1}∥ < C  ∀ F ∈ K.  (8)

For any F let us denote its inverse by G. Let k ∈ K_{A} be an operator with dense image and without kernel. For any invertible G one can define invertible element by the following formula (k^{∗}(G)^{∗}Gk)^{1/2}k^{−1} on the dense set k(l_{2}(A)). This slightly generalize [1] in the present particular case. For this “positive square root” we have the corresponding “polar decomposition”, namely the following unitary operator is defined:

U_{F} := (k^{∗}(G)^{∗}Gk)^{1/2}k^{−1}F.  (9)

Let us define for each vertex F_{j} an operator k_{j} ∈ K_{A} in the following way. We put

k_{j}|_{F_{j}(A_{k})} := F_{j} (\frac{β_{k}}{2^{k}} ) (F_{j})^{−1},  (10)

where β_{k} are defined by (2). So,

k_{j}F_{j}e_{k}α_{k} = F_{j}e_{k}α_{k} \frac{1}{2^{k}} .  (11)

Let

U_{j} := (k_{j}^{∗}(G_{j})^{∗}G_{j}k_{j})^{1/2}(k_{j})^{−1}F_{j}  (12)

be the corresponding unitary.

Now we will chose a subsequence k(s) of k in the following way. Put

ρ := \frac{1}{8 C^{3}} .

Let k(1) = 1. Suppose, k(1), …, k(m) are chosen in a way such that for any s_{0} = 2, …, m, s = 1, …, s_{0} − 1, j = 1, …, N

∥⟨U_{j}e_{k(s_{0})}α_{k(s_{0})}, F_{j}e_{k(s)}α_{k(s)}⟩∥ < ρ · \frac{1}{(s_{0} − 1) · 2^{s_{0}−1}} ,

and

∥⟨U_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(s_{0})}α_{k(s_{0})}⟩∥ < ρ · \frac{1}{(s_{0} − 1) · 2^{s_{0}−1}} .

By Lemma 7 we can choose a large number k(m + 1) such that

∥⟨U_{j}e_{k(m+1)}α_{k(m+1)}, F_{j}e_{k(s)}α_{k(s)}⟩∥ < ρ · \frac{1}{m · 2^{m}} ,  s = 1, …, m, j = 1, …, N, (13)

and (taking into account (11)

∥⟨U_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(m+1)}α_{k(m+1)}⟩∥ =
= ∥⟨(k_{j}^{∗}(G_{j})^{∗}G_{j}k_{j})^{1/2}(k_{j})^{−1}F_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(m+1)}α_{k(m+1)}⟩∥
= ∥⟨(k_{j})^{−1}F_{j}e_{k(s)}α_{k(s)}, (k_{j}^{∗}(G_{j})^{∗}G_{j}k_{j})^{1/2}(k_{j})^{−1}F_{j}e_{k(m+1)}α_{k(m+1)} \frac{1}{2^{k(m+1)}} ⟩∥
= ∥⟨F_{j}e_{k(s)}α_{k(s)} · 2^{k(s)}, U_{j}e_{k(m+1)}α_{k(m+1)} \frac{1}{2^{k(m+1)}} ⟩∥
≤ ∥⟨F_{j}e_{k(s)}α_{k(s)}, U_{j}e_{k(m+1)}α_{k(m+1)}⟩∥
< ρ · \frac{1}{m · 2^{m}} ,  s = 1, …, m, j = 1, …, N,  (14)

Let H_{0} ≅ l_{2}(A) be a submodule with orthogonal complement, constructed as in Lemma 6, by vectors of the form e_{i_{k(s)}}α_{k(s)}ω_{μ}. Of course, the “positive square root” is not positive, but nevertheless we can prove

Lemma 16 For x ∈ H_{0} one has

Re ⟨(k_{j}^{∗}(G_{j})^{∗}G_{j}k_{j})^{1/2}(k_{j})^{−1}F_{j}x, F_{j}x⟩ ≥ \frac{1}{2 C^{3}} ⟨x, x⟩ ≥ \frac{1}{2 C^{5}} ⟨F_{j}x, F_{j}x⟩.  (15)

Proof: By (11) one has

⟨(k_{j}^{∗}(G_{j})^{∗}G_{j}k_{j})^{1/2}(k_{j})^{−1}F_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(s)}α_{k(s)}⟩ =
= 2^{k(s)}⟨(k_{j}^{∗}(G_{j})^{∗}G_{j}k_{j})^{1/2}F_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(s)}α_{k(s)}⟩ ≥

≥ 2^{k(s)} ⟨\frac{1}{C} (k_{j}^{∗}k_{j})^{1/2}F_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(s)}α_{k(s)}⟩ ≥

≥ 2^{k(s)}Re ⟨\frac{1}{C} k_{j}F_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(s)}α_{k(s)}⟩ =

= \frac{1}{C} ⟨F_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(s)}α_{k(s)}⟩ .

An arbitrary element x ∈ H_{0} has the form

x = ∑_{s} e_{k(s)}α_{k(s)}γ_{s} = ∑_{s} e_{k(s)}γ_{s},

⟨x, x⟩ = ∑_{s} γ_{s}^{∗}γ_{s}, this series is norm convergent (ω_{μ} “participate” in γ_{s}). Then by Lemma 8 and by (13) and (14)

Re ⟨(k_{j}^{∗}(G_{j})^{∗}G_{j}k_{j})^{1/2}(k_{j})^{−1}F_{j}x, F_{j}x⟩ =

= Re ∑_{s,t} γ_{s}^{∗}⟨(k_{j}^{∗}(G_{j})^{∗}G_{j}k_{j})^{1/2}(k_{j})^{−1}F_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(t)}α_{k(t)}⟩γ_{t} ≥
≥ ∑_{s} \frac{1}{C} γ_{s}^{∗} ⟨F_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(s)}α_{k(s)}⟩ γ_{s} −

− ∑_{s<t}(∥⟨(k_{j}^{∗}(G_{j})^{∗}G_{j}k_{j})^{1/2}(k_{j})^{−1}F_{j}e_{k(s)}α_{k(s)}, F_{j}e_{k(t)}α_{k(t)}⟩∥

+∥⟨(k_{j}^{∗}(G_{j})^{∗}G_{j}k_{j})^{1/2}(k_{j})^{−1}F_{j}e_{k(t)}α_{k(t)}, F_{j}e_{k(s)}α_{k(s)}⟩∥)(γ_{s}^{∗}γ_{s} + γ_{t}^{∗}γ_{t}) ≥
≥ \frac{1}{C} \frac{1}{C^{2}} ∑_{s} γ_{s}^{∗}γ_{s} − ∑_{t} ∑_{s=1}^{t−1} (2ρ · \frac{1}{t · 2^{t}} ) (γ_{s}^{∗}γ_{s} + γ_{t}^{∗}γ_{t}) ≥
≥ \frac{1}{C^{3}} ∑_{s} γ_{s}^{∗}γ_{s} − ∑_{t} (2ρ · \frac{1}{t · 2^{t}} ) ∑_{s=1}^{∞} γ_{s}^{∗}γ_{s} − ∑_{t} (2ρ · \frac{1}{2^{t}} ) γ_{t}^{∗}γ_{t} ≥
≥ (\frac{1}{C^{3}} − 4ρ) ∑_{s} γ_{s}^{∗}γ_{s} = \frac{1}{2 C^{3}} ∑_{s} γ_{s}^{∗}γ_{s} = \frac{1}{2 C^{3}} ⟨x, x⟩ ≥ \frac{1}{2 C^{5}} ⟨F′_{j}x, F′_{j}x⟩.
□

Let us define the following continuous family of operators D_{F} : H_{0} → l_{2}(A) extending simplex wise linearly U_{j}|_{H_{0}} to the whole complex K:

D_{F} (x) = ∑_{i=1}^{n} t_{i}U_{j(i)}(x), if F = ∑_{i=1}^{n} t_{i}F_{j(i)} ∈ simplex ⟨F_{j(1)}, …, F_{j(n)}⟩, x ∈ H_{0}, (16)

n = dim K = dim S^{n}. It is evident, that these operators are adjointable.

Lemma 17 1) The operators D_{F} are topological injections.

2) There exist a decomposition H_{0} = H′_{0}⊕H′′_{0} and a continuous family V_{F} of adjointable operators such that

V_{F} |_{H′_{0}} = D_{F} |_{H′_{0}}, V_{F} (H′_{0})⊥V_{F} ((H′_{0})^{⊥}).

3) There exists a constant c′ such that

Re ⟨V_{F} (x), F x⟩ ≥ c′⟨x, x⟩, x ∈ H′_{0}.

Proof: Let us find a constant c′ such that

Re ⟨D_{F} (x), F x⟩ ≥ c′⟨x, x⟩, x ∈ H_{0}.

Then 3) will be a particular case of this inequality and 1) will follow from it by Lemma 10.

Suppose, F_{j} and F_{k} are some vertices of one simplex. Then (as in Lemma 10) by Lemma 9

Re ⟨U_{j}(x), (F_{k} − F_{j})x⟩ = Re ⟨∥F_{k} − F_{j}∥^{1/2}U_{j}(x), \frac{1}{∥F_{k} − F_{j}∥^{1/2}} (F_{k} − F_{j})x⟩ ≤

≤ \frac{1}{2} (∥F_{k} − F_{j}∥ ⟨U_{j}(x), U_{j}(x)⟩ + \frac{1}{∥F_{k} − F_{j}∥} ⟨(F_{k} − F_{j})x, (F_{k} − F_{j})x⟩) ≤

≤ ∥F_{k} − F_{j}∥ ⟨x, x⟩

and by Lemma 16

Re ⟨U_{j}(x), F_{k}x⟩ = Re ⟨U_{j}(x), F_{j}x⟩ + Re ⟨U_{j}(x), (F_{k} − F_{j})x⟩ ≥

≥ Re ⟨U_{j}(x), F_{j}x⟩ − ∥F_{k} − F_{j}∥ ⟨x, x⟩ ≥ (\frac{1}{2 C^{3}} − δ) ⟨x, x⟩ = \frac{1}{4 C^{3}} ⟨x, x⟩.

and Hence, for x ∈ H_{0} we have

Re ⟨D_{F} (x), F x⟩ = Re ⟨∑_{i=1}^{n} t_{i}U_{j(i)}(x), ∑_{i=1}^{n} t_{i}F_{j(i)}⟩ = ∑_{i,m=1}^{n} t_{i}t_{m}Re ⟨U_{j(i)}(x), F_{j(m)}⟩ ≥

≥ ∑_{i,m=1}^{n} t_{i}t_{m} \frac{1}{4 C^{3}} ⟨x, x⟩ = \frac{1}{4 C^{3}} ⟨x, x⟩ =: c′ ⟨x, x⟩.

So, 1) and 3) are verified.

To construct a family as in 2) we would like to apply Lemma 15. For this purpose we have only to pass to any submodule H′_{0} ⊂ H_{0}, H_{0} = H′_{0} ⊕ (H′_{0})^{⊥}, H′_{0} ≅ (H′_{0})^{⊥}. Then Condition 2) of Lemma 15 holds, namely (Im D_{F} |_{H′_{0}})^{⊥} ≅ l_{2}(A). Indeed, this is a countable generated Hilbert A-module (as direct summand in l_{2}(A)) and by the Kasparov stabilization theorem [6]

(Im D_{F} |_{H′_{0}})^{⊥} ≅ D_{F} (H′′_{0}) ⊕ (D_{F} (H_{0}))^{⊥} ≅ l_{2}(A) ⊕ (D_{F} (H_{0}))^{⊥} ≅ l_{2}(A).

This completes the proof. □

Definition 18 Let us define a homotopy R_{t},

R_{t}(F) = R(F, t), R : K × [0, 1] → End_{A}(l_{2}(A))

by the formula

R_{t}(F) := \{ \begin{matrix}(1 − t)F + V_{F} & on H′_{0}, \\ V_{F} & on (H′_{0})^{⊥}. \end{matrix}

Due to linearity in t and continuity of V_{F} this family is continuous.

By 3) of Lemma 17 we can apply Lemma 12 to obtain that R_{t}(F) ∈ GL . The family R_{1} is in GL^{∗}. Hence, it is contractible by Cuntz and Higson theorem [3]. To complete the proof of main Theorem 1 it remains to verify that the families R_{0}(F) and F are homotopic to each other. But this fact is just Lemma 14.

□

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Moscow State University Max-Planck-Institut für Mathematik Dept. of Mechanics and Mathematics Vivatgasse 7 Chair of Higher Geometry and Topology D 53111 Bonn 119 899 Moscow Russia Germany

E-mail: troitsky@mech.math.msu.su  (Till 31.05.2001)
Home page URL: http://mech.math.msu.su/~troitsky