Induced map on K theory for certain \Gamma-equivariant maps between Hilbert spaces

Higson-Kapsparov-Trout introduced an infinite-dimensional Clifford algebra of a Hilbert space, and verified Bott periodicity on K-theory. To develop algebraic topology of maps between Hilbert spaces, in this paper we introduce an induced Hilbert Clifford algebra, and construct an induced map between K-theory of the Higson-Kasparov-Trout Clifford algebra and the induced Clifford algebra. We also compute its K-group for some concrete case.


Introduction
Let Γ be a discrete group, and H, H ′ be two Hilbert spaces on which Γ acts linearly and isometrically. Let F = l + c : H ′ → H be a Γequivariant map whose linear part is l, which is also Γ-equivariant. We want to construct is an "induced map" of K-theory of these infinitedimensional spaces. Of course we cannot obtain such a map in the usual sense because these spaces are locally non compact. Thus, we introduce the infinite-dimensional Clifford C * -algebras by Higson, Kasparov and Trout [HKT].
Let E be a finite-dimensional Euclidean space, and let Cl(E) be the complex Clifford algebra. There is a * -homomorphism β : C 0 (R) → C 0 (R)⊗C 0 (E, Cl(E)) called the Bott map, given by the functional calculus f → f (X⊗1 + 1⊗C) where X is an unbounded multiplier of C 0 (R) by X(f )(x) = xf (x), and C, which is called the Clifford operator, is also an unbounded multiplier of C 0 (E, Cl(E)) by C(v) = v. It turns out that β induces an isomorphism on K theory as follows: β * : K * (C 0 (R)) ∼ = K * (C 0 (R)⊗C 0 (E, Cl(E))).
HKT generalized its construction to obtain the Clifford algebra SC(H) for an infinite-dimensional Hilbert space H, and verified the isomorphism β * : K * (C 0 (R)) ∼ = K * (SC(H)). The idea is to use finite-dimensional approximation of the Hilbert space and inductively apply the Bott map.
To develop algebraic topology of maps between Hilbert spaces, our first step is to construct an induced map in K-theory. Let F = l + c : E ′ → E be a proper map such that l : E ∼ = E ′ gives a linear isomorphism, where l is its linear part and c is the non linear part between finite-dimensional Euclidean spaces. Then F induces a map F * : C 0 (E, Cl(E)) → C 0 (E ′ , Cl(E ′ )) given by wherel is the unitary of its polar decomposition. Notice that the image F * (C 0 (E, Cl(E))) ⊂ C 0 (E ′ , Cl(E ′ )) is a C * subalgebra.
It becomes clear why we usel rather than l to construct the infinitedimensional version of this map. Let F = l + c : H ′ → H be a map between two Hilbert spaces. To extend the above pull-back construction to the infinite-dimensional setting we have to impose extra conditions on F . We call such special maps finitely approximable. See Definition 3.1 in Section 3 for more detail. We then obtain the following result.
Proposition 1.1. Suppose F : H ′ → H is finitely approximable. Then there is an induced Clifford C * algebra SC F (H ′ ). This C * algebra coincides with F * (C 0 (E, Cl(E))) above, in finitedimensional case.
If a discrete group Γ acts linearly and isometrically, then it also acts on SC F (H).
The following is our main theorem.
If a discrete group Γ acts on both H ′ and H linearly and isometrically and F is Γ-finitely approximable, then F * is a Γ-equivariant * -homomorphism, that induces a homomorphism between K-groups where the crossed product is full. Suppose F : H ′ → H is strongly finitely approximable. Then by approximating these Hilbert spaces by finite-dimensional linear subspaces, we can obtain its degree deg(F ) ∈ Z. Then the above F * is given by F * : Z → Z which sends 1 to deg (F ) by choosing a suitable orientation.
We also compute the group K(SC F (H ′ ) ⋊ Z) for some concrete cases in Section 6.
In a successive paper, we will apply our construction of the K theoretic induced map to a monopole map that appears in gauge theory. Over a compact oriented four manifold, it turns our that the monopole map is strongly finitely approximable, and its degree coincides with the Bauer-Furuta degree when b 1 = 0 [BF]. We will verify that the covering monopole map on the universal covering space is Γ-finitely approximable, when its linear part gives a linear isomorphism. This produces a higher degree map of Bauer-Furuta type. The idea of the degree goes back to an old result by A.Schwarz [S].
2. Infinite-dimensional Bott periodicity 2.1. Quick review of HKT construction. We review the construction of the Hilbert space Clifford C * -algebras by Higson, Kasparov and Trout [HKT].
Let E be a finite-dimensional Euclidean space, and let Cl(E) be the complex Clifford algebras, where we choose positive sign on the multiplication e 2 = |e| 2 1 for every e ∈ E.
This admits a natural Z 2 -grading. The embedding C : E → Cl(E) gives a map which is called the Clifford operator. Let us denote C(E) = C 0 (E, Cl(E)). Let X : C 0 (R) → C 0 (R) be given by X(f )(x) = xf (x). Then C 0 (R) also admits a natural Z 2 -grading by even or odd functions. Both operators C and X are degree one and essentially self-adjoint unbounded multipliers on C(E) and C 0 (R) respectively. In particular X⊗1 + 1⊗C is a degree one and essentially self-adjoint unbounded multiplier on C 0 (R)⊗C(E).
Let us introduce a * -homomorphism defined by β : f → f (X⊗1 + 1⊗C) through functional calculus. Let E be a separable real Hilbert space, and E a ⊂ E b ⊂ E be a pair of finite-dimensional linear subspaces. We denote the orthogonal complement by E ba : Let us introduce a * -homomorphism passing through this isomorphism coincides with the * -homomorphism Definition 2.1. We denote the direct limit C * -algebras by where the direct limit is taken over all finite-dimensional linear subspaces of E.
It follows from the above construction that we can obtain a * homomorphism Suppose a discrete group Γ acts on E linearly and isometrically. It induces the action on SC(E) by γ(f⊗u)(v) = f⊗γ(u(γ −1 (v))).
Thus, the Bott map is Γ-equivariant. For a Γ-C * -algebra A, let us denote where the right hand side C * algebra is given by the full crossed product of A with Γ.
In particular it induces an isomorphism 2.2. Direct limit C * algebras. Let H be a Hilbert space on which Γ acts linearly and isometrically. Choose exhaustion by finite-dimensional linear subspaces V j ⊂ V j+1 with dense union ∪ j V j ⊂ H. Let 0 < r 1 < · · · < r i < r i+1 < · · · → ∞ be a divergent positive sequence with r i+1 > √ 2r i , and let D j r i ⊂ V j be the open disc with diameter r i .
Consider the diagram of the embeddings : : : : for a finite-dimensional vector space V . Then we have * -homomorphisms where the last embedding is the open inclusion.
Remark 2.3. Trout developed a Thom isomorphism on infinite-dimensional Euclidean bundles. One may regard C 0 (D j r i , Cl(V j )) as the set of continuous sections on the Clifford algebra of the tangent bundle Cl(T D j r i ) vanishing at infinity. Then β j,j ′ can be described as a * -homomorphism ). Then the above Bott map transforms as Lemma 2.4. The direct limit C * -algebra coincides with the Clifford C * -algebra of H. Proof.
Step 1: We claim that the commutativity holds. To make the notations clearer, let us denoteβ j,j ′ : This commutativity allows us to construct the direct limit C * -algebra There is a canonical isometric embedding Step 2: It remains to verify that the image of I is dense. For a linear inclusion V ֒→ H, let β : SC(V ) → SC(H) be the Bott *homomorphism into the Clifford C * algebra. An element a ∈ SC(H) is given as lim j β(a j ) for some a j ∈ SC(V j ). Let χ i ∈ C c ((−r i , r i ); [0, 1]) and ϕ i,j ∈ C c (D j r i ; [0, 1]) be cutoff functions with χ i |(−r i−1 , r i−1 ) ≡ 1 and ϕ i,j |D j r i−1 ≡ 1. Let us set ψ i,j = χ i⊗ ϕ i,j . We claim that b i,j := ψ i,j a j ∈ SC(D j r i ) converges to the same element: lim Choose any j 0 and ǫ > 0. There exists r > 0 such that a j 0 satisfies the estimate Thus, any a j ∈ SC(V j ) with j > j 0 also satisfies the estimate Then we introduce the distance between these planes by The following lemma will not be used later, but may be useful to understand how V ′ andV ′ differ from each other.
Lemma 2.5. Let W ′ i be a family of finite-dimensional linear subspaces with W ′ i ⊂ W ′ i+1 so that the union ∪ i W ′ i ⊂ H ′ is dense. For any finite-dimensional linear subspace V ′ ⊂ H ′ and any small ǫ > 0, there is some i 0 such that for all i ≥ i 0 , ||(1 −pr i )|V ′ || < ǫ holds, wherepr i : H ′ →W ′ i is the orthogonal projection andW ′ i := l −1 (l(W ′ i )). Proof. It is sufficient to verify that for any finite-dimensional linear subspace V ′ ⊂ H ′ and any ǫ > 0, the estimate d(V ′ ,W ′ i ) < ǫ holds for all large i >> 1. ActuallyW ′ = H ′ holds when W ′ = H ′ since the polar decomposition gives the unitary. Thus, for any finite-dimensional linear exhaustion l is asymptotically unitary if for any ǫ > 0, there is a finite-dimensional linear subspace V ⊂ H ′ such that the restriction satisfies the estimate on its operator norm ||(l −l)|V ⊥ || < ǫ wherel is the unitary of the polar decomposition of l : H ′ → H.
Remark 2.6. In a subsequent paper, we will verify that a self-adjoint elliptic operator on a compact manifold is asymptotically unitary between Sobolev spaces.
Lemma 2.7. Let l : H ′ ∼ = H be asymptotically unitary. For any ǫ > 0, there is a finite-dimensional vector subspace be a closed linear subspace, and (V ′ ) ⊥ be its orthogonal complement. Let pr : H ′ → V ′ be the orthogonal projection.
Step 1: Take a finite-dimensional subspace V ′ 0 ⊂ H ′ so that l satisfies the estimate ||(l−l)|(V ′ ) ⊥ || < ǫ for any V ′ ⊃ V ′ 0 . Then the operator norm of the restriction satisfies the estimate Decompose the operatorl * l with respect to V ′ ⊕ (V ′ ) ⊥ , and expressl * l by a matrix form A B C D where both the estimates ||D − id ||, ||B|| < ǫ hold.
Step 2: C = B * holds sincel * l is self-adjoint. Hence the estimate ||C|| < ǫ also holds. Then the conclusion holds because the estimate 2.4. A variant of Clifford C * -algebra. Let us introduce a variant of the HKT construction. Ultimately, the result of the C * -algebra turns out to be * -isomorphic to the original one given by HKT. This variant naturally appears when one considers the induced Clifford C * -algebra we introduce later.
Let l : H ′ ∼ = H be an asymptotically unitary isomorphism. Let E ⊂ H be a finite-dimensional Euclidean space, and denote and introduce a * -homomorphism be a pair of finite-dimensional linear subspaces, and denote the orthogonal complement as Then there is a finite-dimensional vector space V ′ ⊂ H ′ such that there is a canonical * -isomorphism Proof. Let V ′ be the vector subspace given by Lemma 2.7. LetÊ ′ ba ⊂ E ′ b be the orthogonal complement ofĒ ′ a , and consider the orthogonal projectionp r :Ē ′ ba →Ê ′ ba . By the assumption, l * •l is almost unitary on E ′ ba so that the operator norm satisfies the estimate ||(l * • l) − id |E ′ ba || < ǫ. The estimate d(E ′ a ,Ē ′ a ) < ǫ also holds from Lemma 2.7. Thus, the operator norm of the above projection satisfies the estimate ||pr − id |Ē ′ ba || < 2ǫ. In particular the projection gives an isomorphism. Letpr :Ē ′ ba → E ′ ba be the unitary of the polar decomposition. It also satisfies the estimate ||pr − id |Ē ′ ba || < 4ǫ, which induces a * -isomorphism pr : It follows from Lemma 2.8 that there is a canonical * -homomorphism Remark 2.9. Surelypr induces a linear map where Cl 0 (E) is the scalarless part of Cl(E). However it cannot be "almost" * -isomorphic in general, as dim E ′ ba grows. To see this, let us take any . . u ′′ m + other terms Each norm ||u ′′ i || < 1 is strictly less than 1, and hence, the norm of their product in the first term above may degenerate to zero.
Let A a be a family of C * -algebras, and β ba : For v a ∈ A a , introduce the set of equivalent classes Consider the algebra generated by elements of the formv a , where the sum is given byv The direct limit C * -algebra A with respect to the family {β ba : A a → A b } a,b is defined by the closure of the above algebra with the norm We also denote it as Let us set where E ′ a run over all finite-dimensional subspaces, and b ≥ a if and only if E ′ b ⊃ E ′ a holds. It follows from the proof of Lemma 2.8 that the following lemma holds.
Definition 2.3. Let l : H ′ ∼ = H be asymptotically unitary. The direct limit C * -algebra is given by where the norm is given in ( * ) above.
Proposition 2.11. Assume l is asymptotically unitary. Then there is a canonical * -isomorphism between the two Clifford C * algebras. If a group Γ acts on H ′ linearly and isometrically and l is Γ-equivariant, then this * -isomorphism is Γ-equivariant.

Proof.
Step 1: It follows from Lemma 2.7 and the assumption that for any ǫ > 0, there is a finite-dimensional vector space V ′ 0 ⊂ H ′ such that for all E ′ a ⊃ V ′ 0 , the following two estimates hold: ba be the orthogonal projections. Their corresponding unitaries pr i satisfy the bounds They extend to * -isomorphisms . In particular they induce the * -isomorphisms Step 2: Let us consider two Bott maps Denote pr 21 := pr 2⊗ pr 1 . Then this diagram satisfies the estimate ǫ can be arbitrarily small by choosing large E ′ a .
Step 3: Let us take an element . It follows from the above estimate on the diagram that is uniquely defined and independent of choice of x a .
It is easy to check that this assignment gives a * -homomorphism. To see that it is isomorphic, we consider a converse projection, from pr ′ : E ′ a ∼ =Ē ′ a . A parallel argument gives another * -homomorphism pr ′ : SC(H ′ ) → SC l (H ′ ), and their compositions give the required identities.
Step 4: Let us consider Γ-equivariance. Suppose Γ acts on H ′ linearly and isometrically. We claim that pr 1 :Ē ′ a → E ′ a is Γ-equivariant. To see this, notice thatl and hencel * • l are both Γ-equivariant. Then we have the equalities Therefore, As the Bott map is also Γ-equivariant, the process from step 1 to step 3 works equivariantly.

Finite-dimensional approximation
Let F : H ′ → H be a metrically proper map between Hilbert spaces. Then, there is a proper and increasing function g holds for all m ∈ H ′ . Later we analyze a family of maps of the form We say that the family of maps {F i } i is proper, if there are positive and increasing numbers r i , s i → ∞ such that the inclusion holds: Suppose l is surjective and c is compact on each bounded set. Then there is a proper and increasing function f : [0, ∞) → [0, ∞) such that the following holds: for any r > 0 and 1 ≥ δ 0 > 0, there is a finitedimensional linear subspace W ′ 0 ⊂ H ′ such that for any linear subspace and pr is the orthogonal projection to W . Moreover the estimate holds Proof. Let C ⊂ H be the closure of the image c(D r ), which is compact. Hence there are finitely many points w 1 , . . . , w k ∈ c(D r ) such that their δ 0 neighborhoods cover C.
Since g is increasing, we obtain the estimates The function f (x) = g(x + 1) satisfies the desired property. For any other linear subspace Let W ′ i ⊂ H ′ and W i ⊂ H be two families of finite-dimensional linear subspaces. Let us say that a family of linear isomorphisms is an asymptotic unitary family if the following conditions hold: (1) there exists an asymptotically unitary map l : Let us introduce an approximation of F as a family of maps on finitedimensional linear subspaces. Let D ′ r i ⊂ H ′ and D s i ⊂ H be r i and s i balls respectively.
Definition 3.1. Let F = l + c : H ′ → H be a metrically proper map, where l is its linear part and c is a nonlinear term. Let us say that F is finitely approximable if there is an increasing family of finitedimensional linear subspaces there are two sequences s 0 < s 1 < · · · → ∞ and r 0 < r 1 < · · · → ∞ with r i ≥ s i such that the embedding Let us also say that F is strongly finitely approximable if it is finitely approximable, The following restates Lemma 3.1.
Corollary 3.2. Let F = l + c : H ′ → H be a metrically proper map such that l is asymptotically unitary and c is compact on each bounded set. Then F is strongly finitely approximable.
Suppose both H ′ and H admit linear isometric actions by a group Γ and assume that both F and l are Γ-equivariant where F = l + c. Then we say that F is Γ-finitely approximable, if moreover the above family {W ′ i } i satisfies that the union holds. Let us take γ ∈ Γ, and consider the γ shift of the finite approximation data . It is clear that the above shift gives another finite approximation of F .

Induced Clifford C * -algebra
Let F = l + c : H ′ → H be a map. We aim here is to construct an "induced" Clifford C * -algebra SC F (H ′ ). 4.1. Model case. Let us start with a model case that consists of a proper and nonlinear map F = l + c : E ′ → E between finite-dimensional Euclidean spaces, where l is a linear isomorphism. Consider a * -homomorphism , and denote its image by which is a C * -subalgebra in SC(E ′ ), whose norm is denoted by || || SC F .
The induced map is called the induced Clifford operator. We use it to introduce a *homomorphism Now suppose a Hilbert space H ′ is spanned by an infinite family of finite-dimensional Euclidean planes as E ′ 1 ⊕ E ′ 2 ⊕ . . . and assume there is a family of proper maps Lemma 4.1. Let F = (F 1 , F 2 ) be diagonal as above. Then Then the induced Bott map is given by More generally, one can induce . . ). Then the direct limit C * algebra is given by whose norm is given as above.
Notice that SC F (H ′ ) is no longer a C * -subalgebra of SC(H ′ ).
Lemma 4.2. In the case when c i ≡ 0 and hence F i = l i for all i, the induced Clifford C * -algebra admits a canonical * -isomorphism Proof. This follows from Proposition 2.11 with the coincidence where the right-hand side is given in Definition 2.3.

4.2.
Induced Clifford C * -algebra. Assume that F = l + c : H ′ → H is finitely approximable as in Definition 3.1 with respect to the data be the set of continuous functions on (−r, r) vanishing at infinity, and consider the following C * -subalgebras which is a C * -subalgebra with the norm || || Sr i C F i . Let us consider a family of elements Remark 4.3. Consider the induced Clifford operator . Note that the norms satisfy the inequality where the right-hand side is the restriction norm.
For an F -compatible sequence α = {α i } i≥i 0 , the limit exists because both F i and l i converge weakly (see Definition 3.1). Moreover both F * i and β are * -homomorphisms between C * -algebras and so are both norm-decreasing.
Definition 4.1. Let F be finitely approximable. The induced Clifford C * -algebra is given by which is obtained by the norm closure of all F -compatible sequences, where the norm is the above one. (2) When F = l is asymptotically unitary, there is a natural * - where the right-hand side is given in Definition 2.3.

Proof. One can choose W
Let us consider (2), and set F = l. Recall the Bott map which is given above of the Definition 2.3, and denote it as be the unitary of the polar decomposition.
Take an element {α i } i ∈ SC F (H ′ ), with α i = l * i (u i ) and u i = β(u i 0 ) ∈ SC(W i ). Note that the restriction β(u i 0 )| W i 0 = u i 0 holds. Then by the condition of asymptotic unitarity, the restriction of their difference satisfies the estimate Φ is norm-preserving, so it extends to an injective * -homomorphism from SC F (H ′ ). Let us verify that it is surjective. One can follow in a converse way to the above. Take an element δ = β l (δ i 0 ) ∈ SC l (H ′ ) with δ i 0 ∈ SC l (W ′ i 0 ), and set δ i = l * (β(δ i 0 )). Let us set w i 0 = (l * ) −1 (δ i 0 ) ∈ SC(W i 0 ) by v →l(δ i 0 (l −1 (v))). Then we set . The restriction of their difference satisfies the estimate Hence Φ is an isometric * -homomorphism with dense image. This implies that it is surjective.
Let us verify the last property (3). Choose any subindices j i ≥ i for i = 1, 2, . . . , and consider the sub-approximation given by the data {F j i } i . If we replace the original data {F i } i by this subdata, still we obtain the same C * -algebra SC F (H ′ ) as their norms coincide as follows: Let us take two Γ-finite approximations and denote them by F l i : Let us set α ′ i = (F 2 i ) * (u i ). Then it follows from the definition of F -compatible sequence that the convergence holds. Combining this result with the above, we obtain the desired conclusion.
Lemma 4.5. If F is Γ-finitely approximable, then there is a canonical Γ-action on SC F (H ′ ).

Higher degree * -homomorphism
Let F = l + c : H ′ → H be a Γ-equivariant nonlinear map, whose linear part l gives an isomorphism. For a finite-dimensional linear subspace V ⊂ H, denote the orthogonal projection by pr V : H → V . For V ′ = l −1 (V ), we have the modified map Our initial idea was to pull back Let us explain how difficulty arises if one tries to obtain a * -homomorphism in this way. For simplicity, assume l is unitary and the image of c is contained in a finite-dimensional linear subspace V ⊂ H. This will be the simplest situation but already some difficulty appears when we try to construct the induced * -homomorphism by F .
Assume F is metrically proper. This is equivalent to saying that the restriction F : V ′ → V is proper in this particular situation, where V ′ = l −1 (V ) is the finite-dimensional linear subspace. Let us consider the diagram This diagram is far from commutative as the following map c : (W ′ i ) ⊥ ∩ W ′ i+1 → V can affect to control the behavior of F as i → ∞. Thus, the induced maps by F * i will not converge in SC(H ′ ) in general. This is a point where we have account for the nonlinearity of F to construct the target C * -algebra, and is the reason we have to use SC F (H ′ ) instead of SC(H ′ ) below.
5.1. Degree of proper maps. Let E ′ , E be two finite-dimensional vector spaces, and F = l + c : E ′ → E be a proper smooth map whose linear part l : E ′ ∼ = E gives an isomorphism.
Let us reconstruct the degree of F ∈ Z by use of l. Letl : E ′ → E be the unitary corresponding to the polar decomposition. Then,l induces the algebra isomorphisml : Cl(E ′ ) ∼ = Cl(E), and we have the induced * -homomorphism Recall SC F (E ′ ) = F * (SC(E)). Then F * can be described as a *homomorphism F * : SC(E) → SC F (E ′ ). Let us consider the induced homomorphisms between K-groups where both β give the isomorphisms by 2.2. LetF * : K 1 (C 0 (R)) → K 1 (C 0 (R)) be the homomorphism determined uniquely so that the diagram commutes. Let us equip orientations on both E ′ and E so that l preserves them.
Lemma 5.1. Passing through the isomorphism K 1 (C 0 (R)) ∼ = Z, F * : Z → Z is given by multiplication by the degree of F .

Proof.
Step 1: Let us consider the composition of * -homomorphisms where the first map is F * and the second map is given by The latter gives an isomorphism since l is isomorphic. Thus, it is sufficient to see the conclusion for the composition. The composition is given by Step 2: Let l t : E ′ ∼ = E be another family of linear isomorphisms with l 0 =l and l 1 = l. It induces a family of * -homomorphisms Since homotopic * -homomorphisms induce the same maps between their K-groups, it is sufficient to see the conclusion for F * 1 . Noting the equality F • l −1 = 1 + c • l −1 , it is enough to assume l is the identity.
Step 3: When l is the identity, F * : SC(E) → SC(E) is given by whose induced homomorphim on a K-group is given by degree F , passing through the isomorphism where * is 0 or 1 with respect to whether dim E is even or odd. The first isomorphism comes from Proposition 2.2, and the second is the classical Bott periodicity (see [A]).

Induced map for a strongly finitely approximable map.
Let F = l + c : H ′ → H be a strongly finitely approximable map. There are finite-dimensional linear subspaces W ′ i ⊂ W ′ i+1 ⊂ · · · ⊂ H ′ whose union is dense, such that the compositions with the projections pr i • F : W ′ i → W i = l(W i ) consist of a finitely approximable data with the constants r i , s i → ∞.
Let us consider the restriction . Then by definition, the estimate . By contrast, β(f⊗h) = β(f )⊗h and, hence,

Proof.
Step 1: Let us take an element α ∈ SC(H) and its approximation α i ∈ S r i C(D r i ) with lim i→∞ β(α i ) = α ∈ SC(H) by Lemma 2.4.
Assume l : H ′ ∼ = H is unitary, and consider the following two elements: ). Then by Sublemma 5.2 we have the estimates The first term on the right-hand side converges to zero since ||β(α i )|| are uniformly bounded with δ i → 0. The second term also converges to zero. Thus, the * -homomorphisms asymptotically commute with the Bott map. Hence, the sequence β(F * i (α i )) ∈ SC(H ′ ) converges, and gives a * -homomorphism F * : α → F * (α) := lim i β(F * i (α i )). Clearly this assignment is independent of the choice of approximations of α.
Step 2: Let us consider the case when l is not necessarily unitary, but is asymptotically unitary.
Let β l : S → SC l (U ′ i ) be the variant of the Bott map in 2.4. Then the same argument to Sublemma 5.2 verifies the equality . Hence the parallel estimate to step 1 above verifies that the sequence converges β l (F * i (α i )) ∈ SC l (H ′ ). This also gives a * -homomorphism F * : α → F * (α) := lim i β l (F * i (α i )). As SC l (H ′ ) ∼ = SC(H ′ ) are * -isomorphic by Proposition 2.11, we obtain the desired * -homomorphism.
Remark 5.4. Suppose F = l + c satisfies the conditions to be strongly finitely approximable, except that l is not necessarily isomorphic, but the Fredholm index is zero.
We can still construct the induced * -homomorphism F * : SC(H) → SC(H ′ ) as below.
There are finite-dimensional linear subspaces V ′ ⊂ H ′ and V ⊂ H such that the restriction gives an isomorphism l : (V ′ ) ⊥ ∼ = V ⊥ , where V ⊥ ⊂ H is the orthogonal complement. Choose any unitary l ′ : V ′ ∼ = V and take their sum Let us use L to pull back the Clifford algebras and use F itself to pull back the functions. Then we can follow from step 1 and step 2 in the same way.
Definition 5.1. Let F : H ′ → H be a strongly finitely approximable map. Then the induced map is given by multiplication by an integer degree F ∈ Z. We call it the K-theoretic degree of F .

5.3.
Induced map for Γ-finitely approximable map. Let us start from a general property, and let H be a Hilbert space with exhaustion W 0 ⊂ · · · ⊂ W i ⊂ · · · ⊂ H by finite-dimensional linear subspaces. Choose divergent numbers r i < r i+1 < · · · → ∞, and denote r i balls by D r i ⊂ W i . Let S r = C c (−r, r) ⊂ S be the set of compactly supported continuous functions on (−r, r).
The following restates Lemma 2.4 Lemma 5.5. For any α ∈ SC(H), there is a family such that their images by the Bott map converge to α lim i→∞ β(α i ) = α ∈ SC(H).

5.3.1.
Induced * -homomorphism. Let H ′ , H be Hilbert spaces on which Γ act linearly and isometrically, and let F = l + c : H ′ → H be a Γequivariant map such that l : H ′ ∼ = H is a linear isomorphism. Assume that F is Γ-finitely approximable so that there is a family of finite-dimensional linear subspaces W ′ 0 ⊂ W ′ 1 ⊂ · · · ⊂ W ′ i ⊂ · · · ⊂ H ′ with dense union, and a family of maps F i : Moreover the following convergences hold for each i 0 : Recall the induced * -homomorphism i ) and the induced Clifford C * -algebra SC F (H ′ ) in Definition 4.1.
Then it induces the equivariant * -homomorphism Proof. Let us take an element v ∈ SC(H) and its approximation Let us recall the * -homomorphism in 4.2 since both F * i and β are * -homomorphisms. Note that the composition of two * -homomorphisms For a small ǫ > 0, take two sufficiently large i ′ 0 ≥ i 0 >> 1 such that the estimate ||β( Thus, we obtain the assignment v → lim i 0 →∞ F * (v i 0 ), which gives a Γ-equivariant * -homomorphism where {v i } i is any approximation of v.
Definition 5.2. Let F : H ′ → H be a Γ-finitely approximable map. Then, the higher degree of F is given by the induced homomorphism

Computation of K-group of induced Clifford C * -algebras
We compute the equivariant K-group of induced Clifford C * -algebras for some particular cases. This can be a simple model case for further computation of the groups.
6.1. Basics. Let us collect some of basics which we will need. We start from some analytic aspects of Sobolev spaces. We denote by W k,2 as the Sobolev k-norm which is a linear subspace of L 2 . It is a Hilbert space and, hence, complete by the norm which involves derivatives up to the k-th order, and incomplete with respect to the L 2 inner product for k ≥ 1.
The following is well known.
Lemma 6.1. Suppose k ≥ 1. Then (1) The multiplication is compact on each bounded set.
In particular an element in W k,2 (S 1 ) can be regarded as a continuous function.
Later we will consider the non linear map F : W k,2 (S 1 ) → W k,2 (S 1 ) by F (a) = a + a 3 .
(1) Let A be a C * -algebra on which a finite cyclic group Z l acts. Then the crossed product is defined as A ⋊ Z l = {(a g ) g∈Z l } with their product by (a g )(b g ) = ( g 1 g 2 =g∈Z l a g 1 g 1 (b g 2 )). It induces the action by Z on A by using the natural projection π l : Z → Z l . In such situation, there exists a six term exact sequence between K * (A⋊Z) and K * (A⋊Z l ). However this does not seem to contain enough information to apply to our situation. We proceed in a direct way. Recall that an element a ∈ A ⋊ Z can be approximated by a ′ ∈ C c (Z, A).
(2) Let us take an element u ∈ K(A ⋊ Z) and represent it by u = 6.2. Computation of equivariant K-group for a toy model. 6.2.1. Finite cyclic and finite-dimensional case. Consider a Z 2 -equivariant map F : R 2 → R 2 by a b → a + b 3 b + a 3 where the involution acts by the coordinate change.
We claim that this is proper of non-zero degree. In fact, if a+ b 3 = 0, then the equality b + a 3 = b − b 3 2 implies properness.
Consider a Z 2 -equivariant perturbation for t ∈ (0, 1]. If ta + b 3 = 0, then tb + a 3 = tb − t −3 b 3 2 . Thus, this is a family of proper maps. At t = 0, F 0 : R 2 → R 2 is a proper map of degree −1, since it is again Z 2 -equivariantly proper-homotopic to the involution Note that it becomes degree zero, if we replace the exponent 3 by 2.
Next, we generalize slightly as follows. Consider a Z l equivariant map F : where the action is given by cyclic permutation of the coordinates. By the parallel argument as above, this turns out to be a proper map. To compute its degree, consider a perturbation for t ∈ (0, 1]. This is a family of Z l -equivariant proper maps, and at t = 0, F 0 : R l → R l is a proper map of degree ±1, determined by the parity of l. In fact there is a Z l -equivariant proper-homotopy F l t to the cyclic permutation on the equivariant K-theory. Proof. F is Z l -equivariantly properly homotopic to T l above, and so the isomorphism K Z l 1 (SC F (R l )) ∼ = K Z l 1 (SC T l (R l )) holds. T l is a Z l -equivariantly linear isomorphism because Z l is commutative. It follows from Definition 2.2 that a linear isomorphism between finite-dimensional vector spaces is asymptotically unitary. Then by Proposition 2.11 that SC T l (H ′ ) is Z l -equivariantly * -isomorphic to SC(H ′ ). In particular we have the isomorphism The last isomorphism comes from HKT-Bott periodicity for Euclidean space. Then we consider the map F : H ′ → H by F : If we restrict on R 2l+1 ⊂ H ′ by (a −l , . . . , a l ) → (. . . , a −l , . . . , a l , 0, . . . ), then its image is in R 2l+2 ⊂ H. In fact F : Let us consider the map F l : R 2l+1 → R 2l+1 by which moves the last component to the first one. In fact F l is still a proper map as presented in 6.2.1.
Let W ′ l = R 2l+1 be as above. Then the data (F l , W ′ l ) gives the Z-finite approximation in the sense of Definition 3.1.
First, as in the finite cyclic case, we obtain the isomorphism (SC(W ′ l )) on the equivariant K-theory. Notice that this isomorphism heavily depends on the degree being equal to ±1.
Lemma 6.4. The induced * -homomorphism Proof. Injectivity follows from surjectivity of F l , because it has a nonzero degree. It has a closed range since it is isometric embedding. Then, the conclusion follows since it has a dense range.
Second, we obtain the inductive system ). By definition, the equality holds Forgetting the group action, we have the isomorphisms where we used the HKT-Bott periodicity. Now consider the group action by Z. Let F l t be the homotopy in subsection 6.2.1, where F l 1 = F l and F l 0 = T l . Lemma 6.5. There is a * -isomorphism Proof. In fact an element u ∈ SC F l 0 (W ′ l ) is expressed as u = (F l 0 ) * (v 0 ) for some v 0 ∈ SC(W l ). Because F l t has a non zero degree, it follows that v 0 is uniquely determined by u. Then assign v 1 = (F l 1 ) * (v 0 ), and denote its map by . This is a * -homomorphism and, in fact, is an isomorphism, since if we do the same thing, replacing the role of F l 0 and F l 1 , then we can recover u again.
Let v ′ ∈ C({−l, . . . , l}, Mat(SC F l (W ′ l ))) be another approximation and takeṽ ′′ ≡ I −1 l (v ′ ) ∈ C({−l, . . . , l}, Mat(SC T l (W ′ l ))). Then we have the estimates for a small ǫ ′′ > 0. This implies the equality Therefore, we obtain a well defined group homomorphism If we replace the role of F and T and proceed in the same way as above, we obtain another map in a converse direction. By construction, their compositions are both the identities. Therefore, this is an isomorphism on the K-groups.
Since the translation shift T : H ′ ∼ = H ′ is unitary and Z is commutative, there is a * -isomorphism Passing through this isomorphism, we obtain the isomorphism The right-hand side is isomorphic to HKT. 6.3. Nonlinear maps between Sobolev spaces over the circle.
We again consider the non linear map with H ′ = H where the power is taken pointwisely. Then, the map can be written as As we have seen, this is metrically proper. Let k = 1 for simplicity of notation, and consider an element a ∈ W 1,2 (S 1 ) (2πks) and denote Lemma 6.7. Suppose ||a|| W 1,2 ≤ r. Then for any ǫ > 0, there is n = n(r, ǫ) ≥ 0 such that the estimate holds || |k|≥n+1 c k sin(2πks) + d k cos(2πks)|| W 1,2 < ǫ.
Proof. It follows from Lemma 6.1 that W 1,2 (S 1 ) → W 1,2 (S 1 ) by a → a 3 is compact on each bounded set.
Choose divergent numbers as lim i n i = ∞. For each i ∈ N, let V ′ i ⊂ W 1,2 (0, 1) 0 be the finite-dimensional linear subspace spanned by sin(2πks) and cos(2πks) for |k| ≤ n i , and set i as the orthogonal projection. Then, the composition F i ≡ pr i • F : W ′ i → W i gives a strongly finitely approximable data with some s i , r i . Proposition 6.8. There is a Z 2 equivariant * -isomorphism K Z 2 1 (SC F (H ⊕ H)) ∼ = K Z 2 1 (SC (H ⊕ H)). Proof.
Step 1: By the same argument as the toy case, F is metrically proper, and it is Z 2 -equivariantly properly homotopic to the involution I : H ⊕ H ∼ = H ⊕ H by F t .  u(a, b)) with a, b ∈ H.
Step 2: It follows from Lemma 6.7 that holds. As in the toy case, one may assume the same property where F t i = pr i • F t . K-theory is stable under these continuous deformations so that the isomorphism holds Step 3: Recall the induced Clifford C * -algebra SC F (H) whose element {α i } i satisfies the equality α i ) (and l i is the identity in this particular case).
Note that F i |D ′ r i has non-zero degree. We claim that there is a *homomorphism ) which sends α i to α i+1 . In fact α i uniquely determines u i . Suppose the contrary, and choose two elements u i , u ′ i ∈ S s i C(D s i ∩ W i ) with F * i (u i ) = F * i (u ′ i ). If u i = u ′ i could hold, then there exists m ∈ D s i ∩ W i with u i (m) = u ′ i (m). However, since F i has non-zero degree and is hence surjective, there exists x ∈ D r i with F i (x) = m. Then, we have the equality u i (m) = F * i (u i )(x) = F * i (u ′ i )(x) = u ′ i (m), which contradicts to the assumption. Now, since F * i : is an isometric * -embedding, it follows that the inverse Then, Φ i is given by the compositions F * i+1 •β •(F * i ) −1 .
Then, the map can be written as By the same argument as the toy case, this is metrically proper, and its nonlinear part is compact on each bounded set.
By use of F t as above, F is Z l -equivariantly properly homotopic to the cyclic shift T . By a similar argument, we have the following corollary.
Corollary 6.10. There is a Z l -equivariant * -isomorphism K Z l 1 (SC F (H l )) ∼ = K Z l 1 (SC(H l )) ∼ = K Z l 1 (S) where Z l acts on H l by the cyclic permutation of the components.
The above computation is applicable to more general situations of F , and is not restricted to such a specified form of the non linear term. 6.3.3. Infinite cyclic case. It is not so immediate to extend the above finite cyclic case to the infinite case, following the same approach. For example the map l 2 (Z) → l 2 (Z) by {a i } i → {a 3 i+1 } i is not proper. Therefore, we use a very specific approach to compute the Z case. Let H be the Hilbert space identified as H = W k,2 (0, 1) 0 ⊂ W k,2 (R), and let H be the closure of the sum ⊕ i∈Z H i , where H i are the copies of the same H. Then, the T orbit of H, {T n (H)} n∈Z generates H ⊂ W k,2 (R), where T : W k,2 (i, i + 1) 0 ∼ = W k,2 (i + 1, i + 2) 0 is the shift as before.