Induced map on K theory for certain Γ-equivariant maps between Hilbert spaces
aa r X i v : . [ m a t h . K T ] N ov INDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES
TSUYOSHI KATO
Abstract.
Higson-Kapsparov-Trout introduced an infinite-dimensional Clif-ford algebra of a Hilbert space, and verified Bott periodicity on K theory.To develop algebraic topology of maps between Hilbert spaces, in this paperwe introduce an induced Hilbert Clifford algebra, and construct an inducedmap between K -theory of the Higson-Kasparov-Trout Clifford algebra and theinduced 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 whichΓ acts linearly and isometrically. Let F = l + c : H ′ → H be a Γ-equivariant map whose linear part is l , which is also Γ-equivariant. Wewant to construct is an “induced map” of K -theory of these infinite-dimensional spaces. Of course we cannot obtain such a map in the usualsense because these spaces are locally non compact. Thus, we introducethe infinite-dimensional Clifford C ∗ -algebras by Higson, Kasparov andTrout [HKT].Let E be a finite-dimensional Euclidean space, and let Cl ( E ) be thecomplex Clifford algebra. There is a ∗ -homomorphism β : C ( R ) → C ( R ) ˆ ⊗ C ( E, Cl ( E )) called the Bott map, given by the functional cal-culus f → f ( X ˆ ⊗ ⊗ C )where X is an unbounded multiplier of C ( R ) by X ( f )( x ) = xf ( x ),and C , which is called the Clifford operator, is also an unboundedmultiplier of C ( E, Cl ( E )) by C ( v ) = v . It turns out that β inducesan isomorphism on K theory as follows: β ∗ : K ∗ ( C ( R )) ∼ = K ∗ ( C ( R ) ˆ ⊗ C ( E, Cl ( E ))) . HKT generalized its construction to obtain the Clifford algebra S C ( H )for an infinite-dimensional Hilbert space H , and verified the isomor-phism β ∗ : K ∗ ( C ( R )) ∼ = K ∗ ( S C ( H )) . The idea is to use finite-dimensional approximation of the Hilbert spaceand inductively apply the Bott map. Mathematics Subject Classification:
Primary 46L80, Secondary 46L85.
Key words and phrases. K theory, degree, ∞ dimensional Bott periodicity. To develop algebraic topology of maps between Hilbert spaces, ourfirst 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 linearisomorphism, where l is its linear part and c is the non linear partbetween finite-dimensional Euclidean spaces. Then F induces a map F ∗ : C ( E, Cl ( E )) → C ( E ′ , Cl ( E ′ ))given by u (cid:16) v ′ ¯ l ∗ (cid:0) u ( F ( v ′ )) (cid:1)(cid:17) where ¯ l is the unitary of its polar decomposition. Notice that the image F ∗ ( C ( E, Cl ( E ))) ⊂ C ( E ′ , Cl ( E ′ )) is a C ∗ subalgebra.It becomes clear why we use ¯ l rather than l to construct the infinite-dimensional version of this map. Let F = l + c : H ′ → H be a map be-tween two Hilbert spaces. To extend the above pull-back constructionto the infinite-dimensional setting we have to impose extra conditionson F . We call such special maps finitely approximable. See Definition3.1 in Section 3 for more detail. We then obtain the following result. Proposition 1.1.
Suppose F : H ′ → H is finitely approximable. Thenthere is an induced Clifford C ∗ algebra S C F ( H ′ ) . This C ∗ algebra coincides with F ∗ ( C ( E, Cl ( E ))) above, in finite-dimensional case.If a discrete group Γ acts linearly and isometrically, then it also actson S C F ( H ).The following is our main theorem. Theorem 1.2.
Suppose F : H ′ → H is finitely approximable. Then itinduces a ∗ -homomorphism F ∗ : S C ( H ) → S C F ( H ′ ) . In particular it induces a homomorphism between K -groups F ∗ : K ∗ ( S C ( H )) → K ∗ ( S C F ( H ′ )) . If a discrete group Γ acts on both H ′ and H linearly and isomet-rically and F is Γ-finitely approximable, then F ∗ is a Γ-equivariant ∗ -homomorphism, that induces a homomorphism between K -groups F ∗ : K ∗ ( S C ( H ) ⋊ Γ) → K ∗ ( S C F ( H ′ ) ⋊ Γ)where the crossed product is full.Suppose F : H ′ → H is strongly finitely approximable. Then byapproximating these Hilbert spaces by finite-dimensional linear sub-spaces, we can obtain its degree deg( F ) ∈ Z . Then the above F ∗ is NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES3 given by F ∗ : Z → Z which sends 1 to deg ( F ) by choosing a suitable orientation.We also compute the group K ( S C F ( H ′ ) ⋊ Z ) for some concrete casesin Section 6.In a successive paper, we will apply our construction of the K the-oretic induced map to a monopole map that appears in gauge theory.Over a compact oriented four manifold, it turns our that the monopolemap is strongly finitely approximable, and its degree coincides withthe Bauer-Furuta degree when b = 0 [BF]. We will verify that thecovering monopole map on the universal covering space is Γ-finitelyapproximable, when its linear part gives a linear isomorphism. Thisproduces a higher degree map of Bauer-Furuta type. The idea of thedegree goes back to an old result by A.Schwarz [S].2. Infinite-dimensional Bott periodicity
Quick review of HKT construction.
We review the construc-tion of the Hilbert space Clifford C ∗ -algebras by Higson, Kasparov andTrout [HKT].Let E be a finite-dimensional Euclidean space, and let Cl ( E ) bethe complex Clifford algebras, where we choose positive sign on themultiplication e = | e | e ∈ E .This admits a natural Z -grading. The embedding C : E → Cl ( E )gives a map which is called the Clifford operator. Let us denote C ( E ) = C ( E, Cl ( E )). Let X : C ( R ) → C ( R ) be given by X ( f )( x ) = xf ( x ).Then C ( R ) also admits a natural Z -grading by even or odd functions.Both operators C and X are degree one and essentially self-adjointunbounded multipliers on C ( E ) and C ( R ) respectively. In particular X ˆ ⊗ ⊗ C is a degree one and essentially self-adjoint unboundedmultiplier on C ( R ) ˆ ⊗ C ( E ).Let us introduce a ∗ -homomorphism β : C ( R ) → S C ( E ) := C ( R ) ˆ ⊗ C ( E )defined by β : f → f ( X ˆ ⊗ ⊗ 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. Wedenote the orthogonal complement by E ba := E b ∩ E ⊥ a . Then we havethe canonical isomorphism S C ( E b ) ∼ = S C ( E ba ) ˆ ⊗ C ( E a ) of C ∗ algebras.Let us introduce a ∗ -homomorphism passing through this isomorphism β ba = β ˆ ⊗ S C ( E a ) → S C ( E ba ) ˆ ⊗ C ( E a ) = S C ( E b ) TSUYOSHI KATO
Lemma 2.1 (HKT, Proposition 3 . . Let E a ⊂ E b ⊂ E c . Then thecomposition S C ( E a ) β ba −→ S C ( E b ) β cb −→ S C ( E c ) coincides with the ∗ -homomorphism β ca : S C ( E a ) → S C ( E c ) . Definition 2.1.
We denote the direct limit C ∗ -algebras by S C ( E ) = lim −→ a S C ( E a ) where the direct limit is taken over all finite-dimensional linear sub-spaces of E . It follows from the above construction that we can obtain a ∗ homo-morphism β : C ( R ) → S C ( E ) . Suppose a discrete group Γ acts on E linearly and isometrically. Itinduces the action on S C ( E ) by γ ( f ˆ ⊗ u )( v ) = f ˆ ⊗ γ ( u ( γ − ( v ))) . Thus, the Bott map is Γ-equivariant. For a Γ- C ∗ -algebra A , let usdenote K Γ ( A ) := K ( A ⋊ Γ)where the right hand side C ∗ algebra is given by the full crossed productof A with Γ. Proposition 2.2 (HKT, Theorem 3 . . β gives an equivariant asymp-totic equivalence from S to S C ( E ) .In particular it induces an isomorphism β ∗ : K Γ ∗ ( C ( R )) ∼ = K Γ ∗ ( S C ( E )) . Direct limit C ∗ algebras. Let H be a Hilbert space on which Γacts linearly and isometrically. Choose exhaustion by finite-dimensionallinear subspaces V j ⊂ V j +1 with dense union ∪ j V j ⊂ H . Let 0 < r < · · · < r i < r i +1 < · · · → ∞ be a divergent positive sequence with r i +1 > √ r i , and let D jr i ⊂ V j be the open disc with diameter r i . NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES5
Consider the diagram of the embeddings: : : ∩ ∩ ∩· · · ⊂ D jr i ⊂ D jr i +1 ⊂ . . . ⊂ V j ∩ ∩ ∩· · · ⊂ D j +1 r i ⊂ D j +1 r i +1 ⊂ . . . ⊂ V j +1 ∩ ∩ ∩ : : : · · · ⊂ D r i ⊂ D r i +1 ⊂ ⊂ H Let V ⊥ j ⊂ H be the orthogonal complement, and for j ′ ≥ j , denote V j,j ′ := V ⊥ j ∩ V j ′ , E j.j ′ r i := D jr i × V j,j ′ ⊂ V j ′ ,E jr i := D jr i × V ⊥ j ⊂ H. Let us set S C ( D jr i ) ≡ C ( R ) ˆ ⊗ C ( D jr i , Cl ( V j )) . Recall the Bott map β : C ( R ) → S C ( V ) f → f ( X ˆ ⊗ ⊗ C )for a finite-dimensional vector space V . Then we have ∗ -homomorphisms β j,j ′ = β ˆ ⊗ S C ( D jr i ) → SC ( V j,j ′ , Cl ( V j,j ′ )) ˆ ⊗ C ( D jr i ) ∼ = S C ( E j.j ′ r i ) ֒ → S C ( V j ′ )where the last embedding is the open inclusion. Remark 2.3.
Trout developed a Thom isomorphism on infinite-dimensionalEuclidean bundles. One may regard C ( D jr i , Cl ( V j )) as the set of con-tinuous sections on the Clifford algebra of the tangent bundle Cl ( T D jr i ) vanishing at infinity. Then β j,j ′ can be described as a ∗ -homomorphism β j,j ′ = (1 ˆ ⊗ i ∗ ) ◦ ( β E j,j ′ ri ˆ ⊗ D jri id D jri ) where i : E j,j ′ r i ֒ → V ′ j is the open inclusion. See [T] Section . Let S r := C ( − r, r ) ⊂ C ( R ), and set S r C ( D jr ) ≡ C ( − r, r ) ˆ ⊗ C ( D jr , Cl ( V j )) . Then the above Bott map transforms as β j,j ′ = β ˆ ⊗ S r i C ( D jr i ) → S r i +1 C ( D j ′ r i +1 ) . TSUYOSHI KATO
Lemma 2.4.
The direct limit C ∗ -algebra lim i,j →∞ S r i C ( D jr i ) = S C ( H ) coincides with the Clifford C ∗ -algebra of H .Proof. Step 1:
We claim that the commutativity β j,j ′′ = β j ′ ,j ′′ ◦ β j,j ′ holds. To make the notations clearer, let us denote ¯ β j,j ′ : S C ( V j ) → S C ( V j ′ ) as the standard Bott map. Then the commutativity ¯ β j,j ′′ =¯ β j ′ ,j ′′ ◦ ¯ β j,j ′ holds.For a i,j ∈ S r i C ( D jr i ), ¯ β j,j ′′ ( a i,j ) = β j,j ′′ ( a i,j ) holds in S C ( V j ′′ ), pass-ing through the isometric embedding S r i +2 C ( D j ′′ r i +2 ) ֒ → S C ( V j ′′ ). Thisimplies the equalities β j,j ′′ ( a i,j ) = ¯ β j,j ′′ ( a i,j ) = ¯ β j ′ ,j ′′ ◦ ¯ β j,j ′ ( a i,j )= ¯ β j ′ ,j ′′ ◦ β j,j ′ ( a i,j ) = β j ′ ,j ′′ ◦ β j,j ′ ( a i,j ) . This commutativity allows us to construct the direct limit C ∗ -algebralim i,j →∞ S r i C ( D jr i ) . There is a canonical isometric embedding I : lim i,j →∞ S r i C ( D jr i ) ֒ → S C ( H ) . Step 2:
It remains to verify that the image of I is dense. For alinear inclusion V ֒ → H , let β : S C ( V ) → S C ( H ) be the Bott ∗ -homomorphism into the Clifford C ∗ algebra. An element a ∈ S C ( H ) isgiven as lim j β ( a j ) for some a j ∈ S C ( V j ). Let χ i ∈ C c (( − r i , r i ); [0 , ϕ i,j ∈ C c ( D jr i ; [0 , χ i | ( − r i − , r i − ) ≡ ϕ i,j | D jr i − ≡
1. Let us set ψ i,j = χ i ˆ ⊗ ϕ i,j . We claim that b i,j := ψ i,j a j ∈ S C ( D jr i ) converges to the same element:lim i,j →∞ β ( b i,j ) = a ∈ S C ( H ) . Choose any j and ǫ >
0. There exists r > a j satisfies theestimate || a j || C (( D r ) c ) < ǫ . For each f ∈ C ( R ), there is some r > β ( f ) ∈ S C ( H ) satisfies the estimate || β ( f ) || C (( D r ) c ) < ǫ .Thus, any a j ∈ S C ( V j ) with j > j also satisfies the estimate || a j || C (( D √ r ) c ) < ǫ for all large r >>
1. This verifies the claim. (cid:3)
NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES7
Asymptotic unitary operators.
Let H ′ be a Hilbert space. Fortwo finite-dimensional linear subspaces, let us set d ′ ( V ′ , V ′ ) = sup v inf v { || v − v || : || v || = || v || = 1 , v i ∈ V ′ i } .d ′ ( V ′ ; V ′ ) = 0 holds if and only if V ′ contains V ′ . Then we introducethe distance between these planes by d ( V ′ , V ′ ) = min { d ′ ( V ′ ; V ′ ) , d ′ ( V ′ ; V ′ ) } . Let l : H ′ → H be a linear isomorphism between Hilbert spaces, andlet ¯ l = l ◦ √ l ∗ ◦ l − : H ′ → H be the unitary corresponding to the polar decomposition of l . For anyfinite-dimensional linear subspace V ⊂ H , let us compare two linearsubspaces V ′ ≡ l − ( V ) , ¯ V ′ ≡ ¯ l − ( V ) ⊂ H ′ . The following lemma will not be used later, but may be useful tounderstand how V ′ and ¯ V ′ differ from each other. Lemma 2.5.
Let W ′ i be a family of finite-dimensional linear subspaceswith 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 > , there is some i such that for all i ≥ i , || (1 − ¯ pr i ) | ¯ V ′ || < ǫ holds, where ¯ pr i : H ′ → ¯ W ′ i is the orthogonal projection and ¯ W ′ i :=¯ l − ( l ( W ′ i )) .Proof. It is sufficient to verify that for any finite-dimensional linearsubspace V ′ ⊂ H ′ and any ǫ >
0, the estimate d ( ¯ V ′ , ¯ W ′ i ) < ǫ holds for all large i >>
1. Actually ¯ W ′ = H ′ holds when W ′ = H ′ since the polar decomposition gives the unitary. Thus, for anyfinite-dimensional linear exhaustion W ′ i such that ∪ i W ′ i ⊂ H ′ is dense, ∪ i ¯ W ′ i ⊂ H ′ is also dense. Therefore the estimate holds. (cid:3) Definition 2.2.
Let H ′ and H be two Hilbert spaces and l : H ′ → H be a linear isomorphism. l is asymptotically unitary if for any ǫ > , there is a finite-dimensionallinear subspace V ⊂ H ′ such that the restriction l : V ⊥ ∼ = l ( V ⊥ ) TSUYOSHI KATO satisfies the estimate on its operator norm || ( l − ¯ l ) | V ⊥ || < ǫ where ¯ l is the unitary of the polar decomposition of l : H ′ → H . Remark 2.6.
In a subsequent paper, we will verify that a self-adjointelliptic operator on a compact manifold is asymptotically unitary be-tween Sobolev spaces.
Lemma 2.7.
Let l : H ′ ∼ = H be asymptotically unitary. For any ǫ > ,there is a finite-dimensional vector subspace V ′ ⊂ H ′ such that theestimate d ( V ′ , (¯ l ∗ ◦ l )( V ′ )) < ǫ holds for any V ′ ⊃ V ′ .Proof. Let V ′ ⊂ H ′ be a closed linear subspace, and ( V ′ ) ⊥ be its or-thogonal complement. Let pr : H ′ → V ′ be the orthogonal projection. Step 1:
Take a finite-dimensional subspace V ′ ⊂ H ′ so that l satis-fies the estimate || ( l − ¯ l ) | ( V ′ ) ⊥ || < ǫ for any V ′ ⊃ V ′ . Then the operatornorm of the restriction satisfies the estimate || (¯ l ∗ l − pr ◦ ¯ l ∗ l ) | V ′ || < ǫ. Decompose the operator ¯ l ∗ l with respect to V ′ ⊕ ( V ′ ) ⊥ , and express ¯ l ∗ l by a matrix form (cid:18) A BC D (cid:19) where both the estimates || D − id || , || B || < ǫ hold. Step 2: C = B ∗ holds since ¯ l ∗ l is self-adjoint. Hence the estimate || C || < ǫ also holds. Then the conclusion holds because the estimate d ( V ′ , (¯ l ∗ ◦ l )( V ′ )) ≤ || C || holds. (cid:3) A variant of Clifford C ∗ -algebra. Let us introduce a variant ofthe HKT construction. Ultimately, the result of the C ∗ -algebra turnsout to be ∗ -isomorphic to the original one given by HKT. This variantnaturally appears when one considers the induced Clifford C ∗ -algebrawe introduce later.Let l : H ′ ∼ = H be an asymptotically unitary isomorphism. Let E ⊂ H be a finite-dimensional Euclidean space, and denote E ′ = l − ( E ) , ¯ E ′ = ¯ l − ( E ) . NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES9
The map C l ≡ ¯ l ∗ ◦ l : E ′ → ¯ E ′ ֒ → Cl ( ¯ E ′ )is called the induced Clifford operator . Let us denote C l ( E ′ ) = C ( E ′ , Cl ( ¯ E ′ ))and introduce a ∗ -homomorphism β l : C ( R ) → S C l ( E ′ ) ≡ C ( R ) ˆ ⊗ C l ( E ′ )defined by β l : f → f ( X ˆ ⊗ ⊗ C l ) by functional calculus.Let E ′ a ⊂ E ′ b ⊂ H ′ be a pair of finite-dimensional linear subspaces,and denote the orthogonal complement as E ′ ba = E ′ b ∩ ( E ′ a ) ⊥ . Lemma 2.8.
Let l : H ′ ∼ = H be asymptotically unitary.Then there is a finite-dimensional vector space V ′ ⊂ H ′ such thatthere is a canonical ∗ -isomorphism I ba : S C l ( E ′ ba ) ˆ ⊗ C l ( E ′ a ) ∼ = S C l ( E ′ b ) if the inclusion V ′ ⊂ E ′ a holds.Proof. Let V ′ be the vector subspace given by Lemma 2.7. Let ˆ E ′ ba ⊂ ¯ E ′ b be the orthogonal complement of ¯ E ′ a , and consider the orthogonalprojection ˆpr : ¯ E ′ ba → ˆ E ′ ba . By the assumption, l ∗ ◦ l is almost unitary on E ′ ba so that the operatornorm satisfies the estimate || ( l ∗ ◦ l ) − id | E ′ ba || < ǫ . The estimate d ( E ′ a , ¯ E ′ a ) < ǫ also holds from Lemma 2.7. Thus, the operator norm ofthe above projection satisfies the estimate || ˆpr − id | ¯ E ′ ba || < ǫ. In particular the projection gives an isomorphism. Let ˆ pr : ¯ E ′ ba → ˆ E ′ ba be the unitary of the polar decomposition. It also satisfies theestimate || ˆ pr − id | ¯ E ′ ba || < ǫ , which induces a ∗ -isomorphismˆ pr : Cl ( ¯ E ′ ba ) ∼ = Cl ( ˆ E ′ ba ) . It extends to the ∗ -isomorphismˆ pr ˆ ⊗ Cl ( ¯ E ′ ba ) ˆ ⊗ Cl ( ¯ E ′ a ) ∼ = Cl ( ˆ E ′ ba ) ˆ ⊗ Cl ( ¯ E ′ a ) ∼ = Cl ( ¯ E ′ b )which induces the desired ∗ -isomorphism S C l ( E ′ ba ) ˆ ⊗ C l ( E ′ a ) ∼ = S C l ( E ′ b ) . (cid:3) It follows from Lemma 2.8 that there is a canonical ∗ -homomorphism β ba = β l ˆ ⊗ S C l ( E ′ a ) → S C l ( E ′ ba ) ˆ ⊗ C l ( E ′ a ) ∼ = S C l ( E ′ b ) . Remark 2.9.
Surely ˆ pr induces a linear map ˆ pr : Cl ( ¯ E ′ ba ) → Cl ( ˆ E ′ ba ) by setting u = u + u ∈ Cl ( ˆ E ′ ba ) ⊕ Cl ( ¯ E ′ a ) to u ∈ Cl ( ˆ E ′ ba ) , where Cl ( 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 ′ ∈ ¯ E ′ ba and set u ′′′ = u ′ − ˆ pr ( u ′ ) := u ′ − u ′′ . For any orthonormalbasis { u ′ , u ′ , . . . } of ¯ E ′ ba , consider their product u ′ u ′ . . . u ′ m ∈ Cl ( ¯ E ′ ba ) . u ′ u ′ . . . = ( u ′′ + u ′′′ )( u ′′ + u ′′′ )( u ′′ + u ′′′ ) . . . ( u ′′ m + u ′′′ m )= u ′′ u ′′ . . . u ′′ m + other termsEach norm || u ′′ i || < is strictly less than , and hence, the norm oftheir product in the first term above may degenerate to zero. Let A a be a family of C ∗ -algebras, and β ba : A a → A b be a family of ∗ -homomorphisms, where { a } is a semi ordered set. The family { β ba } b,a is asymptotically commutative if for any ǫ >
0, there is a such that forany triplet c ≥ b ≥ a ≥ a , the estimate || β ca − β cb ◦ β ba || < ǫ holds.For v a ∈ A a , introduce the set of equivalent classes¯ v a := { β ba ( v a ) } b ≥ a divided by all elements ¯ v ′ a withlim b || β ba ( v ′ a ) || = 0 . Consider the algebra generated by elements of the form ¯ v a , where thesum is given by ¯ v a + ¯ v a ′ = { β ba ( v a ) + β ba ′ ( v a ′ ) } b ≥ a,a ′ and the multiplication is given by¯ v a · ¯ v a ′ = { β ba ( v a ) · β ba ′ ( v a ′ ) } b ≥ a,a ′ . 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 || ¯ v a || := lim b || β ba ( v a ) || ( ∗ )We also denote it as A := lim −→ a A a . Let us set β ba := I ba ◦ β l : S C l ( E ′ a ) → S C l ( E ′ ba ) ˆ ⊗ C l ( E ′ a ) ∼ = S C l ( E ′ b ) NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES11 where E ′ a run over all finite-dimensional subspaces, and b ≥ a if andonly if E ′ b ⊃ E ′ a holds.It follows from the proof of Lemma 2.8 that the following lemmaholds. Lemma 2.10.
The family { β ba : S C l ( E ′ a ) → S C l ( E ′ b ) } b,a is asymptoti-cally commutative. Definition 2.3.
Let l : H ′ ∼ = H be asymptotically unitary. The directlimit C ∗ -algebra is given by S C l ( H ′ ) = lim −→ a S C l ( E ′ a ) where the norm is given in ( ∗ ) above. Proposition 2.11.
Assume l is asymptotically unitary.Then there is a canonical ∗ -isomorphism S C l ( H ′ ) → S C ( H ′ ) 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 thatfor any ǫ >
0, there is a finite-dimensional vector space V ′ ⊂ H ′ suchthat for all E ′ a ⊃ V ′ , the following two estimates hold: d ( E ′ a , ¯ l ∗ ◦ l ( E ′ a )) < ǫ,d (( E ′ a ) ⊥ , ¯ l ∗ ◦ l (( E ′ a ) ⊥ )) < ǫ. Take another E ′ b ⊃ E ′ a with E ′ ba , and let pr : ¯ E ′ a ∼ = E ′ a and pr :¯ E ′ ba ∼ = E ′ ba be the orthogonal projections. Their corresponding unitaries pr i satisfy the bounds || pr i − id || < ǫ. They extend to ∗ -isomorphisms pr : Cl ( ¯ E ′ a ) ∼ = Cl ( E ′ a ) , pr : Cl ( ¯ E ′ ba ) ∼ = Cl ( E ′ ba ) . In particular they induce the ∗ -isomorphisms pr : C ( E ′ a , Cl ( ¯ E ′ a )) ∼ = C ( E ′ a , Cl ( E ′ a )) , pr : C ( E ′ ba , Cl ( ¯ E ′ ba )) ∼ = C ( E ′ ba , Cl ( E ′ ba )) . Step 2:
Let us consider two Bott maps β : C ( R ) → S C l ( W ′ ) , β ( f ) = f ( X ˆ ⊗ ⊗ C l ) ,β : C ( R ) → S C ( W ′ ) , β ( f ) = f ( X ˆ ⊗ ⊗ C )and the diagram S C l ( E ′ a ) β −−−→ S C l ( E ′ ba ) ˆ ⊗ C ( E ′ a , Cl ( ¯ E ′ a )) ⊗ pr y ⊗ pr ˆ ⊗ pr y S C ( E ′ a ) β −−−→ S C ( E ′ ba ) ˆ ⊗ C ( E ′ a , Cl ( E ′ a ))Denote pr := pr ˆ ⊗ pr . Then this diagram satisfies the estimate || ⊗ pr ◦ β − β ◦ ⊗ pr || < ǫ.ǫ can be arbitrarily small by choosing large E ′ a . Step 3:
Let us take an element x ∈ S C l ( H ′ ), and choose x a ∈ S C l ( E ′ a ) with lim a || β ( x a ) − x || = 0, where β ( x a ) ∈ S C l ( H ′ ). Itfollows from the above estimate on the diagram that pr : S C l ( H ′ ) → S C ( H ′ ) , pr ( x ) = lim a β (1 ˆ ⊗ pr ( x a ))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 ∼ = ¯ E ′ a . A parallel argument gives another ∗ -homomorphism pr ′ : S C ( H ′ ) → S C l ( H ′ ), and their compositions give the requiredidentities. Step 4:
Let us consider Γ-equivariance. Suppose Γ acts on H ′ lin-early and isometrically. We claim that pr : ¯ E ′ a → E ′ a is Γ-equivariant.To see this, notice that ¯ l and hence ¯ l ∗ ◦ l are both Γ-equivariant.Then we have the equalities γ ( ¯ E ′ a ) = γ (¯ l ∗ ◦ l ( E ′ a )) = ¯ l ∗ ◦ l ( γE ′ a )) = γE ′ a . Therefore, pr ( γ ( ¯ E ′ a )) = pr ( γE ′ a ) = γ ( E ′ a ) = γ ( pr ( ¯ E ′ a )) . As the Bott map is also Γ-equivariant, the process from step 1 to step3 works equivariantly. (cid:3)
NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES13 Finite-dimensional approximation
Let F : H ′ → H be a metrically proper map between Hilbert spaces.Then, there is a proper and increasing function g : [0 , ∞ ) → [0 , ∞ )such that the lower bound g ( || F ( m ) || ) ≥ || m || holds for all m ∈ H ′ . Later we analyze a family of maps of the form F i : B ′ i → W i , where W i ⊂ H is a finite-dimensional linear subspaceand B ′ i ⊂ W ′ i ⊂ H ′ is a closed and bounded set in a finite-dimensionallinear space.Let D t ⊂ H be a t ball. We say that the family of maps { F i } i is proper , if there are positive and increasing numbers r i , s i → ∞ suchthat the inclusion holds: F − i ( D s i ∩ W i ) ⊂ D r i ∩ W ′ i . Denote F = l + c where l is its linear part and c is a non linear term. Lemma 3.1.
Let F = l + c : H ′ → H be a metrically proper map.Suppose l is surjective and c is compact on each bounded set. Thenthere is a proper and increasing function f : [0 , ∞ ) → [0 , ∞ ) such thatthe following holds: for any r > and ≥ δ > , there is a finite-dimensional linear subspace W ′ ⊂ H ′ such that for any linear subspace W ′ ⊂ W ′ ⊂ H ′ , the composed mappr ◦ F : D r ∩ W ′ → W also satisfies the bound f ( || pr ◦ F ( m ) || ) ≥ || m || for any m ∈ D r ∩ W ′ , where W = l ( W ′ ) and pr is the orthogonalprojection to W .Moreover the estimate holds sup m ∈ D r ∩ W ′ || F ( m ) − pr ◦ F ( m ) || ≤ δ . Proof.
Let C ⊂ H be the closure of the image c ( D r ), which is compact.Hence there are finitely many points w , . . . , w k ∈ c ( D r ) such that their δ neighborhoods cover C .Choose w ′ i ∈ H ′ with l ( w ′ i ) = w i for 1 ≤ i ≤ k , and let W ′ be thelinear span of these w ′ i .The restriction pr ◦ F : D r ∩ W ′ → W satisfies the equalitypr ◦ F = l + pr ◦ c where W = l ( W ′ ). Then for any m ∈ D r ∩ W ′ , there is some w ′ i with || c ( m ) − c ( w ′ i ) || ≤ δ , and the estimate || F ( m ) − pr ◦ F ( m ) || ≤ δ holds.Since g is increasing, we obtain the estimates g ( || pr ◦ F ( m ) || + δ ) ≥ g ( || F ( m ) || ) ≥ || m || . The function f ( x ) = g ( x + 1) satisfies the desired property.For any other linear subspace W ′ ⊂ W ′ ⊂ H ′ , the same propertyholds for pr ◦ F : D r ∩ W ′ → W with W = l ( W ′ ). (cid:3) Let W ′ i ⊂ H ′ and W i ⊂ H be two families of finite-dimensional linearsubspaces. Let us say that a family of linear isomorphisms l i : W ′ i ∼ = W i is an asymptotic unitary family if the following conditions hold:(1) there exists an asymptotically unitary map l : H ′ ∼ = H ,(2) for each i , lim i || l − l i || W ′ i = 0 holds, where l, l i : W ′ i → H , and(3) uniform bounds C − || l || ≤ || l i || ≤ C || l || hold on their norms,where C is independent of i .Let us introduce an approximation of F as a family of maps on finite-dimensional 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 finite-dimensional linear subspaces W ′ ⊂ W ′ ⊂ · · · ⊂ W ′ i ⊂ · · · ⊂ H ′ and a family of maps F i = l i + c i : W ′ i → W i , where W i = l i ( W ′ i ) , suchthat (1) the union ∪ i ≥ W ′ i ⊂ H ′ is dense, (2) there are two sequences s < s < · · · → ∞ and r < r < · · · →∞ with r i ≥ s i such that the embedding F − i ( D s i ∩ W i ) ⊂ D ′ r i ∩ W ′ i holds for all i , (3) for each i , lim i →∞ sup m ∈ D ′ ri ∩ W ′ i || F ( m ) − F i ( m ) || = 0 , (4) l i : W ′ i ∼ = W i is an asymptotic unitary family with respect to l . NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES15
Let us also say that F is strongly finitely approximable if it is finitelyapproximable, l i = l | W ′ i and c i = pr i ◦ c , such thatlim i →∞ || (1 − pr i ) ◦ c | D ′ r i || = 0where pr i : H → W i is the orthogonal projection.The following restates Lemma 3.1. Corollary 3.2.
Let F = l + c : H ′ → H be a metrically proper mapsuch that l is asymptotically unitary and c is compact on each boundedset. Then F is strongly finitely approximable. Suppose both H ′ and H admit linear isometric actions by a groupΓ 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 abovefamily { W ′ i } i satisfies that the union ∪ i { γ ( W ′ i ) ∩ W ′ i } ⊂ H ′ is dense for any γ ∈ Γ.Note that the above family { F i } i satisfies convergence for any γ ∈ Γlim i →∞ sup m || γF i ( m ) − F i ( γm ) || = 0where m ∈ D ′ r i ∩ W ′ r i ∩ γ − ( W ′ r i ) because the estimate || γF i ( m ) − F i ( γm ) || ≤ || γF ( m ) − γF i ( m ) || + || γF ( m ) − F i ( γm ) || = || F ( m ) − F i ( m ) || + || F ( γm ) − F i ( γm ) || holds.Let us take γ ∈ Γ, and consider the γ shift of the finite approximationdata γ ( W ′ i ) , γ ∗ ( F i ) , γ ∗ ( l i ) . It is clear that the above shift gives another finite approximation of F .4. Induced Clifford C ∗ -algebra Let F = l + c : H ′ → H be a map. We aim here is to construct an“induced” Clifford C ∗ -algebra S C F ( H ′ ). Model case.
Let us start with a model case that consists of aproper and nonlinear map F = l + c : E ′ → E between finite-dimensional Euclidean spaces, where l is a linear isomor-phism. Consider a ∗ -homomorphism F ∗ : S C ( E ) → S C ( E ′ ) = C ( R ) ˆ ⊗ C ( E ′ , Cl ( E ′ ))defined by F ∗ ( f ˆ ⊗ u )( v ′ ) := f ˆ ⊗ ¯ l − ( u ( F ( v ′ )), and denote its image by S C F ( E ′ ) = F ∗ ( S C ( E ))which is a C ∗ -subalgebra in S C ( E ′ ), whose norm is denoted by || || S C F .The induced map C F ≡ ¯ l − ◦ F : E ′ → E ′ ֒ → Cl ( E ′ )is called the induced Clifford operator . We use it to introduce a ∗ -homomorphism β F : C ( R ) → S C F ( E ′ )defined by β F : f → f ( X ˆ ⊗ ⊗ C F ) by functional calculus.Now suppose a Hilbert space H ′ is spanned by an infinite family offinite-dimensional Euclidean planes as E ′ ⊕ E ′ ⊕ . . . and assume there is a family of proper maps F i = l i + c i : E ′ i → E i which extends to a map F = ( F , F , . . . ) = l + c : H ′ → H where H is spanned by E ⊕ E ⊕ . . . . Assume l = ( l , l , . . . ) : H ′ ∼ = H is asymptotically unitary. Lemma 4.1.
Let F = ( F , F ) be diagonal as above. Then C F ( E ′ ⊕ E ′ ) ∼ = C F ( E ′ ) ˆ ⊗ C F ( E ′ ) . Proof.
By definition C F i ( E ′ i ) = F ∗ i ( C ( E ′ i )) holds for i = 1 ,
2. Hence wehave the isomorphisms C F ( E ′ ⊕ E ′ ) ∼ = F ∗ ( C ( E ′ ⊕ E ′ )) ∼ = ( F ˆ ⊗ F ) ∗ C ( E ′ ) ˆ ⊗ C ( E ′ ) ∼ = F ∗ ( C ( E ′ )) ˆ ⊗ F ∗ ( C ( E ′ )) ∼ = C F ( E ′ ) ˆ ⊗ C F ( E ′ ) . (cid:3) NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES17
Then the induced Bott map is given by β F i +1 : S C F ( E ′ ⊕ · · · ⊕ E ′ i ) → S C F ( E ′ i +1 ) ˆ ⊗ C F ( E ′ ⊕ · · · ⊕ E ′ i ) ∼ = S C F ( E ′ ⊕ · · · ⊕ E ′ i +1 )by use of C F i +1 .More generally, one can induce β i,j : S C F ( E ′ ⊕ · · · ⊕ E ′ i ) → S C F ( E ′ ⊕ · · · ⊕ E ′ j )by use of a canonical extension C ( F i +1 ,...,F j ) : E ′ i +1 ⊕ · · · ⊕ E ′ j → E ′ i +1 · · · ⊕ E ′ j ⊂ Cl ( E ′ i +1 · · · ⊕ E ′ j ) . Let u ∈ S C F ( E ′ ⊕ · · · ⊕ E ′ i ) for some i . Then the limit || u || ≡ lim j →∞ || β i,j ( u ) || S C F exists, which gives a norm on S C F ( E ′ ⊕ E ′ ⊕ . . . ). Then the directlimit C ∗ algebra is given by S C F ( H ′ ) = lim j S C F ( E ′ ⊕ · · · ⊕ E ′ j )whose norm is given as above.Notice that S C F ( H ′ ) is no longer a C ∗ -subalgebra of S C ( H ′ ). Lemma 4.2.
In the case when c i ≡ and hence F i = l i for all i , theinduced Clifford C ∗ -algebra admits a canonical ∗ -isomorphism S C F ( H ′ ) ∼ = S C ( H ′ ) . Proof.
This follows from Proposition 2.11 with the coincidence S C F ( H ′ ) = S C l ( H ′ )where the right-hand side is given in Definition 2.3. (cid:3) Induced Clifford C ∗ -algebra. Assume that F = l + c : H ′ → H is finitely approximable as in Definition 3.1 with respect to the data W ′ ⊂ · · · ⊂ W ′ i ⊂ · · · ⊂ H ′ with open disks D ′ r i ⊂ W ′ i and D s i ⊂ W i ,and F i = l i + c i : W ′ i → W i .Let S r = C ( − r, r ) ⊂ S be the set of continuous functions on ( − r, r )vanishing at infinity, and consider the following C ∗ -subalgebras S r i ˆ ⊗ C ( D ′ r i , Cl ( W ′ i )) ≡ S r i C ( D ′ r i ) . Since the inclusion F − i ( D s i ) ⊂ D ′ r i holds, it induces a ∗ -homomorphism F ∗ i : S s i C ( D s i ) → S r i C ( D ′ r i )given by F ∗ i ( h )( v ′ ) := ¯ l − i ( h ( F i ( v ′ ))). Denote its image by S r i C F i ( D ′ r i ) := F ∗ i ( S s i C ( D s i )) which is a C ∗ -subalgebra with the norm || || S ri C Fi .Let us consider a family of elements α i ∈ S r i C F i ( D ′ r i ) , i ≥ i for some i . Let us say that the family is F - compatible if there is anelement u i ∈ S s i C ( D s i ) such that α i = F ∗ i ( u i ) ∈ S r i C F i ( D ′ r i )holds for any i ≥ i , where u i = β ( u i ) ∈ S s i C ( D s i ) with the standardBott map β . Remark 4.3.
Consider the induced Clifford operator C F i ≡ ¯ l − i ◦ F i : D ′ r i → W i ֒ → Cl ( W ′ i ) and introduce a ∗ -homomorphism β F i : S r i → S r i C ( D ′ r i ) defined by β F i : f → f ( X ˆ ⊗ ⊗ C F i ) by functional calculus.Then, F ∗ i ( β ( f )) = β F i ( f ) , for all f ∈ S r i For an element α i ∈ S r i C F i ( D ′ r i ) and for i ≤ i , we denote its restric-tion α i | D ′ r i ∈ S r i ˆ ⊗ C b ( D ′ r i ) ˆ ⊗ Cl ( W ′ i ) . Note that the norms satisfy the inequality || α i || S ri C Fi ( D ′ ri ) ≥ || α i | D ′ r i || where the right-hand side is the restriction norm.For an F -compatible sequence α = { α i } i ≥ i , the limit || { α i } i || := lim j →∞ lim i →∞ || α i | D ′ r j || exists because both F i and l i converge weakly (see Definition 3.1).Moreover both F ∗ i and β are ∗ -homomorphisms between C ∗ -algebrasand so are both norm-decreasing. Definition 4.1.
Let F be finitely approximable. The induced Clifford C ∗ -algebra is given by S C F ( H ′ ) = { { α i } i ; F -compatible sequences } which is obtained by the norm closure of all F -compatible sequences,where the norm is the above one. NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES19
Lemma 4.4. (1)
In the model case, S C F ( H ′ ) coincides with S C F ( H ′ ) in subsection 4.1 (2) When F = l is asymptotically unitary, there is a natural ∗ -isomorphism Φ : S C F ( H ′ ) ∼ = S C l ( H ′ ) where the right-hand side is given in Definition 2.3. (3) Suppose F is Γ -finitely approximable. Then S C F ( H ′ ) is indepen-dent of choice of Γ -finite approximations.Proof. One can choose W ′ i = E ⊕ · · · ⊕ E ′ i . Then (1) follows from theequality F ∗ i ◦ β = β F i : S s i → S r i C F i ( D ′ r i )by Remark 4.3, with Lemma 2.4.Let us consider (2), and set F = l . Recall the Bott map which isgiven above of the Definition 2.3, and denote it as β l : S C l ( W ′ i ) → S C l ( H ′ ) . Let { l i } i be asymptotically unitary, and denote l i ( W i ) = ˜ W i and l ( W i ) = W i . For each i and ǫ >
0, there is some i ′ >> i such that || pr i ,i − id || < ǫ holds for any i ≥ i ′ , where pr i ,i : W i → l i ( W i ) is the orthogonalprojection. Let pr i ,i : W i ∼ = l i ( W i ) be the unitary of the polardecomposition.Take an element { α i } i ∈ S C F ( H ′ ), with α i = l ∗ i ( u i ) and u i = β ( u i ) ∈ S C ( ˜ W i ). Note that the restriction β ( u i ) | W i = u i holds.Then by the condition of asymptotic unitarity, the restriction of theirdifference satisfies the estimate || l ∗ ( β ( pr ∗ i ,i ( u i ))) − α i || W ′ i < ǫ where β ( pr ∗ i ,i ( u i )) ∈ S C ( W i ). Then we setΦ( { α i } i ) = lim i →∞ β l ( l ∗ ( β ( pr ∗ i ,i ( u i )))) . Φ is norm-preserving, so it extends to an injective ∗ -homomorphismfrom S C F ( H ′ ).Let us verify that it is surjective. One can follow in a converse way tothe above. Take an element δ = β l ( δ i ) ∈ S C l ( H ′ ) with δ i ∈ S C l ( W ′ i ),and set δ i = l ∗ ( β ( δ i )). Let us set w i = ( l ∗ ) − ( δ i ) ∈ S C ( W i ) by v → ¯ l ( δ i ( l − ( v ))). Then we set u i = ( pr − i ,i ) ∗ ( w i ) ∈ S C ( ˜ W i ) and α i = l ∗ i ( β ( u i )) ∈ S C l i ( W ′ i ). The restriction of their differencesatisfies the estimate || l ∗ ( β ( w i )) − α i || W ′ i < ǫ where β ( w i ) ∈ S C ( W i ). The estimate || l ∗ ( β ( w i )) − δ i || W ′ i < ǫ issatisfied because || l ∗ ( w i ) − δ i || < ǫ holds. This implies that Φ( { α i } i ) = δ ∈ S C l ( H ′ ).Hence Φ is an isometric ∗ -homomorphism with dense image. Thisimplies that it is surjective.Let us verify the last property (3). Choose any subindices j i ≥ i for i = 1 , , . . . , 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 weobtain the same C ∗ -algebra S C F ( H ′ ) as their norms coincide as follows:lim i →∞ || α i | D ′ r ji || = lim i →∞ || α j i | D ′ r ji || . Let us take two Γ-finite approximations and denote them by F li :( D ′ s i ) l → W i,l for l = 1 , F -compatible sequence α = { α i } i ≥ i with respect to F i :( D ′ r i ) → W i, , where α i = ( F i ) ∗ ( u i ) and u i = β ( u i ) ∈ S s i C ( D s i ). Letus take subindices j i ≥ i for i = 1 , , . . . so that lim i →∞ d ′ ( W j i , , W i, ) =0 holds (see 2.3).Let us set α ′ i = ( F i ) ∗ ( u i ). Then it follows from the definition of F -compatible sequence that the convergencelim i →∞ || α j i | D ′ r ji − α ′ i | D ′ r ji || = 0holds. Combining this result with the above, we obtain the desiredconclusion. (cid:3) Lemma 4.5. If F is Γ -finitely approximable, then there is a canonical Γ -action on S C F ( H ′ ) .Proof. Recall that if { W ′ i , F i , l i } i is a finite approximation data, thenso is { γ ( W ′ i ) , γ ∗ ( F i ) , γ ∗ ( l i ) } i (see the last sentence in section 3).Take an F -compatible sequence α = { α i } i ≥ i with respect to F i : D ′ r i → W i , where α i = ( F i ) ∗ ( u i ) and u i = β ( u i ) ∈ S s i C ( D s i ). Then { γ ∗ α i } i is an γ ∗ ( F )-compatible sequence as, for m ′ ∈ γ ( D ′ r i ), γ ∗ ( α i )( m ′ ) = γ ∗ (( F i ) ∗ ( u i ))( m ′ ) = γu i ( F i ( γ − ( m ′ )))= γβ ( u i )( F i ( γ − ( m ′ ))) = β ( γu i )( F i ( γ − ( m ′ )))= β ( γu i γ − )( γF i ( γ − ( m ′ ))) = ( γ ∗ F i ) ∗ ( γ ∗ ( u i ))( m ′ ) . Thus, γ ∗ ( α i ) = ( γ ∗ F i ) ∗ ( β ( γ ∗ ( u i ))) holds. (cid:3) NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES21 Higher degree ∗ -homomorphism Let F = l + c : H ′ → H be a Γ-equivariant nonlinear map, whoselinear part l gives an isomorphism. For a finite-dimensional linearsubspace V ⊂ H , denote the orthogonal projection by pr V : H → V .For V ′ = l − ( V ), we have the modified map F V = l + pr V ◦ c : V ′ → V. The restriction map F V → F U satisfies the formula F U = pr U ◦ F V | U ,for a linear subspace U ⊂ V .Our initial idea was to pull back W i = l ( W ′ i ) by F W i and com-bine them all. For F i = pr i ◦ F : W ′ i → W i , consider the induced ∗ -homomorphism F ∗ i : S C ( W i ) → S C ( W ′ i ). Let us explain how diffi-culty arises if one tries to obtain a ∗ -homomorphism in this way. Forsimplicity, assume l is unitary and the image of c is contained in afinite-dimensional linear subspace V ⊂ H . This will be the simplestsituation but already some difficulty appears when we try to constructthe induced ∗ -homomorphism by F .Assume F is metrically proper. This is equivalent to saying that therestriction F : V ′ → V is proper in this particular situation, where V ′ = l − ( V ) is the finite-dimensional linear subspace. Let us considerthe diagram S C ( W i ) F ∗ i −−−→ S C ( W ′ i ) β y β l y S C ( W i +1 ) F ∗ i +1 −−−→ S C ( W ′ i +1 )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 inducedmaps by F ∗ i will not converge in S C ( H ′ ) in general. This is a pointwhere we have account for the nonlinearity of F to construct the target C ∗ -algebra, and is the reason we have to use S C F ( H ′ ) instead of S C ( H ′ )below.5.1. Degree of proper maps.
Let E ′ , E be two finite-dimensionalvector spaces, and F = l + c : E ′ → E be a proper smooth map whoselinear part l : E ′ ∼ = E gives an isomorphism.Let us reconstruct the degree of F ∈ Z by use of l . Let ¯ l : E ′ → E bethe unitary corresponding to the polar decomposition. Then, ¯ l inducesthe algebra isomorphism ¯ l : Cl ( E ′ ) ∼ = Cl ( E ), and we have the induced ∗ -homomorphism F ∗ : S C ( E ) = C ( E, Cl ( E )) → S C ( E ′ ) F ∗ ( h )( v ) = ¯ l − ( h ( F ( v ))) . Recall S C F ( E ′ ) = F ∗ ( S C ( E )). Then F ∗ can be described as a ∗ -homomorphism F ∗ : S C ( E ) → S C F ( E ′ ) . Let us consider the induced homomorphisms between K -groups K ( S C ( E )) F ∗ −−−→ K ( S C F ( E ′ )) inc ∗ −−−→ K ( S C ( E ′ )) β x x β K ( C ( R )) K ( C ( R ))where both β give the isomorphisms by 2.2.Let ˜ F ∗ : K ( C ( R )) → K ( C ( R )) be the homomorphism determineduniquely so that the diagram commutes. Let us equip orientations onboth E ′ and E so that l preserves them. Lemma 5.1.
Passing through the isomorphism K ( C ( R )) ∼ = Z , ˜ F ∗ : Z → Z is given by multiplication by the degree of F .Proof. Step 1:
Let us consider the composition of ∗ -homomorphisms C ( E, Cl ( E )) → C ( E ′ , Cl ( E ′ )) ∼ = C ( E, Cl ( E ))where the first map is F ∗ and the second map is given by(¯ l − ) ∗ ( h ′ )( v ) ≡ ¯ l ( h ′ (¯ l − ( v ))) . The latter gives an isomorphism since l is isomorphic. Thus, it issufficient to see the conclusion for the composition. The compositionis given by h → { v h ( F ◦ ¯ l − ( v )) } Step 2:
Let l t : E ′ ∼ = E be another family of linear isomorphismswith l = ¯ l and l = l . It induces a family of ∗ -homomorphisms F ∗ t : C ( E, Cl ( E )) → C ( E, Cl ( E )) h → { v → h ( F ◦ ( l t ) − ( v )) } . Since homotopic ∗ -homomorphisms induce the same maps betweentheir K -groups, it is sufficient to see the conclusion for F ∗ . Notingthe equality F ◦ l − = 1 + c ◦ l − , it is enough to assume l is theidentity. NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES23
Step 3:
When l is the identity, F ∗ : S C ( E ) → S C ( E ) is given byid × F ∗ : S C ( E ) ∼ = ( S ˆ ⊗ Cl ( E )) ⊗ C ( E ) → ( S ˆ ⊗ Cl ( E )) ⊗ C ( E )whose induced homomorphim on a K -group is given by degree F , pass-ing through the isomorphism K ( S ˆ ⊗ Cl ( E ) ⊗ C ( E )) ∼ = K ( S ) ∼ = K ∗ ( C ( E )) ∼ = Z where ∗ is 0 or 1 with respect to whether dim E is even or odd. Thefirst isomorphism comes from Proposition 2.2, and the second is theclassical Bott periodicity (see [A]). (cid:3) 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 projectionspr i ◦ F : W ′ i → W i = l ( W i ) consist of a finitely approximable data withthe constants r i , s i → ∞ .Let us consider the restriction F i +1 : D ′ r i ∩ W ′ i +1 → W i +1 . Decompose W ′ i ⊕ U ′ i = W ′ i +1 , and define F i +1 : W ′ i +1 → W i +1 by F i +1 ( w ′ + u ′ ) = F i ( w ′ ) + l ( u ′ ) . Then by definition, the estimatesup m ∈ D ′ ri ∩ W ′ i +1 || F i +1 ( m ) − F i +1 ( m ) || < δ i holds, where 0 < δ i → Sublemma 5.2.
Suppose l : H ′ ∼ = H is unitary. Let β : S C ( W ′ i ) → S C ( W ′ i +1 ) be the Bott map. Then the equality holds β ◦ F ∗ i = ( F i +1 ) ∗ ◦ β : S C ( W i ) → S C ( W ′ i +1 ) . Proof.
Take f ˆ ⊗ h ∈ S C ( W i ) with ( β ◦ F ∗ i )( f ˆ ⊗ h ) = β ( f ) ˆ ⊗ F ∗ i ( h ). Bycontrast, β ( f ˆ ⊗ h ) = β ( f ) ˆ ⊗ h and, hence,( F i +1 ) ∗ ◦ β ( f ˆ ⊗ h ) = ( l ⊕ F i ) ∗ ◦ β ( f ) ˆ ⊗ h = l ∗ ( β ( f )) ˆ ⊗ F ∗ i ( h )where l ∗ : S C ( U i ) ∼ = S C ( U ′ i ) with U i = l ( U ′ i ). Since l is unitary, theequality holds l ∗ ( β ( f )) = β ( f ) ∈ S C ( U ′ i ) . (cid:3) Proposition 5.3.
Let F = l + c : H ′ → H be a strongly finitelyapproximable map. Then the family { F ∗ i } i induces a ∗ -homomorphism F ∗ : S C ( H ) → S C ( H ′ ) . Proof.
Step 1:
Let us take an element α ∈ S C ( H ) and its approxi-mation α i ∈ S r i C ( D r i ) with lim i →∞ β ( α i ) = α ∈ S C ( H ) by Lemma2.4.Assume l : H ′ ∼ = H is unitary, and consider the following two ele-ments: β ( F ∗ i ( α i )) , F ∗ i +1 ( α i +1 ) ∈ S C ( W ′ i +1 ) . Then by Sublemma 5.2 we have the estimates || β ( F ∗ i ( α i )) − F ∗ i +1 ( α i +1 ) || = || ( F i +1 ) ∗ ( β ( α i )) − F ∗ i +1 ( α i +1 ) |||| ( F i +1 ) ∗ ( β ( α i )) − F ∗ i +1 ( β ( α i )) + || F ∗ i +1 ( β ( α i )) − F ∗ i +1 ( α i +1 ) ||≤ δ i || β ( α i ) || + || β ( α i ) − α i +1 || 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 tozero. Thus, the ∗ -homomorphisms asymptotically commute with theBott map. Hence, the sequence β ( F ∗ i ( α i )) ∈ S C ( H ′ ) converges, andgives a ∗ -homomorphism F ∗ : α → F ∗ ( α ) := lim i β ( F ∗ i ( α i )). Clearlythis 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 → S C l ( U ′ i ) be the variant of the Bott map in 2.4. Thenthe same argument to Sublemma 5.2 verifies the equality β l ◦ F ∗ i = ( F i +1 ) ∗ ◦ β : S C ( W i ) → S C l ( W ′ i +1 ) . Hence the parallel estimate to step 1 above verifies that the sequenceconverges β l ( F ∗ i ( α i )) ∈ S C l ( H ′ ) . This also gives a ∗ -homomorphism F ∗ : α → F ∗ ( α ) := lim i β l ( F ∗ i ( α i )).As S C l ( H ′ ) ∼ = S C ( H ′ ) are ∗ -isomorphic by Proposition 2.11, we obtainthe desired ∗ -homomorphism. (cid:3) Remark 5.4.
Suppose F = l + c satisfies the conditions to be stronglyfinitely approximable, except that l is not necessarily isomorphic, butthe Fredholm index is zero.We can still construct the induced ∗ -homomorphism F ∗ : S C ( H ) → S C ( 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 L ≡ l ⊕ l ′ : ( V ′ ) ⊥ ⊕ V ′ ∼ = V ⊥ ⊕ V. NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES25
Let us use L to pull back the Clifford algebras and use F itself to pullback the functions. Then we can follow from step and step in thesame way. Definition 5.1.
Let F : H ′ → H be a strongly finitely approximablemap. Then the induced map F ∗ : K ( S C ( H )) ∼ = Z → K ( S C ( H ′ )) ∼ = Z is given by multiplication by an integer degree F ∈ Z . We call it the K -theoretic degree of F . Induced map for Γ -finitely approximable map. Let us startfrom a general property, and let H be a Hilbert space with exhaustion W ⊂ · · · ⊂ 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 supportedcontinuous functions on ( − r, r ).The following restates Lemma 2.4 Lemma 5.5.
For any α ∈ S C ( H ) , there is a family α i ∈ S r i ˆ ⊗ C ( D r i , Cl ( W i )) := S r i C ( D r i ) such that their images by the Bott map converge to α lim i →∞ β ( α i ) = α ∈ S C ( H ) . Induced ∗ -homomorphism. Let H ′ , H be Hilbert spaces on whichΓ act linearly and isometrically, and let F = l + c : H ′ → H be a Γ-equivariant map such that l : H ′ ∼ = H is a linear isomorphism.Assume that F is Γ-finitely approximable so that there is a familyof finite-dimensional linear subspaces W ′ ⊂ W ′ ⊂ · · · ⊂ W ′ i ⊂ · · · ⊂ H ′ with dense union, and a family of maps F i : W ′ i → W i = l i ( W ′ i ) withthe inclusions F − i ( D s i ) ⊂ D ′ r i . Moreover the following convergenceshold for each i :lim i →∞ sup m ∈ D ′ ri || F ( m ) − F i ( m ) || = 0 ( ∗ )lim i →∞ || ( l − l i ) | W ′ i || = 0 ( ∗ ) . Recall the induced ∗ -homomorphism F ∗ i : S C ( D s i ) → S C F i ( D ′ r i )and the induced Clifford C ∗ -algebra S C F ( H ′ ) in Definition 4.1. Theorem 5.6.
Let F = l + c : H ′ → H be Γ -finitely approximable.Then it induces the equivariant ∗ -homomorphism F ∗ : S C ( H ) → S C F ( H ′ ) . Proof.
Let us take an element v ∈ S C ( H ) and its approximation v =lim i →∞ v i with v i ∈ S s i C ( D s i ) = C ( − s i , s i ) ˆ ⊗ C ( D s i , Cl ( W i )).Let us recall the ∗ -homomorphism in 4.2 F ∗ i : S s i C ( D s i ) → S r i C F i ( D ′ r i ) . Let us fix i and let u i = β ( v i ) ∈ S s i C ( D s i ) be the image of thestandard Bott map. Then the family { F ∗ i ( u i ) } i ≥ i determines an element in S C F ( H ′ ), which gives a ∗ -homomorphism F ∗ : S s i C ( D s i ) → S C F ( H ′ )since both F ∗ i and β are ∗ -homomorphisms. Note that the compositionof two ∗ -homomorphisms S s i C ( D s i ) β −−−→ S s i ′ C ( D s i ′ ) F ∗ −−−→ S C F ( H ′ )coincides with F ∗ : S s i C ( D s i ) → S C F ( H ′ ).For a small ǫ >
0, take two sufficiently large i ′ ≥ i >> || β ( v i ) − v i ′ || < ǫ holds, and set u ′ i = β ( v i ′ ) ∈ S s i C ( D s i )for i ≥ i ′ . Since F ∗ is norm-decreasing, the estimate || F ∗ i ( u i ) − F ∗ i ( u ′ i ) || < ǫ holds for all i ≥ i ′ . Hence, the estimate || F ∗ ( v i ) − F ∗ ( v i ′ ) || < ǫ holds.Thus, we obtain the assignment v → lim i →∞ F ∗ ( v i ), which gives aΓ-equivariant ∗ -homomorphism F ∗ : S C ( H ) → S C F ( H ′ )where { v i } i is any approximation of v . (cid:3) 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 F ∗ : K ∗ +1 ( C ∗ (Γ)) → K ∗ ( S C F ( H ′ ) ⋊ Γ) . NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES27 Computation of K -group of induced Clifford C ∗ -algebras We compute the equivariant K -group of induced Clifford C ∗ -algebrasfor some particular cases. This can be a simple model case for furthercomputation of the groups.6.1. Basics.
Let us collect some of basics which we will need. We startfrom some analytic aspects of Sobolev spaces. We denote by W k, asthe Sobolev k -norm which is a linear subspace of L . It is a Hilbertspace and, hence, complete by the norm which involves derivatives upto the k -th order, and incomplete with respect to the L inner productfor k ≥ Lemma 6.1.
Suppose k ≥ . Then (1) The multiplication W k, ( S ) ⊗ W k, ( S ) → W k, ( S ) is compact on each bounded set. (2) The continuous embedding W k, ( S ) ֒ → C ( S ) holds. In particular an element in W k, ( S ) can be regarded as a continuousfunction.Later we will consider the non linear map F : W k, ( S ) → W k, ( S )by F ( a ) = a + a . Remark 6.2. (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 ) = ( P g g = g ∈ Z l a g g ( b g )) . It induces theaction by Z on A by using the natural projection π l : Z → Z l . In suchsituation, there exists a six term exact sequence between K ∗ ( A ⋊ Z ) and K ∗ ( A ⋊ Z l ) . However this does not seem to contain enough informationto apply to our situation. We proceed in a direct way. Recall that anelement 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 =[ p ] − [ π ( p )] , where π : ¯ A = A ⊕ C → C is the projection. Recall that [ p ] − [ π ( p )] = [ q ] − [ π ( q )] , if and only if there is some v ∈ M n,m ( A ⋊ Z ) such that p ⊕ a = v ∗ v, vv ∗ = q ⊕ b for some a, b ≥ . Computation of equivariant K -group for a toy model. Finite cyclic and finite-dimensional case.
Consider a Z -equivariantmap F : R → R by (cid:18) ab (cid:19) → (cid:18) a + b b + a (cid:19) where the involution acts by the coordinate change.We claim that this is proper of non-zero degree. In fact, if a + b = 0,then the equality b + a = b − b implies properness.Consider a Z -equivariant perturbation F t (cid:18) ab (cid:19) = (cid:18) ta + b tb + a (cid:19) for t ∈ (0 , ta + b = 0, then tb + a = tb − t − b . Thus, this isa family of proper maps. At t = 0, F : R → R is a proper map ofdegree −
1, since it is again Z -equivariantly proper-homotopic to theinvolution I : (cid:18) ab (cid:19) → (cid:18) ba (cid:19) . Note that it becomes degree zero, if we replace the exponent 3 by 2.Next, we generalize slightly as follows. Consider a Z l equivariantmap F : R l → R l by a . . .a l → a + a l a + a . . .a l + a l − where the action is given by cyclic permutation of the coordinates. Bythe parallel argument as above, this turns out to be a proper map. Tocompute its degree, consider a perturbation a . . .a l → ta + a l ta + a . . .ta l + a l − for t ∈ (0 , Z l -equivariant proper maps, and at t = 0, F : R l → R l is a proper map of degree ±
1, determined by theparity of l . In fact there is a Z l -equivariant proper-homotopy F lt to thecyclic permutation T l ≡ F l a a . . .a l → a l a . . .a l − . NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES29
Corollary 6.3. F induces a Z l -equivariant isomorphism K Z l ( S C F ( R l )) ∼ = K Z l ( S C ( R l )) ∼ = R ( Z l ) on the equivariant K -theory.Proof. F is Z l -equivariantly properly homotopic to T l above, and sothe isomorphism K Z l ( S C F ( R l )) ∼ = K Z l ( S C T l ( R l )) holds. T l is a Z l -equivariantly linear isomorphism because Z l is commu-tative. It follows from Definition 2.2 that a linear isomorphism be-tween finite-dimensional vector spaces is asymptotically unitary. Thenby Proposition 2.11 that S C T l ( H ′ ) is Z l -equivariantly ∗ -isomorphic to S C ( H ′ ). In particular we have the isomorphism K Z l ( S C T l ( R l )) ∼ = K Z l ( S C ( R l )).The last isomorphism comes from HKT-Bott periodicity for Eu-clidean space. (cid:3) Infinite cyclic case.
Let H ′ = H be the closure of R ∞ withthe standard metric. It admits an isometric action of Z by the shift T : H ′ ∼ = H ′ T : ( . . . , a − , a , a , . . . ) ∼ = ( . . . , a − , a − , a , . . . ) . Then we consider the map F : H ′ → H by F : . . .a − a a . . . → . . .a − + a − a + a − a + a . . . . If we restrict on R l +1 ⊂ H ′ by ( a − l , . . . , a l ) → ( . . . , a − l , . . . , a l , , . . . ),then its image is in R l +2 ⊂ H . In fact F : . . . a − l a − l +1 . . .a l . . .. . . → . . . a − l a − l +1 + a − l . . .a l + a l − a l . . . . Let us consider the map F l : R l +1 → R l +1 by F l : a − l a − l +1 . . .a l → a − l + a l a − l +1 + a − l . . .a l + a l − which moves the last component to the first one. In fact F l is still aproper map as presented in 6.2.1.Let W ′ l = R l +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 K Z l +1 ( S C F l ( W ′ l )) ∼ = K Z l +1 ( S C ( W ′ l ))on the equivariant K -theory. Notice that this isomorphism heavilydepends on the degree being equal to ± Lemma 6.4.
The induced ∗ -homomorphism ( F l ) ∗ : S C ( W l ) → S C F l ( W ′ l ) is in fact, an isomorphism.Proof. Injectivity follows from surjectivity of F l , because it has a non-zero degree. It has a closed range since it is isometric embedding.Then, the conclusion follows since it has a dense range. (cid:3) Second, we obtain the inductive systemΦ l ≡ ( F l +1 ) ∗ ◦ β ◦ (( F l ) ∗ ) − : S C F l ( W ′ l ) → S C F l +1 ( W ′ l +1 ) . By definition, the equality holds S C F ( H ′ ) = lim l S C F l ( W ′ l ) . Forgetting the group action, we have the isomorphisms K ( S C F ( H ′ )) = lim l K ( S C F l ( W ′ l )) ∼ = lim l K ( S C ( W ′ l ))= K ( S C ( H ′ )) ∼ = K ( S ) ∼ = Z where we used the HKT-Bott periodicity.Now consider the group action by Z . Let F lt be the homotopy insubsection 6.2.1, where F l = F l and F l = T l . Lemma 6.5.
There is a ∗ -isomorphism I l : S C F l ( W ′ l ) ∼ = S C F l ( W ′ l ) . NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES31
Proof.
In fact an element u ∈ S C F l ( W ′ l ) is expressed as u = ( F l ) ∗ ( v )for some v ∈ S C ( W l ). Because F lt has a non zero degree, it followsthat v is uniquely determined by u . Then assign v = ( F l ) ∗ ( v ), anddenote its map by I l : S C F l ( W ′ l ) ∼ = S C F l ( W ′ l ) . This is a ∗ -homomorphism and, in fact, is an isomorphism, since if wedo the same thing, replacing the role of F l and F l , then we can recover u again. (cid:3) Proposition 6.6.
There is an isomorphism K ( S C F ( H ′ ) ⋊ Z ) ∼ = Z . Proof.
Denote by ¯ A the unitization of A . Take an element u ∈ K ( A ⋊Z ) and represent it by u = [ p ] − [ π ( p )]. Approximate p ∈ M at ( S C F ( H ′ ) ⋊ Z )by an element p ′ = ( p ′ g ) g ∈ B ∈ M at ( C c ( Z , S C F ( H ′ ))) where B ⊂ Z is afinite set. There is some l such that each p ′ g can be approximiated by p ′ g,l ∈ S C F l ( W ′ l ). Therefore, p can be approximated by an element p ′′ = ( p ′ g,l ) ∈ C ( {− l, . . . , l } , M at ( S C F l ( W ′ l ))) . Let us put I − l ( p ′′ ) ∈ C ( {− l, . . . , l } , M at ( S C T l ( W ′ l )))where S l = F l is the cyclic permutation, and I l is in Lemma 6.5.˜ p ′′ ≡ I − l ( p ′′ ) + ( I − l ( p ′′ )) ∗ ∈ M at ( S C T ( H ′ ) ⋊ Z )is an “almost” projection, in the sense that || (˜ p ′′ ) − ˜ p ′′ || < ǫ for a small ǫ >
0. Then there is a projection ˜ p ∈ M at ( S C T ( H ′ ) ⋊ Z )with the estimate || ˜ p − ˜ p ′′ || < ǫ ′ for a small ǫ ′ > u = [ p ] − [ π ( p )] = [ q ] − [ π ( q )]. Inthe same way, we obtain a projection ˜ q ∈ M at ( S C T ( H ′ ) ⋊ Z ). Recallthat there is some v ∈ M n,m ( S C F ( H ′ ) ⋊ Z ) such that p ⊕ a = v ∗ v, vv ∗ = q ⊕ b for some a, b ≥ Let v ′ ∈ C ( {− l, . . . , l } , M at ( S C F l ( W ′ l ))) be another approximationand take ˜ v ′′ ≡ I − l ( v ′ ) ∈ C ( {− l, . . . , l } , M at ( S C T l ( W ′ l ))). Then we havethe estimates || (˜ v ′′ ) ∗ ˜ v ′′ − ˜ p ⊕ a || , || ˜ v ′′ (˜ v ′′ ) ∗ − ˜ q ⊕ b || < ǫ ′′ for a small ǫ ′′ >
0. This implies the equality[˜ p ] − [ π (˜ p )] = [˜ q ] − [ π (˜ q )] ∈ K ( S C T ( H ′ ) ⋊ Z ) . Therefore, we obtain a well defined group homomorphism K ( S C F ( H ′ ) ⋊ Z ) → K ( S C T ( H ′ ) ⋊ Z ) . 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, theircompositions are both the identities. Therefore, this is an isomorphismon the K -groups.Since the translation shift T : H ′ ∼ = H ′ is unitary and Z is commu-tative, there is a ∗ -isomorphism S C T ( H ′ ) ⋊ Z ∼ = S C ( H ′ ) ⋊ Z . Passing through this isomorphism, we obtain the isomorphism K ( S C F ( H ′ ) ⋊ Z ) → K ( S C ( H ′ ) ⋊ Z ) . The right-hand side is isomorphic to K ( C ∗ Z ) ∼ = K ( S ) ∼ = Z by HKT. (cid:3) Nonlinear maps between Sobolev spaces over the circle.
Involution.
Consider the space S = R / Z = [0 , / { ∼ } and W k, ( S ) which is generated by sin( πks ) and cos( πks ) for k ∈ Z .Consider the Sobolev spaces W k, ( S ) , W k, ( S ) ⊂ W k, ( S )which are generated by W k, (0 , and W k, (1 , respectively. Here, W k, ( S ) i is naturally isometric to W k, ( S ) i − by the shift operator T : u → u , u ( s ) = u ( s + 1)mod 2. Note that T is the identity. Therefore, we can identify bothHilbert spaces by the same symbol H and, hence, the following inclu-sion holds: H ⊕ H ⊂ W k, ( S ) . NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES33
We again consider the non linear map with H ′ = HF : H ′ ⊕ H ′ → H ⊕ H by F ( a ) = a + T ( a ) , where the power is taken pointwisely. Then, themap can be written as (cid:18) ab (cid:19) → (cid:18) a + b b + a (cid:19) As we have seen, this is metrically proper.Let k = 1 for simplicity of notation, and consider an element a ∈ W , ( S ) a = ∞ X k = −∞ a k sin(2 πks ) + b k cos(2 πks )and denote a = ∞ X k = −∞ c k sin(2 πks ) + d k cos(2 πks ) Lemma 6.7.
Suppose || a || W , ≤ r . Then for any ǫ > , there is n = n ( r, ǫ ) ≥ such that the estimate holds || X | k |≥ n +1 c k sin(2 πks ) + d k cos(2 πks ) || W , < ǫ. Proof.
It follows from Lemma 6.1 that W , ( S ) → W , ( S ) by a → a is compact on each bounded set. (cid:3) Choose divergent numbers as lim i n i = ∞ . For each i ∈ N , let V ′ i ⊂ W , (0 , be the finite-dimensional linear subspace spanned bysin(2 πks ) and cos(2 πks ) for | k | ≤ n i , and set W ′ i = V ′ i ⊕ T ( V ′ i ) ⊂ H ′ ⊕ H ′ . Denote pr i : H ⊕ H = H ′ ⊕ H ′ → W i = W ′ 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 equivariant ∗ -isomorphism K Z ( S C F ( H ⊕ H )) ∼ = K Z ( S C ( H ⊕ H )) . Proof.
Step 1:
By the same argument as the toy case, F is metricallyproper, and it is Z -equivariantly properly homotopic to the involution I : H ⊕ H ∼ = H ⊕ H by F t . Sublemma 6.9.
There is a Z -equivariant ∗ -isomorphism S C I ( H ⊕ H ) ∼ = S C ( H ⊕ H ) . Proof.
By construction, S C I ( H ⊕ H ) = { ˜ u ; u ∈ S C ( H ⊕ H ) } where ˜ u ( a, b ) = I ∗ ( u ( a, b )) with a, b ∈ H . (cid:3) Step 2:
It follows from Lemma 6.7 that F − i ( D s i ∩ W i ) ⊂ D ′ r i ∩ W ′ i holds. As in the toy case, one may assume the same property( F ti ) − ( D s i ∩ W i ) ⊂ D r i ∩ W ′ i ≡ D ′ r i where F ti = pr i ◦ F t . K -theory is stable under these continuous deformations so that theisomorphism holds K Z ( S r i C F i ( D ′ r i )) ∼ = K Z ( S r i C F i ( D ′ r i )) . Step 3:
Recall the induced Clifford C ∗ -algebra S C F ( H ) whose ele-ment { α i } i satisfies the equality α i = F ∗ i β ( u i ) for some u i = β ( u i ) ∈ S s i C ( D s i ∩ W i ) and all i ≥ i . Here, S r i C F i ( D ′ r i ) is defined as the imageof F ∗ i : S s i C ( D s i ∩ W i ) → S r i C l i ( D ′ r i ) = S r i C ( D ′ r i ) (and l i is the identityin this particular case).Note that F i | D ′ r i has non-zero degree. We claim that there is a ∗ -homomorphism Φ i : S r i C F i ( D ′ r i ) → S r i +1 C F i +1 ( D ′ r i +1 )which sends α i to α i +1 . In fact α i uniquely determines u i . Supposethe 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 ishence surjective, there exists x ∈ D r i with F i ( x ) = m . Then, wehave the equality u i ( m ) = F ∗ i ( u i )( x ) = F ∗ i ( u ′ i )( x ) = u ′ i ( m ), whichcontradicts to the assumption.Now, since F ∗ i : S s i C ( D s i ∩ W i ) → S r i C F i ( D ′ r i ) ⊂ S r i C l i ( D ′ r i ) is anisometric ∗ -embedding, it follows that the inverse( F ∗ i ) − : S r i C F i ( D ′ r i ) → S s i C ( D s i ∩ W i )is ∗ -isomorphic. Then, Φ i is given by the compositions F ∗ i +1 ◦ β ◦ ( F ∗ i ) − . Step 4:
Then, K Z (( S C F ( H ⊕ H )) ∼ = lim i K Z ( S r i C F i ( D r i )) ∼ = lim i K Z ( S r i C F i ( D r i )) ∼ = K Z ( S C I ( H ⊕ H )) . By Sublemma 6.9, we have the desired isomorphism. (cid:3)
NDUCED MAP ON K THEORY FOR CERTAIN Γ − EQUIVARIANT MAPS BETWEEN HILBERT SPACES35
Finite cyclic case.
Consider the space S l = R /l Z = [0 , l ] / { ∼ l } and W k, ( S l ) which is generated by sin(2 π kl s ) and cos(2 π kl s ) for k ∈ Z .Consider the Sobolev spaces W k, ( S l ) , W k, ( S l ) , · · · , W k, ( S l ) l − ⊂ W k, ( S l )which are, respectively, generated by W k, ( i, i + 1) . Then, W k, ( S l ) i is naturally isometric to W k, ( S l ) i +1 , by the shift T : u i → u i +1 by u i +1 ( s ) := u i ( s − l . Thus, we can identify these Hilbertspaces by the same symbol H and, so the inclusion H ⊕ H ⊕ · · · ⊕ H ⊂ W k, ( S l ) holds.We again consider the non linear map F : H l = H ⊕ H ⊕ · · · ⊕ H → H ⊕ H ⊕ · · · ⊕ H,F ( a , . . . , a l ) = ( a + T ( a l ) , a + T ( a ) , . . . , a l + T ( a l − ) ) . Then, the map can be written as a a . . .a l → a + a l a + a . . .a l + a l − By the same argument as the toy case, this is metrically proper, andits nonlinear part is compact on each bounded set.By use of F t as above, F is Z l -equivariantly properly homotopicto the cyclic shift T . By a similar argument, we have the followingcorollary. Corollary 6.10.
There is a Z l -equivariant ∗ -isomorphism K Z l ( S C F ( H l )) ∼ = K Z l ( S C ( H l )) ∼ = K Z l ( 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 abovefinite cyclic case to the infinite case, following the same approach. Forexample the map l ( Z ) → l ( Z ) by { a i } i → { a 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, (0 , ⊂ W k, ( R ), andlet H be the closure of the sum ⊕ i ∈ Z H i , where H i are the copies of thesame H . Then, the T orbit of H , { T n ( H ) } n ∈ Z generates H ⊂ W k, ( R ),where T : W k, ( i, i + 1) ∼ = W k, ( i + 1 , i + 2) is the shift as before. Consider the map F : H → H by F : . . .a − a a . . . → . . .a − + a − a + a − a + a . . . . Let H ′ l be spanned by the vectors ( a − l , . . . , a l ). As in the toy modelcase, consider the approximation F l : H ′ l → H l by shifting the lastcomponent F l : a − l a − l +1 . . .a l → a − l + a l a − l +1 + a − l . . .a l + a l − . There is a finite-dimensional linear subspace W ′ i ⊂ H ′ i with r i , s i > F i , W ′ i , D ′ r i ) gives a Z -finitely approximable data with l i =id.An element u ∈ K ( S C F ( H ) ⋊ Z ) has a representative as u = [ p ] − [ π ( p )], where p ∈ M at ( S C F ( H ) ⋊ Z ) . Here, p can be approximated as p ′ ∈ M at ( C ( {− l, . . . , l } , S C F l ( W ′ l ))) . The rest of the process is parallel to the toy model case, and so onecan proceed in the same way, and then obtain an isomorphism K ( S C F ( H ) ⋊ Z ) ∼ = K ( S C ( H ) ⋊ Z ) ∼ = K ( C ∗ ( Z )) = K ( S ) ∼ = Z . References [A]
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