Homotopy type of the unitary group of the uniform Roe algebra on \mathbb{Z}^n
aa r X i v : . [ m a t h . A T ] F e b HOMOTOPY TYPE OF THE UNITARY GROUP OF THE UNIFORM ROEALGEBRA ON Z n TSUYOSHI KATO, DAISUKE KISHIMOTO, AND MITSUNOBU TSUTAYAA bstract . We study the homotopy type of the space of the unitary group U ( C ∗ u ( | Z n | )) of the uni-form Roe algebra C ∗ u ( | Z n | ) of Z n . We show that the stabilizing map U ( C ∗ u ( | Z n | )) → U ∞ ( C ∗ u ( | Z n | ))is a homotopy equivalence. Moreover, when n = ,
2, we determine the homotopy type ofU ( C ∗ u ( | Z n | )), which is the product of the unitary group U ( C ∗ ( | Z n | )) (having the homotopy typeof U ∞ ( C ) or Z × B U ∞ ( C ) depending on the parity of n ) of the Roe algebra C ∗ ( | Z n | ) and rationalEilenberg–MacLane spaces.
1. I ntroduction
For a C ∗ -algebra A , let GL d ( A ) and U d ( A ) denote the space of the invertible and unitary ma-trices with entries in A , respectively. It it well-known that they always have the same homotopytype. We will often refer only to U d ( A ) but most statements are valid for GL d ( A ) as well. Therehave been a lot of works on the homotopy theory of U d ( A ) and some of them have importantapplications. For finite-dimensional case, the complex-valued unitary matrices U d ( C ) is justthe usual unitary group acting linearly on C d . For inifinite-dimensional case, Kuiper [Kui65]proved that the space of all unitary operators on an inifinite-dimensional Hilbert space is con-tractible. This result is basic in the Atiyah–Singer index theory. This kind of contractibilityresult has been extended to U d ( A ) of some other algebras A while A is all the bounded operatorson a infinite dimensional Hilbert space in the original result. Of course it is not always the casefor U d ( A ) of other infinite-dimensional C ∗ -algebras A . In general, it is hard to determine thehomotopy type of U d ( A ).Let us use the notationGL ∞ ( A ) = lim d →∞ GL d ( A ) and U ∞ ( A ) = lim d →∞ U d ( A ) . It is well-known that U ∞ ( A ) has the same homotopy type as U ( A ⊗ K ) where K is the space ofcompact operators. The K -theory K i ( A ) ( i = ,
1) is a basic homotopy invariant of A , which ischaracterized as K ( A ) = π (U ∞ ( A )) and K ( A ) = π (U ∞ ( A )) . Since U d ( A ) is not necessarily homotopy equivalent to U ∞ ( A ), K i ( A ) is not a so strong invariantin general. But sometimes the natural map U d ( A ) → U ∞ ( A ), which we will call the stabilizingmap , becomes a homotopy equivalence. Study on such stability seems to trace back to the workof Bass [Bas64]. There have been a various works on this kind of stability. Rie ff el introduced the topological stable rank in [Rie83] and applied it to show the stability of the non-commutative Mathematics Subject Classification.
Key words and phrases. uniform Roe algebra, Roe algebra, unitary group, homotopy type, operator K -theory.Kato was supported by JSPS KAKENHI 17K18725 and 17H06461. Kishimoto was supported by JSPS KAK-ENHI 17K05248 and 19K03473. Tsutaya was supported by JSPS KAKENHI 19K14535. torus in [Rie87], which is a key tool in the present work. It is di ffi cult in general to determinehow stable a given C ∗ -algebra is.In the present paper, we study the stability of the uniform Roe algebra C ∗ u ( | Z n | ) on Z n andinvestigate its homotopy type. The uniform Roe algebra C ∗ u ( X ) of a metric space X introducedby Roe in [Roe88] to establish an index theory on open manifolds, where the index lives in the K -theory K ∗ ( C ∗ u ( X )). The algebra C ∗ u ( X ) itself is also important since it encodes a kind of “largescale geometry” of X . Studying the homotopy type of U d ( C ∗ u ( X )) will provide more insightsfrom a homotopy theoretic viewpoint, which cannot be obtained only from its K -theory. Butthere are only a few works on the homotopy type of U d ( C ∗ u ( X )) yet. For example, Manuilov andTroitsky [MT21] studied some condition for U d ( C ∗ u ( X )) being contractible. In the present work,we observe the other extreme, that is, U d ( C ∗ u ( | Z n | )) has a highly nontrivial homotopy type.We give some comment on the relation with our previous work [KKT] on the space U offinite propagation unitary operators on Z . Note that U ( C ∗ u ( | Z | )) can be viewed as a kind ofcompletion of U . We determined the homotopy type of U there. But it is not clear whether U has the same homotopy type as U ( C ∗ u ( | Z | )). Actually, they turn out to have the same homotopytype (Theorem 1.2). Also, the method there does not seem to be extended to Z n when n ≥ d ( C ∗ u ( | Z n | )) to the one to show the surjectivityof the homomorphism on K -theory K ∗ ( C ∗ u ( | Z n | )) → K ∗ ( C ∗ ( | Z n | )) induced from the inclusion(Proposition 7.4), where C ∗ ( | Z n | ) denotes the Roe algebra of Z n .For stability, we show the following theorem in Section 3. Theorem 1.1.
For any integer n ≥ , the stabilizing maps GL ( C ∗ u ( | Z n | )) → GL ∞ ( C ∗ u ( | Z n | )) , U ( C ∗ u ( | Z n | )) → U ∞ ( C ∗ u ( | Z n | )) , GL ( C ∗ ( | Z n | )) → GL ∞ ( C ∗ ( | Z n | )) , U ( C ∗ ( | Z n | )) → U ∞ ( C ∗ ( | Z n | )) between the spaces of invertible and unitary elements are homotopy equivalences. This impliesthat these maps induce the following isomorphisms on homotopy groups for all i ≥ : π i (GL ( C ∗ u ( | Z n | ))) (cid:27) π i (U ( C ∗ u ( | Z n | ))) (cid:27) K ( C ∗ u ( | Z n | )) i is even,K ( C ∗ u ( | Z n | )) i is odd, π i (GL ( C ∗ ( | Z n | ))) (cid:27) π i (U ( C ∗ ( | Z n | ))) (cid:27) K ( C ∗ ( | Z n | )) i is even,K ( C ∗ ( | Z n | )) i is odd. Let K ( V , i ) denote the Eilenberg–MacLane space of type ( V , i ) and B U ∞ ( C ) denote the clas-sifying space of the unitary group U ∞ ( C ). Also, for based spaces X i ( i = , , . . . ), define ◦ Y i ≥ X i = lim k →∞ ( X × X × · · · × X k ) . For the homotopy type of U ( C ∗ u ( | Z n | )), we show the following results when n = , Theorem 1.2.
There exist homotopy equivalences of infinite loop spaces GL ( C ∗ u ( | Z | )) ≃ U ( C ∗ u ( | Z | )) ≃ Z × B U ∞ ( C ) × ◦ Y i ≥ K ( ℓ ∞ ( Z , Z ) S , i − . OMOTOPY TYPE OF THE UNITARY GROUP OF THE UNIFORM ROE ALGEBRA ON Z n Theorem 1.3.
There exist homotopy equivalences of infinite loop spaces GL ( C ∗ u ( | Z | )) ≃ U ( C ∗ u ( | Z | )) ≃ V × U ∞ ( C ) × ◦ Y i ≥ ( K ( V , i − × K ( V , i )) , where V , V are the rational vector spaces given byV = ker[ K ( C ∗ u ( | Z | )) → K ( C ∗ ( | Z | ))] , V = K ( C ∗ u ( | Z | )) and the product factor V is a discrete space. More detailed descriptions of the vector spaces V and V appear in the proof of Lemma 7.7.We will see the existence of a homotopy section of the inclusion U ( C ∗ u ( | Z | )) → U ( B SW )in Section 6, where U ( B SW ) is the Segal–Wilson restricted unitary group [SW85] having thehomotopy type of Z × B U ∞ ( C ). This implies Theorem 1.2. We also show in Section 7 that,for any integer n ≥
1, the inclusion U ( C ∗ u ( | Z n | )) → U ( C ∗ ( | Z n | )) admits a homotopy section ifand only if the homomorphism K ∗ ( C ∗ u ( | Z n | )) → K ∗ ( C ∗ ( | Z n | )) is surjective. Since we can see it issurjective when n = ,
2, Theorems 1.2 again and 1.3 follows. If one could show the surjectivityfor n ≥
3, then a similar homotopy decomposition will immediately follow.This paper is organized as follows. We fix our notation in Section 2. In Section 3, werecall Rie ff el’s results on stability and show Theorem 1.1. In Section 4, we recall the Bottperiodicity realized as a ∗ -homomorphism. In Section 5, we recall the Segal–Wilson restrictedunitary group and show its stability. In Section 6, we show Theorem 1.2 using the Segal–Wilsonrestricted unitary group. In Section 7, we discuss the homotopy type of U d ( C ∗ u ( | Z n | )) for general n ≥ otation The C ∗ -algebra of bounded operators on a Hilbert space V is denoted by B ( V ) and the sub-algebra of compact operators by K ( V ). We write the operator norm of T ∈ B ( V ) as k T k . TheHilbert space of square summable sequences indexed by a discrete group Γ will be written as ℓ ( Γ ) = { ( v g ) g | X g ∈ Γ | v g | < ∞} . We also consider the tensor product Hilbert space ℓ ( Γ ) ⊗ H with an infinite dimensional sepa-rable Hilbert space H .A bounded operator T ∈ B ( ℓ ( Γ )) can be expressed in the matrix form as T = ( T g , h ) g , h , T g , h ∈ C . For T ∈ B ( ℓ ( Γ ) ⊗ H ), we also have a similar expression T = ( T g , h ) g , h with T g , h ∈ B ( H ). Definition 2.1.
Let Γ be a finitely generated group and d denote the word metric with respectto some finite set of generators. We say that a bounded operator T ∈ B ( ℓ ( Γ )) has finite propa-gation if prop( T ) = sup { d ( g , h ) | T g , h , } is finite. We define finite propagation for T = ( T g , h ) g , h ∈ B ( ℓ ( Γ ) ⊗ H ) similarly. TSUYOSHI KATO, DAISUKE KISHIMOTO, AND MITSUNOBU TSUTAYA
Example 2.2.
The shift S x ∈ B ( ℓ ( Γ )) by x ∈ Γ is defined by S x = (( S x ) g , h ) g , h , S g , h = g − h = x , S x is a unitary operator with prop( S x ) = d ( x , T ) depends on the word metric. Since we haveprop( S T ) ≤ prop( S ) + prop( T ) , prop( T ∗ ) = prop( T ) , prop(1) = S , T ∈ B ( ℓ ( Γ )), the subset of finite propagation opera-tors becomes a unital ∗ -subalgebra of B ( ℓ ( Γ )). Similar properties hold for finite propagationoperators S , T ∈ B ( ℓ ( Γ ) ⊗ H ) such that the components T g , h and S g , h are compact operators. Definition 2.3.
The uniform Roe algebra C ∗ u ( | Γ | ) of Γ is the norm closure of the algebra of finitepropagation operators in B ( ℓ ( Γ )). Definition 2.4.
The
Roe algebra C ∗ ( | Γ | ) of Γ is the norm closure of the algebra of finite propa-gation operators T ∈ B ( ℓ ( Γ ) ⊗ H ) such that each component T g , h is a compact operator. Remark . We follow the usual notation C ∗ ( | Γ | ) for the Roe algebra of Γ to distinguish it fromthe group C ∗ -algebra of Γ though we do not consider the latter here.We will consider the uniform Roe algebra C ∗ u ( | Γ | ) is a subalgebra of the Roe algebra C ∗ ( | Γ | )with respect to some inclusion C ⊂ H .We use the symbol ℓ ∞ ( Γ , C ) to express the Banach algebra of C -valued bounded sequencesindexed by Γ rather than the simpler symbol ℓ ∞ ( Γ ) since we also consider the abelian group of Z -valued bounded sequences ℓ ∞ ( Γ , Z ).The group Γ acts on the algebras ℓ ∞ ( Γ , C ) and ℓ ∞ ( Γ , K ( H )) by right translation. The actionby x ∈ Γ is compatible with the conjugation by S x through the diagonal inclusion ℓ ∞ ( Γ , C ) → C ∗ u ( | Γ | ) or ℓ ∞ ( Γ , K ( H )) → C ∗ ( | Γ | ) given by( t g ) g ( T g , h ) g , h , T g , h = t g g = h , ℓ ∞ ( Γ , C ) ⋊ Γ (cid:27) C ∗ u ( | Γ | ) , ℓ ∞ ( Γ , K ( H )) ⋊ Γ (cid:27) C ∗ ( | Γ | )from the reduced crossed products of C ∗ -algebras. For example, see [Roe03, Theorem 4.28].The d × d -matrix algebra M d ( A ) of a C ∗ -algebra A is again a C ∗ -algebra. The spaces ofinvertible elements and unitary elements in M d ( A ) will be denoted as GL d ( A ) and U d ( A ). The stabilizing maps are given asGL ( A ) → GL ∞ ( A ) = lim d →∞ GL d ( A ) , U ( A ) → U ∞ ( A ) = lim d →∞ U d ( A ) , where the inductive limits are taken along the inclusions GL d ( A ) ⊂ GL d + ( A ) and U d ( A ) ⊂ U d + ( A ). The inductive limit spaces GL ∞ ( A ) and U ∞ ( A ) are well-known to be homotopy equiv-alent to the spaces GL ( A ⊗ K ( H )) and U ( A ⊗ K ( H )). OMOTOPY TYPE OF THE UNITARY GROUP OF THE UNIFORM ROE ALGEBRA ON Z n
3. S tability
The aim of this section is to prove Theorem 1.1. Once the assumption of the following resultby Rie ff el [Rie87] is verified, the theorem will immediately follow. Theorem 3.1 (Rie ff el) . Let A be a unital C ∗ -algebra. If A is tsr-boundedly divisible, then thestabilizing maps GL ( A ) → GL ∞ ( A ) and U ( A ) → U ∞ ( A ) are homotopy equivalences.Remark . The original statement of Theorem 4.13 in [Rie87] is involved only with homotopygroups. But what is actually proved there is slightly stronger as above.For the definitions of the topological stable rank tsr( A ) ∈ Z ≥ , see [Rie83]. A C ∗ -algebra A is said to be tsr-boundedly divisible [Rie87] if there is a constant K such that for any integer m ,there exists an integer d ≥ m such that A is isomorphic to M d ( B ) for some C ∗ -algebra B withtsr( B ) ≤ K . To verify the assumption, we need the following two lemmas. Lemma 3.3.
The topological stable ranks of C ∗ u ( | Z n | ) and C ∗ ( | Z n | ) are estimated as tsr( C ∗ u ( | Z n | )) ≤ n + and tsr( C ∗ ( | Z n | )) ≤ n + . Remark . We will see that both C ∗ u ( | Z n | ) and C ∗ ( | Z n | ) are tsr-boundedly divisible using thislemma. Thus we will actually obtain the estimates tsr( C ∗ u ( | Z n | )) ≤ C ∗ ( | Z n | )) ≤ Proof.
Let A = C or K ( H ). Since the invertible elements in ℓ ∞ ( Z n , C ) and C ⊕ ℓ ∞ ( Z n , K ( H ))are dense, we have tsr( ℓ ∞ ( Z n , A )) = Z m ⊂ Z n on the first m factorsof Z n , we obtain the isomorphism ℓ ∞ ( Z n , A ) ⋊ Z m + (cid:27) ( ℓ ∞ ( Z n , A ) ⋊ Z m ) ⋊ Z . Thus, by [Rie83, Theorem 7.1], we get the desired estimates on tsr( C ∗ u ( | Z | )) and tsr( C ∗ ( | Z n | )). (cid:3) Lemma 3.5.
For any integer d ≥ , there exist isomorphisms φ : C ∗ u ( | Z n | ) (cid:27) M d ( C ∗ u ( | Z n | )) and φ : C ∗ ( | Z n | ) (cid:27) M d ( C ∗ ( | Z n | )) . Proof.
Let V = C or H . According to the decomposition ℓ ( Z n ) ⊗ V = M ( i ,..., i n ) ∈ Z n V ( i ,..., i n ) , V ( i ,..., i n ) (cid:27) V , we have the matrix expression for T ∈ B ( ℓ ( Z n ) ⊗ V ) T = ( T ( i ,..., i n )( j ,..., j n ) ) ( i ,..., i n )( j ,..., j n ) , T ( i ,..., i n )( j ,..., j n ) : V ( j ,..., j n ) → V ( i ,..., i n ) . Consider the map φ : B ( ℓ ( Z n ) ⊗ V ) → M d ( B ( ℓ ( Z n ) ⊗ V )) given by φ ( T ) ( i ,..., i n )( j ,..., j n ) = T ( di ,..., i n )( d j ,..., j n ) · · · T ( di ,..., i n )( d j + d − ,..., j n ) ... . . . ... T ( di + d − ,..., i n )( d j ,..., j n ) · · · T ( di + d − ,..., i n )( d j + d − ,..., j n ) ∈ M d ( B ( V )) . TSUYOSHI KATO, DAISUKE KISHIMOTO, AND MITSUNOBU TSUTAYA
The restrictions to C ∗ u ( | Z n | ) ⊂ B ( ℓ ( Z n )) and C ∗ ( | Z n | ) ⊂ B ( ℓ ( Z n ) ⊗ H ) are desired isomorphisms. (cid:3) Remark . When n =
1, the map φ is just taking the block matrix of which each block is a d × d -matrix. Proof of Theorem 1.1.
By Lemmas 3.3 and 3.5, we can apply Theorem 3.1 to C ∗ u ( | Z n | ) and C ∗ ( | Z n | ). This completes the proof of the theorem. (cid:3)
4. B ott periodicity
Let us recall the Bott periodicity of C ∗ -algebras here. Let A be a C ∗ -algebra, which mightbe non-unital. The direct sum C ⊕ A is considered as the unitization with unit (1 , ∈ C ⊕ A .Define the unitary group U ′ d ( A ) byU ′ d ( A ) = { U ∈ U n ( C ⊕ A ) | U − ( I d , ∈ M d ( A ) } . If A is already unital, we have a canonical isomorphism U ′ d ( A ) (cid:27) U d ( A ). So we use the samesymbol U d ( A ) for U ′ d ( A ) even if A is not unital.Consider the following space of continuous functions: C ( R m , A ) = { T : R m → A | T is continuous and lim | z |→∞ T ( z ) = } . This is a C ∗ -algebra without unit. Notice that C ( R m , A ) is isomorphic to the space Ω m A ofbased maps from the m -sphere S m to A where the basepoint ∗ ∈ S m is mapped to 0 ∈ A .Set the element p B ( z ) = + | z | | z | z ¯ z ! ∈ M ( C ⊕ C ( R , C )) ( z ∈ R ) , where we identify R (cid:27) C in the matrix entries. The Bott map β : A → M ( A ⊕ C ( R , A )) is a ∗ -homomorphism defined by β ( a ) = p B a a ! = a a ! p B . Then we have the commutative square of unital C ∗ -algebras C ⊕ A ǫ / / β (cid:15) (cid:15) C η (cid:15) (cid:15) M ( C ⊕ A ⊕ C ( R , A )) ǫ / / M ( C ⊕ A )where ǫ : C ⊕ A → C and ǫ : M ( C ⊕ A ⊕ C ( R , A )) → M ( C ⊕ A ) are the projections and η : C → M ( C ⊕ A ) is the unit map. This square induces the ∗ -homomorphism between thekernels of ǫ : β : A → M ( C ( R , A )) . OMOTOPY TYPE OF THE UNITARY GROUP OF THE UNIFORM ROE ALGEBRA ON Z n We call this β the Bott map as well. It is natural in the following sense: if f : A → B is a ∗ -homomorphism between C ∗ -algebras, then the following square commutes: A f / / β (cid:15) (cid:15) B β (cid:15) (cid:15) M ( C ( R , A )) f ∗ / / M ( C ( R , B )) Proposition 4.1.
The Bott map β : A → M ( C ( R , A )) induces an isomorphism on K-theory.Remark . This can be seen as a formulation of the Bott periodicity. If you wish to deducethis proposition from the results appearing in [Bla86], it follows from the observation 9.2.10 onthe generator of K ( C ( R , C )) and the K¨unneth theorem for tensor products (Theorem 23.1.3).The Bott periodicity provides the natural homotopy equivalenceU ∞ ( A ) β −→ U ∞ ( M ( C ( R , A ))) ≃ Ω U ∞ ( A ) , which is a group homomorphism. Thus we obtain the following proposition on infinite loopstructure. Proposition 4.3.
The unitary group U ∞ ( A ) of a C ∗ -algebra A is equipped with a canonical infi-nite loop space structure such that the map U ∞ ( A ) → U ∞ ( B ) induced from a ∗ -homomorphismA → B is an infinite loop map. Moreover, the underlying loop structure of U ∞ ( A ) coincideswith the group structure of U ∞ ( A ) .Remark . The last sentence in the proposition means that there exists a homotopy equivalence B U ∞ ( A ) ≃ Ω U ∞ ( A ) from the classifying space B U ∞ ( A ) of the topological group U ∞ ( A ).5. S egal –W ilson restricted unitary group To study the homotopy type of U ( C ∗ u ( | Z | )), we will relate it with other spaces. One is theSegal–Wilson restricted unitary group U ( B SW ) and the other is the unitary group of the Roealgebra U ( C ∗ ( | Z | )). We recall the former in this section.We have another matrix expression for T ∈ B ( ℓ ( Z )) as T = T −− T − + T + − T ++ ! , where T −− : ℓ ( Z < ) → ℓ ( Z < ) , T − + : ℓ ( Z ≥ ) → ℓ ( Z < ) , T + − : ℓ ( Z < ) → ℓ ( Z ≥ ) , T ++ : ℓ ( Z ≥ ) → ℓ ( Z ≥ ) . Definition 5.1.
We define the C ∗ -algebra B SW by B SW : = { T ∈ B ( H ) | T − + , T + − are compact } . The symbol “SW” stands for Segal–Wilson. The unitary group U ( B SW ) is called the re-stricted unitary group in the work of Segal and Wilson [SW85]. They used it as a model of theinfinite Grassmannian. TSUYOSHI KATO, DAISUKE KISHIMOTO, AND MITSUNOBU TSUTAYA
Lemma 5.2 (Segal–Wilson) . The space U ( B SW ) has the homotopy type of Z × B U ∞ ( C ) . More-over, the map π (U ( B SW )) → Z , [ U ] ind( U ++ ) , is bijective, where ind( U ++ ) denotes the Fredholm index of the Fredholm operator U ++ . Let S = S + ∈ B SW the shift operator as in Example 2.2. We have ind S n = n .The goal of this section is to see the following. Proposition 5.3.
The stabilizing maps GL ( B SW ) → GL ∞ ( B SW ) and U ( B SW ) → U ∞ ( B SW ) are homotopy equivalences. To show this, we do not use a kind of stability as in Section 3.
Lemma 5.4.
For any integer d ≥ , the inclusion U ( B SW ) → U d ( B SW ) induces an isomorphism on π .Proof. Consider the composite of the inclusion and the isomorphism φ : B SW → M d ( B SW ) sim-ilar to the one in the proof of Lemma 3.5:U ( B SW ) → U d ( B SW ) φ − −−→ U ( B SW ) . It is easy to see that the image of the shift S ∈ U ( B SW ) under this composite again has index 1.This implies the lemma. (cid:3) Lemma 5.5.
The K-theory of B SW is computed asK i ( B SW ) (cid:27) i = , Z i = , where K ( B SW ) is generated by the shift S ∈ U ( B SW ) .Proof. This follows from the isomorphisms K ( B SW ) (cid:27) lim d →∞ π (U d ( B SW )) and K ( B SW ) (cid:27) lim d →∞ π (U d ( B SW ))and Lemmas 5.2 and 5.4. (cid:3) Lemma 5.6.
For any i ≥ , there exists an integer m ≥ such that the iterated Bott map β i : U d ( B SW ) → U d ( M i ( C ( R i , B SW ))) induces an isomorphism on π if d ≥ m.Proof. From the isomorphisms π (U d ( M i ( C ( R i , B SW ))) (cid:27) π i (U i d ( B SW )) (cid:27) π i (U ( B SW )) (cid:27) Z and K ( M i ( C ( R i , B SW ))) (cid:27) lim d →∞ π (U d ( M i ( C ( R i , B SW ))) (cid:27) K ( B SW ) (cid:27) Z , OMOTOPY TYPE OF THE UNITARY GROUP OF THE UNIFORM ROE ALGEBRA ON Z n we can find an integer m ≥ π (U d ( M i ( C ( R i , B SW ))) → K ( C ( M i ( R i , B SW )))is an isomorphism if d ≥ m . Consider the commutative diagram π (U d ( B SW )) (cid:27) / / ( β i ) ∗ (cid:15) (cid:15) K ( B SW ) ( β i ) ∗ (cid:15) (cid:15) π (U d ( M i ( C ( R i , B SW ))) (cid:27) / / K ( M i ( C ( R i , B SW ))where the top arrow is an isomorphism by Lemma 5.4 and the right Bott map β i is an isomor-phism by Proposition 4.1. Then the lemma follows. (cid:3) Proof of Proposition 5.3.
Take an integer i ≥
0. We can find an integer m ≥ π i (U d ( B SW )) → π i (U ∞ ( B SW )) (cid:27) Z is an isomorphism if d ≥ m . Consider the following commutative diagram:U ( C ( R i , B SW )) (cid:27) / / (cid:15) (cid:15) U ( M i ( C ( R i , B SW ))) (cid:15) (cid:15) U ( B SW ) β i o o isom. on π (cid:15) (cid:15) U d ( C ( R i , B SW )) (cid:27) / / U d ( M i ( C ( R i , B SW ))) U d ( B SW ) β i o o where the left horizontal arrows are the isomorphisms similar to the one in Lemma 3.5 and thevertical arrows are the inclusions. Since the compositeU ( B SW ) → U d ( B SW ) β i −→ U d ( M i ( C ( R i , B SW )))induces an isomorphism on π , the middle vertical arrowU ( M i ( C ( R i , B SW ))) → U d ( M i ( C ( R i , B SW )))induces a surjection on π . But it is indeed an isomorphism as their π are isomorphic to Z .Then the map U ( C ( R i , B SW )) → U d ( C ( R n , B SW ))induces an isomorphism on π . This implies that the mapU ( B SW ) → U d ( B SW )induces an isomorphism on π i . Thus the mapU ( B SW ) → U ∞ ( B SW )induces an isomorphism on π i . This completes the proof. (cid:3)
6. H omotopy type of U ( C ∗ u ( | Z | ))The goal of this section is to prove Theorem 1.2. The components T − + and T + − of a finitepropagation operator T ∈ B ( H ) are finite rank operators. This implies the inclusion C ∗ u ( | Z | ) ⊂ B SW . This map is a key to the proof of Theorem 1.2.We begin with computing the K -theory. Proposition 6.1.
The following isomorphism holds:K ∗ ( C ∗ u ( | Z | )) (cid:27) ℓ ∞ ( Z , Z ) S i = , Z i = , where ℓ ∞ ( Z , Z ) S = ℓ ∞ ( Z , Z ) / { a − S a | a ∈ ℓ ∞ ( Z , Z ) } is the coinvariant by the shift S : ℓ ∞ ( Z , Z ) → ℓ ∞ ( Z , Z ) .Proof. Applying the Pimsner–Voiculescu exact sequence [PV80] to the crossed product C ∗ u ( | Z | ) (cid:27) ℓ ∞ ( Z , C ) ⋊ Z , we get the six-term cyclic exact sequence: K ( ℓ ∞ ( Z , C )) − S / / K ( ℓ ∞ ( Z , C )) / / K ( C ∗ u ( | Z | )) (cid:15) (cid:15) K ( C ∗ u ( | Z | )) O O K ( ℓ ∞ ( Z , C )) o o K ( ℓ ∞ ( Z , C )) − S o o As is well-known, we have K i ( ℓ ∞ ( Z , C )) (cid:27) ℓ ∞ ( Z , Z ) i = , i = , where the induced homomorphism S : ℓ ∞ ( Z , Z ) → ℓ ∞ ( Z , Z ) is the shift as well. Thus we cancompute K ∗ ( C ∗ u ( | Z | )) by the previous exact sequence. (cid:3) We saw the homotopy stabilities as in Theorem 1.1 and Proposition 5.3. Then it is su ffi cientto investigate the inclusion U ∞ ( C ∗ u ( | Z | )) → U ∞ ( B SW ). Lemma 6.2.
The inclusion U ∞ ( C ∗ u ( | Z | )) → U ∞ ( B SW ) induces isomorphisms on π i for i ≥ .Proof. By Lemma 5.5, K ( B SW ) is isomorphic to Z and generated by the shift S ∈ B SW . Since S ∈ C ∗ u ( | Z | ) and K ( C ∗ u ( | Z | )) (cid:27) Z , the map K ( C ∗ u ( | Z | )) → K ( B SW ) is an isomorphism. Thus themap π i (U ∞ ( C ∗ u ( | Z | ))) → π i (U ∞ ( B SW )) is also an isomorphism. (cid:3) Let F be the homotopy fiber of the inclusion U ∞ ( C ∗ u ( | Z | )) → U ∞ ( B SW ). Proposition 6.3.
The space F has the homotopy type of the product of Eilenberg–MacLanespaces ◦ Y i ≥ K ( ℓ ∞ ( Z , Z ) S , i − . where ℓ ∞ ( Z , Z ) S is a rational vector space of uncountable dimension. OMOTOPY TYPE OF THE UNITARY GROUP OF THE UNIFORM ROE ALGEBRA ON Z n Proof.
Observing the homotopy exact sequence · · · → π i ( F ) → π i (U ∞ ( C ∗ u ( | Z | ))) → π i (U ∞ ( B SW )) → π i − ( F ) → · · · , we can see that the homotopy fiber inclusion F → U ∞ ( C ∗ u ( | Z | )) induces an isomorpshim on π i − and π i ( F ) = π i − (U ( B SW )) =
0. By Proposition6.1, we have π i − ( F ) (cid:27) ℓ ∞ ( Z , Z ) S . The abelian group ℓ ∞ ( Z , Z ) S is a rational vector spaceof uncountable dimension as seen in [KKT, Section 5]. By [KKT, Lemma 5.4], F has thehomotopy type of the product of Eilenberg–MacLane spaces as above. (cid:3) The following easy lemma is useful to study the homotopy type of the unitary group of a C ∗ -algebra. Let p r ∈ M d ( C ) denote the projection of rank r . Lemma 6.4.
Let A be a C ∗ -algebra, where we do not require the existence of unit. For anyelement u ∈ K ( A ) , there exists a (non-unital in general) ∗ -homomorphism f : C → M d ( A ) suchthat f ∗ [ p ] ∈ K ( M d ( A )) (cid:27) K ( A ) equals to u.Proof. We can find a projection p ∈ M d ( C ⊕ A ) and r ≥ u = [ p ] − [ p r ] in K ( A ).Define a ∗ -homomorphism f : C → M d ( A ) by f (1) = p . This is the desired map. (cid:3) Proposition 6.5.
The inclusion U ∞ ( C ∗ u ( | Z | )) → U ∞ ( B SW ) admits a homotopy section, which isan infinite loop map.Proof. Consider the inclusion of based loop spaces U ∞ ( C ( R , C ∗ u ( | Z | ))) → U ∞ ( C ( R , B SW )).By Proposition 4.3, Lemma 6.4 and K ( C ( R , C ∗ u ( | Z | ))) (cid:27) Z , there exists an infinite loop map f : U ∞ ( C ) → U ∞ ( C ( R , C ∗ u ( | Z | ))) which induces an isomorphism on π i − for any i ≥
1. Itfollows from this and Lemma 6.2 that the compositeU ∞ ( C ) f −→ U ∞ ( C ( R , C ∗ u ( | Z | ))) → U ∞ ( C ( R , B SW ))is a homotopy equivalence. Then the inclusion of based loop spaces U ∞ ( C ( R , C ∗ u ( | Z | ))) → U ∞ ( C ( R , B SW )) admits a homotopy section. This implies that the inclusion of the double loopspace U ∞ ( C ( R , C ∗ u ( | Z | ))) → U ∞ ( C ( R , B SW )) also admits a homotopy section. Thus the in-clusion U ∞ ( C ∗ u ( | Z | )) → U ∞ ( B SW ) admits a homotopy section by Bott periodicity, which is againan infinite loop map. (cid:3) Proof of Theorem 1.2.
By Proposition 6.5, we have a homotopy equivalenceU ∞ ( C ∗ u ( | Z | )) ≃ U ∞ ( B SW ) × F as infinite loop spaces. The homotopy types of the spaces U ∞ ( B SW ) and F are determined inLemma 5.2 and Proposition 6.3, respectively. Together with the homotopy stability in Theorem1.1, this completes the proof of the theorem. (cid:3)
7. G eneralization
In this section, we study the relation between the homotopy type of U ( C ∗ u ( | Z n | )) and theinclusion U ( C ∗ u ( | Z n | )) ⊂ U ( C ∗ ( | Z n | )) for general n ≥
2. In view of Theorem 1.2, we proposethe following question.
Question 7.1.
Does the inclusion U d ( C ∗ u ( | Γ | )) → U d ( C ∗ ( | Γ | )) admits a homotopy section? Arethe homotopy groups of its homotopy fiber are rational vector spaces? Let us see the case when
Γ = Z n in view of this question. Lemma 7.2.
The K-theory of the Roe algebra C ∗ ( | Z n | ) is computed asK i ( C ∗ ( | Z n | )) (cid:27) Z i ≡ n mod , i . n mod .Proof. Let A m = ℓ ∞ ( Z n , K ( H )) ⋊ Z m with respect to the action of Z m ( m ≤ n ) on the first m factors of Z n . Let S j denote the shift onthe j -th factor. Then by the Pimsner–Voiculescu exact sequence K ( A m − ) − S m / / K ( A m − ) / / K ( A m ) (cid:15) (cid:15) K ( A m ) O O K ( A m − ) o o K ( A m − ) − S m o o for A m = A m − ⋊ S m Z , we obtain the short exact sequence0 → K i ( A m − ) S m → K i ( A m ) → K − i ( A m − ) S m → i = ,
1, where K i ( A m − ) S m and K i ( A m − ) S m denote the coinvariant and the invariant by S m ,respectively. Since A = ℓ ∞ ( Z n , K ( H )) and we have the well-known isomorphism K i ( ℓ ∞ ( Z n , K )) (cid:27) Z Z n i = , i = , where Z Z n is the group of all Z -valued sequences over Z n , we obtain K i ( A m ) (cid:27) Z Z n − m i ≡ m mod 2,0 i . m mod 2,by induction on m . The lemma is just the case when m = n . (cid:3) Together with the previous lemma, the homotopy type of U ∞ ( C ∗ ( | Z n | )) is determined by thefollowing lemma. Lemma 7.3.
Let A be a C ∗ -algebra, where we do not require the existence of unit. Consider thefollowing two conditions on K-theory:(i) K i ( A ) (cid:27) Z i = , i = , (ii) K i ( A ) (cid:27) i = , Z i = . If (i) holds, then U ∞ ( A ) has the homotopy type of U ∞ ( C ) as an infinite loop space. If (ii) holds,then U ∞ ( A ) has the homotopy type of Z × B U ∞ ( C ) as an infinite loop space.Proof. Suppose the condition (i). By Lemma 6.4, there exists a homotopy equivalence f : U ∞ ( C ) → U ∞ ( A ), which is an infinite loop map. When the condition (ii) holds, apply the result for thecondition (i) to the algebra C ( R , A ). This implies that U ∞ ( C ( R , A )) is homotopy equivalent toU ∞ ( C ). By the Bott periodicity, U ∞ ( A ) is homotopy equivalent to Ω U ∞ ( C ) ≃ Z × B U ∞ ( C ). (cid:3) Proposition 7.4.
The inclusion U ∞ ( C ∗ u ( | Z n | )) → U ∞ ( C ∗ ( | Z n | )) admits a homotopy section as aninfinite loop map if and only if the homomorphism K ∗ ( C ∗ u ( | Z n | )) → K ∗ ( C ∗ ( | Z n | )) is surjective. OMOTOPY TYPE OF THE UNITARY GROUP OF THE UNIFORM ROE ALGEBRA ON Z n Proof.
The only if part is obvious. For the if part, when n is odd, this follows from Lemma7.2 and the same argument as in the proof of Proposition 6.5. When n is even, apply the sameargument to the map on the based loop spaces U ∞ ( C ( R , C ∗ u ( | Z n | ))) → U ∞ ( C ( R , C ∗ ( | Z n | ))).Then the proposition follows from the existence of the homotopy section of the map on thedouble loop spaces U ∞ ( C ( R , C ∗ u ( | Z n | ))) → U ∞ ( C ( R , C ∗ ( | Z n | ))) and the Bott periodicity. (cid:3) Now all we have to do is to see that the homomorphism K ∗ ( C ∗ u ( | Z n | )) → K ∗ ( C ∗ ( | Z n | )) issurjective. Let B m = ℓ ∞ ( Z n , C ) ⋊ Z m with respect to the action Z m ( m ≤ n ) on the first m factors of Z n and S j denote the shift on the j -th factor. We obtain the short exact sequences similar to (1)0 → K i ( B m − ) S m → K i ( B m ) → K − i ( B m − ) S m → i = ,
1. For n = ,
2, we can see the surjectivity as follows.
Lemma 7.5.
The homomorphism K ( C ∗ u ( | Z | )) → K ( C ∗ ( | Z | )) is an isomorphism.Proof. Consider the commutative square K ( C ∗ u ( | Z | )) (cid:27) / / (cid:15) (cid:15) ℓ ∞ ( Z , Z ) S (cid:27) (cid:15) (cid:15) K ( C ∗ ( | Z | )) (cid:27) / / ( Z Z ) S obtained from the exact sequences (1) and (2). Thus the lemma follows. (cid:3) Lemma 7.6.
The homomorphism K ( C ∗ u ( | Z | )) → K ( C ∗ ( | Z | )) is surjective.Proof. When n =
2, we can compute K ∗ ( B ) by the exact sequence (2) as follows: K i ( B ) (cid:27) ℓ ∞ ( Z , Z ) S i = ,ℓ ∞ ( Z , Z ) S i = . Again by the exact sequences (1) and (2) for m =
2, we have the commutative diagram0 / / ℓ ∞ ( Z , Z ) S S / / (cid:15) (cid:15) K ( C ∗ u ( | Z | )) / / (cid:15) (cid:15) ℓ ∞ ( Z , Z ) S S / / (cid:27) (cid:15) (cid:15) / / / / K ( C ∗ ( | Z | )) (cid:27) / / ( Z Z ) S S / / K ( C ∗ u ( | Z | )) → K ( C ∗ ( | Z | )) is surjective by the right square. (cid:3) To determine the homotopy type of C ∗ u ( | Z | ), we also need its K -theory. Lemma 7.7.
The K-theory K ( C ∗ u ( | Z | )) and the kernel of the homomorphism K ( C ∗ u ( | Z | )) → K ( C ∗ ( | Z | )) are rational vector spaces of uncountable dimension.Proof. As seen in the proof of Lemma 7.6, the latter group is isomorphic to ℓ ∞ ( Z , Z ) S S . Thecoinvariant ℓ ∞ ( Z , Z ) S can be seen to be a rational vector space of uncountable dimension bythe same argument as in [KKT, Section 5]. Then, since S : ℓ ∞ ( Z , Z ) S → ℓ ∞ ( Z , Z ) S is a linear map on a rational vector space, the coinvariant ℓ ∞ ( Z , Z ) S S is a rational vector space ofuncountable dimension. For K ( C ∗ u ( | Z | )), we obtain the exact sequence0 → ( ℓ ∞ ( Z , Z ) S ) S → K ( C ∗ u ( | Z | )) → ( ℓ ∞ ( Z , Z ) S ) S → ℓ ∞ ( Z , Z ) S ) S (cid:27) ℓ ∞ ( Z , Z ) S and ℓ ∞ ( Z , Z ) S are rational vector spaces, K ( C ∗ u ( | Z | ))is also a rational vector space of uncountable dimension. (cid:3) Proof of Theorem 1.3.
By Proposition 7.4 and Lemma 7.6, the inclusion U ∞ ( C ∗ u ( | Z | )) → U ∞ ( C ∗ ( | Z | ))admits a homotopy section as an infinite loop map. Let F be the homotopy fiber of the inclu-sion. Then we have a homotopy equivalenceU ∞ ( C ∗ u ( | Z | )) ≃ U ∞ ( C ∗ ( | Z | )) × F as infinite loop spaces. By Lemmas 7.2 and 7.3, U ∞ ( C ∗ ( | Z | )) is homotopy equivalent to U ∞ ( C )as an infinite loop space. By the naturality of the Bott maps β : U ∞ ( C ∗ u ( | Z | )) ≃ −→ U ∞ ( C ( R , C ∗ u ( | Z | ))) and β : U ∞ ( C ∗ ( | Z | )) ≃ −→ U ∞ ( C ( R , C ∗ ( | Z | ))) , we have the homotopy equivalence ˜ β : F ≃ −→ Ω F as well. The homotopy group of F can be computed by Lemma 7.6: π i ( F ) (cid:27) V i is even, V i is odd,where V = ker[ K ( C ∗ u ( | Z | )) → K ( C ∗ ( | Z | ))] , V = K ( C ∗ u ( | Z | ))are rational vector spaces by Lemma 7.7. Again as in the proof of [KKT, Lemma 5.4], we canfind maps ◦ Y i ≥ K ( V , i − → F and ◦ Y i ≥ K ( V , i − → Ω F inducing isomorphisms on the odd degree homotopy groups. Then, using the homotopy equiv-alence ˜ β , we obtain the homotopy equivalence ◦ Y i ≥ ( K ( V , i − × K ( V , i )) → F . This completes the proof of the theorem. (cid:3)
Moreover, Lemma 7.5 provides another proof of Theorem 1.2 in a similar manner.R eferences [Bas64] H. Bass. K -theory and stable algebra. Inst. Hautes ´Etudes Sci. Publ. Math. , (22):5–60, 1964.[Bla86] Bruce Blackadar.
K-theory for operator algebras , volume 5 of
Mathematical Sciences Research InstitutePublications . Springer-Verlag, New York, 1986.[KKT] T. Kato, D. Kishimoto, and M. Tsutaya. Homotopy type of the space of finite propagation unitary opera-tors on Z . preprint, arxiv: 2007.06787 .[Kui65] Nicolaas H. Kuiper. The homotopy type of the unitary group of Hilbert space. Topology , 3:19–30, 1965.
OMOTOPY TYPE OF THE UNITARY GROUP OF THE UNIFORM ROE ALGEBRA ON Z n [MT21] Vladimir Manuilov and Evgenij Troitsky. On Kuiper type theorems for uniform Roe algebras. LinearAlgebra Appl. , 608:387–398, 2021.[PV80] M. Pimsner and D. Voiculescu. Exact sequences for K -groups and Ext-groups of certain cross-product C ∗ -algebras. J. Operator Theory , 4(1):93–118, 1980.[Rie83] Marc A. Rie ff el. Dimension and stable rank in the K -theory of C ∗ -algebras. Proc. London Math. Soc. (3) ,46(2):301–333, 1983.[Rie87] Marc A. Rie ff el. The homotopy groups of the unitary groups of noncommutative tori. J. Operator Theory ,17(2):237–254, 1987.[Roe88] John Roe. An index theorem on open manifolds. I, II.
J. Di ff erential Geom. , 27(1):87–113, 115–136,1988.[Roe03] John Roe. Lectures on coarse geometry , volume 31 of
University Lecture Series . American MathematicalSociety, Providence, RI, 2003.[SW85] Graeme Segal and George Wilson. Loop groups and equations of KdV type.
Inst. Hautes ´Etudes Sci.Publ. Math. , (61):5–65, 1985.D epartment of M athematics , K yoto U niversity , K yoto , 606-8502, J apan Email address : [email protected] D epartment of M athematics , K yoto U niversity , K yoto , 606-8502, J apan Email address : [email protected] F aculty of M athematics , K yushu U niversity , F ukuoka apan Email address ::