Shape invariance of K-stability of C *-algebras（Kawamura）

Shape invariance of K-stability of C ∗ -algebras
Kazuhiro Kawamura

Abstract
The present note records an observation that the K-stability of C ∗ -algebras is invariant undershape equivalence in the sense of Blackadar [1].
1Introduction

For a unital C ∗ -algebra A, GLn (A) denotes the group of all invertible n × n-matrices over A. It
is, as a topological space, homotopy equivalent to the group of unitaries Un (A). The canonicalinclusion
GLn (A) → GLn+1 (A)(1)

is deﬁned by a → diag(a, 1A ), a ∈ GLn (A). A unital C ∗ -algebra A is said to be K-stable ([7]) if foreach n ≥ 1 and for each k ≥ 0, the inclusion (1) induces an isomorphism between k-th homotopy
groups:πk (GLn (A)) ∼= πk (GLn+1 (A)),(2)

in other words, the inclusion map (1) is a homotopy equivalence for each n ≥ 1. The irrationalrotation algebra, the Cuntz algebras, and simple AF algebras are examples of K-stable algebras([3], [4], [7]). Also for each ﬁnite dimensional compact metrizable space X, a C(X)-algebra (see[8, Appendix C]) A is K-stable if the ﬁber A(x) is K-stable for each x ∈ X ([5]). For each such
algebra A, we have an isomorphism

πk (GLn (A)) ∼={K0 (A)K1 (A)

k is odd,
k is even,for each n ≥ 1.

The author is supported by JSPS KAKENHI Grant Number 20K03577.
Keywords: K-stability, shape theory for C ∗ -algebras.
MSC. 46L80, 46L85.
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Two ∗-homomorphisms φ, ψ : A → B between C ∗ -algebras A and B are said to be homotopic,written as ϕ  ψ, if there exists a one-parameter family (φt : A → B)t∈[0,1] of ∗-homomorphisms(called a homotopy) such that φ0 = ϕ, φ1 = ψ, and for each a ∈ A, the map [0, 1] → B deﬁned byt → φt (a)is continuous.

Two C ∗ -algebras A and B are said to be homotopy equivalent if there exist∗-homomorphisms ϕ : A → B, ψ : B → A such that ϕ ◦ ψ  idB , ψ ◦ ϕ  idA . The K-stabilityof a C ∗ -algebra is a homotopy invariant property: if A and B are homotopy equivalent, then A isK-stable if and only if B is K-stable.
Topological shape theory, originated by K. Borsuk, studies topological spaces under the notionof shape equivalence. It is aimed to study spaces that exhibit ”wild” local behavior from homotopytheoretic view point. Two topological spaces are homotopy equivalent, then they are always shapeequivalent, but the converse does not hold in general (see [2]). The topological shape theory
has been most successful for the class of compact metrizable spaces. Seeking a non-commutative
analogue of topological shape theory, Blackadar [1] systematically developed shape theory forseparable C ∗ -algebras. The theory reduces to topological shape theory for compact metrizable
spaces when restricted to commutative unital C ∗ -algebras. As in topological shape theory, twoC ∗ -algebras A and B are shape equivalent in the sense of [1] if they are homotopy equivalent.
The present note records an observation that the K-stability is invariant under shape equivalence.
The proof is very simple and straightforward, yet the fact seems unnoticed in the literature. Theauthor hopes that the observation enlarges the class of K-stable C ∗ -algebras and gives an insightto the non-stable K-theory ([7]).
2ResultFirst we recall the notion of shape equivalence of [1]. Throughout all C ∗ -algebras and all∗-homomorphisms are assumed to be unital for simplicity. The standard modiﬁcation describedin [7] allows us to extend the result for general (not necessarily unital) C ∗ -algebras. All ideals ofC ∗ algebras are assumed to be closed and two-sided. For two ideals I, J of a C ∗ -algebra A withI ⊂ J, πIJ : A/I → A/J denotes the canonical projection.
Deﬁnition 2.1. A ∗-homomorphism ϕ : A → B is said to be semiprojective if, for each C ∗ -algebra

C and for each increasing sequence (Jn ) of ideals with ∪n Jn = J and for each ∗-homomorphism

σ : B → C/J, there exist an integer n and a ∗-homomorphism ψ : A → C/Jn such that σ ◦ ϕ =− 248 −Shape invariance of K-stability of C *-algebras（Kawamura）

πJn J ◦ ψ. If ψ can always be chosen as a ∗-homomorphism A → C, then ϕ is said to be projective.
A C ∗ -algebra A is said to be projective (resp. semiprojective) if idA is projective (resp. semipro-jective).

C
C/Jnψ πJJn
 C/JA φ Bσ
For a compact metrizable space X, the commutative C ∗ -algebra C(X) is semiprojective, with allalgebras involved in the above deﬁnition being unital commutative C ∗ -algebras, if and only if Xis an absolute neighborhood retract (ANR) [1, Proposition 2.11]. This indicates that the notionof semiprojective C ∗ -algebras is a non-commutative analogue of that of ANR spaces. The next
theorem corresponds to the fact that every compact metrizable space is the limit of an inverse
sequence of compact ANR’s [2].
Theorem 2.2. [1, Theorem 4.3] For every separable (unital) C ∗ -algebra A, there exists an inductive sequence of C ∗ -algebras and ∗-homomorphisms
A1γ1

 A2

γ2

 · · · γi−1  Aiγi

 Ai+1

 ···(3)such that A ∼(A , γ ) and each γi is semiprojective.
= lim−→i i iIn the above sequence, for integers i, j with i < j, let γij be the composition given by γij =γj−1 ◦ · · · ◦ γi : Ai → Aj . The above theorem allows us to introduce the notion of shape equivalence

as follows.

Deﬁnition 2.3. Let A, B be two C ∗ -algebras and let (Ai , γi : Ai → Ai+1 ) and (Bi , δi : Bi →Bi+1 ) be inductive sequences as in (3) such that A ∼(A , γ ) and B ∼(B , δ ). We say= lim= lim−→i i i−→i i ithat A and B are shape equivalent if there exist subsequences (mk ), (nk ) of positive integers and∗-homomorphisms (αk : Amk → Bnk )k , (βk : Bnk → Amk +1 )k such thatβk ◦ αk  γmk mk+1 ,αk+1 ◦ βk  δnk nk+1
for each k.
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第 98 号γmk mk+1

 Amk+1
αk+1αk  βk BnB nkk+1Am k

δnk nk+1

It follows from [1, Theorem 4.8] that the above condition does not depend on the choice of theinductive sequences (Ai , γi ) and (Bi , δi ). Now our result is stated as follows.
Main Theorem. Let A, B be two unital shape equivalent C ∗ -algebra. Then A is K-stable if andonly if B is K-stable.
We start with the next lemma.
Lemma 2.4. Let (ϕt : A → B)0≤t≤1 be a one-parameter family that gives a homotopy between ϕ0
and ϕ1 , and deﬁne Φ : A × [0, 1] → B byΦ(a, t) = ϕt (a), (a, t) ∈ A × [0, 1].
Then the map Φ is continuous.
Proof. Assume that a sequence {(an , tn )} of points of A×[0, 1] converges to a point (a, t) ∈ A×[0, 1].

We haveΦ(an , tn ) − Φ(a, t) ≤ Φ(an , tn ) − Φ(a, tn ) + Φ(a, tn ) − Φ(a, t)
= ϕtn (an ) − ϕtn (a) + ϕtn (a) − ϕt (a)≤ an − a + ϕtn (a) − ϕt (a),where the last inequality follows from the fact that each ∗-homomorphism is norm non-increasing.

By the hypothesis, the last term converges to zero as n → ∞ and we see that the sequence{Φ(an , tn )} converges to Φ(a, t).

The above lemma implies that the map GLn (Φ) : GLn (A) × [0, 1] → GLn (B); ((aij ), t) →(ϕt (aij )) ((aij ) ∈ GLn (A)), is continuous for each positive integer n.
Proof of Main Theorem.

Let A, B be two shape equivalent C ∗ -algebras and let(Ai , γi ), (Bi , δi ) be inductive sequences as in (3) with A = lim(Ai , γi ), B = lim(Bi , δi ).
−→−→There exist sequences (mk ), (nk ) of positive integers and ∗-homomorphisms (αk : Amk → Bnk )k ,(βk : Bnk → Amk +1 )k required in (4). Passing to the j-th homotopy groups with the aid of− 250 −Shape invariance of K-stability of C *-algebras（Kawamura）

Lemma 2.4, we obtain the following commutative diagram, where the induced homomorphisms
are indicated by the same symbol for simplicity:
γmk mk+1

 πj (GLn (Am ))
k+1



nn

αk+1

αk
 n
 βk πj (GLn (Bn ))πj (GLn (Bnk ))k+1

πj (GLn (Amk ))δnk nk+1

γmk+1 ∞

 lim πj (GLn (Am ))k−→klim αnk−→k lim πj (GLn (Bn ))k→kδn k ∞ −We can directly verify that the limit homomorphism limk αkn : limk πj (GLn (Ank )) →−→−→
limk πj (GLn (Bnk )) is an isomorphism of groups with the inverse homomorphism limk βkn .
−→−→B
The inclusions iA
n,k : GLn (Amk ) → GLn+1 (Amk ) and in,k : GLn (Bnk ) → GLn+1 (Bnk ) induce
homomorphisms of homotopy groups. Passing to the inductive limit, we obtain homomorphismslimk iA: lim π (GLn (Amk )) → limk πj (GLn+1 (Amk )) and limk iB: lim π (GLn (Bnk )) →−→ n,k −→k j−→−→ n,k −→k jlimk πj (GLn+1 (Bnk )). It is straightforward to verify that the following diagram is commutative:−→
limk πj (GLn (Amk ))−→lim αnklim iAn,k−→k  lim π (GLn+1 (Amk ))−→k jlim αn+1k−→k−→k lim πj (GLn+1 (Bn ))limk πj (GLn (Bnk ))k−→−→klim iB−→k n,kSince limk αkn and limk αkn+1 are both isomorphisms, we see that−→−→is an isomorphism if and only if limk iBis(∗) limk iA−→ n,k−→ n,k
an isomorphism.
π (GLn (Amk )) in such a
By [6, Proposition 1.4], we have an isomorphism πj (GLn (A)) ∼= lim−→k j
way that the diagram below is commutative:
πj (GLn (A))

iAn

∼
=

 πj (GLn+1 (A))∼=


lim iA−→k n,k lim πj (GLn+1 (Am ))limk πj (GLn (Amk ))k−→−→kwhere iAn denotes the homomorphism induced by the inclusion. The same holds for the inducedAhomomorphism iBn : πj (GLn (B)) → πj (GLn+1 (B)). Combining these with (∗), we see that in is

an isomorphism if and only if iB
n is an isomorphism, for each n ≥ 1 and for each k ≥ 0.
This proves Main Theorem.

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References[1] B. Blackadar, Shape theory for C ∗ -algebras, Math. Scand. 56 (1985), 249-275.
[2] S. Mardešić and J. Segal, Shape theory- the inverse system approach, North Holland Math. Lib. 26(1982).
[3] M.A. Rieﬀel, Dimension and stable rank in the K-theory of C ∗ -algebras, Proc. London Math. Soc.46(1983), 301-333.
[4] M.A. Rieﬀel, The homotopy group of the unitary groups of noncommutative tori, J. Op. Th. 17(1987), 237-254.
[5] A. Seth and R. Vaidyanathan, K-stability of continuous C(X)-algebras, Proc. Amer. Math. Soc. 148(2020), 3897-3909.
[6] A. Seth and R. Vaidyanathan, AF algebras and rational homotopy theory, New York J. Math. 26(2020), 931-949.
[7] K. Thomsen, Nonstable K-theory for operator algebras, K theory 4 (1991), 245-267.
[8] D.P. Williams, Crossed products of C ∗ -algebras, Math.Surv.Mono. 134 (2007), A.M.S.
Kazuhiro Kawamura
Department of Mathematics
University of Tsukuba
Tsukuba, Ibaraki 305-8071
Japankawamura@math.tsukuba.ac.jp− 252 −