Stable rank for a pair of $C^*$-algebras ($C^*$-algebras and its applications to topological dynamical systems)

Stable

rank

for

a

pair of

$\mathrm{C}^{*}$

-algebras

立命館大学理工

大坂 博幸

(Hiroyuki Osaka)

1

Introduction and Main Result

The (topological) stable rank of $\mathrm{R}\mathrm{i}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{l}[19]$ is noncommutative generalization of the

di-mension ofa compact Hausdorffspace. In fact, when$X$ is a compact Hausdorffspace, the

stable rank of $C(X)$ is $[ \frac{\dim X}{2}]+1$, where $\dim X$ is covering dimension of$X$. Recall that

a unital $\mathrm{C}^{*}$-algebra $A$ has stable rank _$n$ if for any element

$a_{1},$ $a_{2},$$\cdots,$$a_{n}$ and $\epsilon>0$ there

exist $b_{1},$$b_{2},$

$\cdots,$$b_{n}$ in $A$ such that

(1)$||a_{i}-b_{i}||<\epsilon$

(2) $\Sigma_{i=1i}^{n}b*bi>0$.

The condition (2) is equivalent to that there exist $c_{1},$$c_{2,n},$$\cdots C$ in$A$such that $\Sigma_{i=1}^{n}C_{i}b_{i}=1$.

If$A$has no umital, we define stable rank of$A$ as stable rank ofthe unitaization of$A$. Note

that stable rank one condition is equivelent to that the set of invertible elemenets is dense

ina given $\mathrm{C}^{*}$-algebra.

Many mathematicians tried to determine stable rank of interesting $\mathrm{C}^{*}$-algebras, in

par-ticular, simple umital $\mathrm{C}^{*}$-algebras ([5] [6] [8] [10] [11] [12] [13] [14] [15] [18] [20] [21] [22]

etc). For examples, AF $\mathrm{C}^{*}$-algebras and non-commutative tori have stable rank one ([18]),

Toeplitz algebra has stable rank two, and Cuntz algebra has an infinity ([19]).

It has been a problem of considerable interest to determine stable rank of a crossed

product algebra $A\cross_{\alpha}G$ ofa unital $\mathrm{C}^{*}$-algebra $A$ with stable rank one by a finite group

$G$. Blackadar presented this problem in the case that $A$ is an AF $\mathrm{C}^{*}$-algebra ([2]), and

constructeda symmetry $\alpha$ on $A=C[0,1]\otimes UHF$ whose crossed product algebra $A\cross_{\alpha}Z_{2}$

has stable rank two. So, to consider the above problem, we need the assumption of the

simplicity on a given $\mathrm{C}^{*}$-algebra $A$.

Inthis directionJeongand the author conclude ([10] [11]) that a crossed product algebra

$A$ $\mathrm{X}_{\alpha}G$ has the cancellation property if $A$ is simple with stable rank one and the

SP-property. Recall that a $\mathrm{C}^{*}$-algebra _$A$ is said to have the $\mathrm{S}\mathrm{P}$-property if any non-zero

hereditary subalgebra of$A$ has non-zero projection. For example, an AF $\mathrm{C}^{*}$-algebra has

the$\mathrm{S}\mathrm{P}$-property. Therefore, wecould concludeby [1] that acrossed product algebra$A\cross_{\alpha}G$

has stable rank one ifwe addreal rank zero condition to this crossed product algebra, that

is, the set of self-adjoint elements with finite spectra in $A$ $\mathrm{X}_{\alpha}G$ is dense in the set of

rank one, however, we

can

not always hope that

a

given crossed

_produc.t

algebra $\mathrm{h}..\mathrm{a}\mathrm{s}$ real

rank zero.

In thuis talk wetry to estimate stable rankofa given umital $\mathrm{C}^{*}$-algebra _$B$ by stable rank

of a $\mathrm{C}^{*}$-subalgebra $A$ with

common

unit. In

case

that _$B$ is a crossed product algebra of

$A$ by a finite group _{$G,$ $sr(B)\leq sr(A)\cross|G|([11])$}. More generally, we have the following

result:

Theorem 1 Let $1\in A\subset B$ be unital $C^{*}$-algebras. Suppose that _$B$ is a finitely generated

lefl

$A$-module, thatis, there are some_$n$ elements

$v_{1},$$v_{2},$$\cdots,$$v_{n}$ in$B$ such that$\sum_{i=1}^{n}Av_{i}\Leftarrow B$.

Then, $sr(B)\leq sr(A)\cross n$.

2

Stable

rank

We prove main theorem with using the techmique of matrix algebras. To this end the

following lemma is needed.

Lemma 2 (Spatial case of $\mathrm{R}\mathrm{i}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{l}[19]$) Let $n\in \mathrm{N}$.

$sr(M_{n}(A))\leq sr(A)$.

Proof. We will give a sketch of the proof. Suppose that $sr(A)=m$. Take $m$ elements

$T_{1},$

$\cdots,$$T_{m}$ from $M_{n}(A)$. Set $S=(T_{1}, T_{2}, \cdots, T_{m})^{t}$ in $M_{nm,m}(A)$. Let $(a_{1}, a_{2}, \cdots, a_{nm})$ be

thefirst rowin $S$. Since $sr(A)=m$, we may assumethat there exist

$c_{2)},$$\cdots c_{m+}1$ suchthat

$c_{2}a_{2}+C_{3}a_{3}+\cdot\cdot$$,$ $+cm+1am+1=1-a_{1}$. Consider $(1$ $c_{2}1$ $..$. $c_{m+1}..$

.

$.0.$ . $101^{S}$.

Then, the new first row is $(1, b_{2,n}\ldots, b)^{t}m$. Doing the iteration there is an invertible

matrix $R\in M_{nm}(A)$ such that $RS=di\mathrm{a}g(1, S/),$ $S’\in M_{nm-1,n-1}$. By induction there is

$U\in M_{n-1,nm-1}$ such that $US’=I_{n-1}$. Note that $||R^{-1}di\mathrm{a}g(1, s’)-S||$ is small.

Write

$R^{-1}\mathrm{d}i\mathrm{a}g(1, S’)=(s_{1\backslash /}, \cdots s_{m})^{t}$

diag$(1, U)R=(U_{1}, \cdots, U_{m})$,

where $s_{1},$

$\cdots,$$s_{m},$$U_{1},$$\cdots,$$U_{m}$ are in $M_{n}(A)$. Then, we have $||T_{i}-S_{i}||$ is small, and

Definition 3

_Define

$Lg_{n}(A)= \{(a_{1}, a_{2}, *\cdot\cdot, a_{n})\in A^{n}|\sum^{n}i=1Aai=A\}$.

Then, $sr(A)\leq n$

if

and only

if

$Lg_{n}(A)$ is dense in $A^{n}$.

Proof of Theorem 1.

We give only the proof of the case of$sr(A)=1$ and $G=\mathrm{Z}_{2}$. That is, wewill show that

$sr(A\cross_{\alpha}Z_{2})\leq 2$ for a unital $\mathrm{C}^{*}$-algebra $A$. In general case we can guess it from the proof

of Lemma 2.

Take $a_{0}+a_{1}u,$ $b_{0}+b_{1}u$ in $A\cross_{\alpha}Z_{2}$, where $u$ is a unitary implementing $\alpha$. Let $\epsilon>0$ be

given. Consider

$=$

.

Since $sr(M_{2}(A))=1$ by Lemma 2, there exists an invertible element

$\in M_{n}(A)$

such that

$||-||< \frac{\epsilon}{2}$.

Consider

$=$

.

Then, $(c_{0}+c_{1}u, d_{0}+d_{1}u)\in Lg_{2}(A\cross_{\alpha}\mathrm{Z}_{2})$, and $||\mathit{0}_{0}+a_{1}u-(c_{0}+c_{1}u)||<\epsilon,$

$||b_{0}+b_{1}u-\coprod$ $(d_{0}+d_{1}u)||<\in \mathrm{i}$. Hence, $sr(A\cross_{\alpha}\mathrm{Z}_{2})\leq 2$.

Corollary 4 Let $1\in A\subset B$ be a pair

of

unital $C^{*}- algebra\mathit{8}$, and $E:Barrow A$ be a

faithful

conditional expectation

_{of index-finite}

type. That is, there $exi_{\mathit{8}}t_{S}$ a quasi-basis $\{v_{i}^{*}, v_{i}\}_{i=}^{n}1$

such that $x= \sum_{i=1}E(xv_{i}^{*}n)v_{i},$ $\forall x\in B$. Then, $sr(B)\leq sr(A)\mathrm{x}n$.

Corollary 5 Let $1\in A$ be a unital $C^{*}$-algebra and $G$ be a

finite

group. $Then_{f}$

$sr(A\cross_{\alpha}G)\leq sr(A)\cross|G|$.

3

Application

Question 6 Let$A$ be a $AFC^{*}$-algebra and $G$ be a

finite

group. Then $sr(A\mathrm{x}_{\alpha}c)\leq 1$.

Theorem 7 $(\mathrm{J}\mathrm{e}\mathrm{o}\mathrm{n}\mathrm{g}-\mathrm{o}\mathrm{S}\mathrm{a}\mathrm{k}\mathrm{a}[11])$ Let $A$ be a simple unital$C^{*}$-algebra with $sr(A)=1$ and

$SP$-property.

If

$G$ is a

finite

group and$\alpha$ is an action

of

$G$ on A then the $cro\mathit{8}\mathit{8}ed$product

$A\cross_{\alpha}G$ has cancellation.

Here, a $C^{*}$-algebra has $\mathrm{S}\mathrm{P}$-property if each of

its

non-zero

hereditary $C^{*}$-subalgebras

contains a

non-zero

projection.

In particular,

Corollary 8 Under the $a\mathit{8}sumpti_{\mathit{0}}n\mathit{8}$

of

the above theorem,

if

_{$A\cross_{\alpha}Gha\mathit{8}$} real rank

zero, that is, any $\mathit{8}elf$-adjoint element can be approximated by a self-adjint

element with

_finite

$spectra_{f}$ then $sr(A\cross_{\alpha}G)=1$.

Remark 9 Generally, we can _{not hope that a given simple crossedproduct algebra}$A\cross_{\alpha}G$

has real rank zero, even

_if

A $\dot{u}UHF$ , and$G=\mathrm{Z}_{2}[7]$.

If one consider a crossed product by the integer group $\mathrm{Z}$ then there is no

conditional

expectation ofindex-finite type fromthe crossed product $A\cross_{\alpha}\mathrm{Z}$ onto $A$, but

we

have the

following cancellation theorem:

Theorem 10 $(\mathrm{J}\mathrm{e}\mathrm{o}\mathrm{n}\mathrm{g}-\mathrm{o}_{\mathrm{S}\mathrm{a}}\mathrm{k}\mathrm{a}[11])$ Let $A$ be a simple unital $C^{*}$-algebra with $sr(A)=1$

and $SP$-property.

If

$\alpha$ is an outer action

of

the integer group $\mathrm{Z}$ on$A_{\mathit{8}u}ch$ that _{$\alpha_{*}=id$} on

$K_{0}(A)$ then the crossed product $A\cross_{\alpha}\mathrm{Z}$ has cancellation.

Example 11 Simple $AFC^{*_{a}}-lgebra\mathit{8}$ and non-commutative tori $A_{\theta}$ are _{$example\mathit{8}$}

for

$C^{*}-$

algebras in $Theorem\mathit{8}7$ and 10.

$*\vee’\vee\doteqdot \mathrm{X}\ovalbox{\tt\small REJECT}$

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