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Normal subgroups of invertibles and of unitaries in a C*-algebra

Leonel Robert

Abstract

We investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity of a C*-algebra. By relating normal subgroups to closed two-sided ideals we obtain a “sandwich condition” describing all the closed normal subgroups both in the invertible and in the unitary case. We use this to prove a conjecture by Elliott and Rørdam: in a simple C*-algebra, the group of approximately inner automorphisms induced by unitaries in the connected component of the identity is topologically simple. Turning to non-closed subgroups, we show, among other things, that in a simple unital C*-algebra the commutator subgroup of the group of invertibles in the connected component of the identity is a simple group modulo its center. A similar result holds for unitaries under a mild extra assumption.

A Appendix

Here we prove Theorems 5.1 and 5.2.

A.1 Proof of Theorem 5.1

In order to justify analytically some formal manipulations we sometimes work in the setting of a free Banach algebra. Let E=. Let us denote by X and Y the canonical basis of E. Let 𝒜n(X,Y)=En for n1. Let

𝒜[X,Y]=n=1𝒜n(X,Y),
𝒜[[X,Y]]=n=1𝒜n(X,Y),

𝒜[X,Y] is the free non-unital algebra in the variables X and Y. We regard the elements of 𝒜[X,Y] as polynomials in the noncommuting variables X and Y. We regard the elements of 𝒜[[X,Y]] as formal power series in X and Y.

Let us endow E with the 1 norm and then 𝒜n(X,Y)=En with the projective tensor product norm (i.e., the 1 norm after identifying En with 2n). Let 𝒜(X,Y) denote the Banach algebra of 1 convergent series in 𝒜[[X,Y]] endowed with the 1 norm. We have the inclusions 𝒜[X,Y]𝒜(X,Y)𝒜[[X,Y]]. The Banach algebra 𝒜(X,Y) is the free Banach algebra on the vector space E (cf. [26]). Given Z𝒜(X,Y) and elements in U,V in some Banach algebra A and of norm at most 1, we can evaluate Z(U,V) via the universal property of 𝒜(X,Y), i.e., by extending the assignment XU, YV to a contractive algebra homomorphism. The function (U,V)Z(U,V) is defined for U1 and V1, and is given by a normally convergent power series in U,V. In particular, it is continuous and has Fréchet derivatives of all orders for U<1 and V<1.

Let t. Define λt:EE by λt(v)=tv for all vE and extend it to an algebra homomorphism on 𝒜[X,Y]. Since λt maps 𝒜n[X,Y] to itself, we can further extend it to 𝒜[[X,Y]]. We call λt the scaling automorphism. Notice that λt maps the Banach algebra 𝒜(X,Y) to itself for |t|1.

Let n(X,Y)𝒜n(X,Y) denote the span of the n-iterated commutators in X and Y, i.e., [v1,[v2,,[vn-1,vn]]] with viE. Define also

[X,Y]=n=1n(X,Y),
[[X,Y]]=n=1n(X,Y);

[X,Y] is the free Lie algebra on X,Y. Finally, let (X,Y) denote the closure of [X,Y] inside the Banach algebra 𝒜(X,Y).

Let νn:𝒜n(X,Y)n(X,Y) denote the linear operator

ν(v1vn)=1n[v1,[v2,,[vn-1,vn]]]

for v1,,vnE. The Dynkin–Specht–Wever theorem asserts that the map νn is the identity on n(X,Y). We can estimate that νn2n. From this we deduce that λ1/2νn is a contractive map. Define ν:𝒜[X,Y]𝒜[X,Y] by ν=n=1νn. Observe then λ1/2ν extends to a contractive map from 𝒜(X,Y) to (X,Y).

Lemma A.1.

Let Z=n=1ZnA(X,Y) be such that ZnLn(X,Y) for all n (i.e., ZL[[X,Y]]). Then λ1/2ZL(X,Y).

Proof.

We have νnZn=Zn for all n. So λ1/2Z=n=1(λ1/2νn)Zn(X,Y). ∎

Lemma A.2.

Let ZL[[X,Y]]. Then there exist P,QL[[X,Y]] such that

Z=Z1+[X,P]+[Y,Q].

If ZL(X,Y), then λ1/2P and λ1/2Q are in L(X,Y).

Proof.

Let Z=n=1Zn with Znn(X,Y) for all n. For each n2 we can write Zn=XPn+YQn for some Pn,Qn𝒜n-1(X,Y). We remark that, from the definition of the norm on 𝒜n(X,Y), we have that PnZn-1 and QnZn-1. Applying νn on both sides of Zn=XPn+YQn we get Zn=[X,νnPn]+[Y,νnQn], where now νnPn and νnQn are in n-1(X,Y). Let us set P=n=2νnPn and Q=n=2νnQn. Then

Z=Z1+[X,P]+[Y,Q],

as desired. Since λ1/2νn is a contraction, λ1/2νnPnZn-1 and λ1/2νnQnZn-1 for all n2. Hence, if n=1Zn<, then λ1/2P,λ1/2Q(X,Y). ∎

Let A be a Banach algebra and X,YA. Define by holomorphic functional calculus

V(X,Y)=log(eXeY).

Note that V(X,Y) is well defined for X+Y<log2.

Theorem A.3.

Let A be a Banach algebra. Let 0<δ<log28. There exist continuous functions (X,Y)A(X,Y) and (X,Y)B(X,Y) defined for X,Yδ such that for all X,Y,

(A.1)V(X,Y)=X+Y+[X,A(X,Y)]+[Y,B(X,Y)].

If A is a C*-algebra and X and Y are skewadjoint, then so are A(X,Y) and B(X,Y)

Proof.

We first work in the Banach algebra 𝒜(X,Y). Since 4δX+4δY<log2, it follows that V(4δX,4δY)𝒜(X,Y). By the Campbell–Baker–Hausdorff Theorem, we obtain that V(4δX,4δY)[[X,Y]]. Thus, by Lemma A.1, we have V(2δX,2δY)(X,Y). Then, by Lemma A.2, there exist P,Q(X,Y) such that

(A.2)V(δX,δY)=δX+δY+[X,P]+[Y,Q].

Now let A be a Banach algebra. For X,YA such that X,Yδ define

A(X,Y)=P(1δX,1δY)andB(X,Y)=Q(1δX,1δY).

Applying the assignment δXX, δYY in (A.2), we get equation (A.1). The functions (X,Y)A(X,Y) and (X,Y)B(X,Y) are given by normally convergent series of iterated Lie brackets. In particular, they are continuous.

Suppose now that A is a C*-algebra and that X,Y are skewadjoint elements of A. Observe that the involution σZ=-Z* is a continuous Lie algebra homomorphism; i.e., we have σ[Z1,Z2]=[σZ1,σZ2]. Since A(X,Y) is a convergent series of Lie brackets on X,Y we get that σA(X,Y)=A(σX,σY)=A(X,Y), i.e., A(X,Y) is skewadjoint. The same argument applies to B(X,Y). ∎

A.2 Proof of Theorem 5.2

Theorem 5.2 follows essentially from the collection formulas related to the Kashiwara–Vergne equations, as developed in [28, Sections 1.1–1.5]. We have pieced together the various parts of the argument from this reference (still referring the reader to [28] for some computations). Besides having tailored the statement of Theorem 5.2 to our purposes, our objective here has been to make it explicit that these formulas are valid in the infinite-dimensional setting of a Banach algebra (the setting in [28] is either formal or finite-dimensional one).

Let A be a Banach algebra. Let X,YA and 0<δ<log28. We have demonstrated in Theorem A.3 the existence of A(X,Y) and B(X,Y) for X,Yδ such that

(A.3)V(X,Y)=X+Y+[X,A(X,Y)]+[Y,B(X,Y)].

Moreover, A(X,Y) and B(X,Y) are expressible as normally convergent series of Lie brackets in X and Y.

Let adX:AA denote the map adX(Z)=[X,Z] for all ZA. Let x=adX and y=adY. Define

F(X,Y)=xe-x-1B(Y,X),G(X,Y)=yey-1A(Y,X).

(Our notation matches that of [28, Sections 1.1–1.5], with the following exception: our A(X,Y) and B(X,Y) are B(Y,X) and A(Y,X), respectively, in [28, Section 1.5.2].) From (A.3) we deduce that

(A.4)V(Y,X)=X+Y-(1-ex)F(X,Y)-(ey-1)G(X,Y).

This is the first Kashiwara–Vergne equation. Since e-x-1x and ey-1y are invertible operators for x,y<2π (hence, for X,Y<π), F(X,Y) and G(X,Y) are defined for X,Yδ and are normally convergent series of Lie brackets on this domain. (More formally, we can first define F(δX,δY) and G(δX,δY) in the Banach–Lie algebra (X,Y) and then evaluate them at X,YA such that X,Yδ via the assignments δXX and δYY.)

Let us define Vt(X,Y)=1tV(tX,tY) for all X,YA and t non-zero and small enough (depending on X and Y). Define

V0(X,Y)=X+Y.

Let v(t)=adVt(X,Y)=log(etxety), which is defined also for small enough t. (We sometimes omit reference to X and Y to simplify notation.) Let us also define Ft(X,Y)=t-1F(tX,tY) and Gt(X,Y)=t-1G(tX,tY). Then (A.4) implies that

(A.5)Vt(Y,X)=X+Y-(1-etx)Ft(X,Y)-(ety-1)Gt(X,Y)

for small enough t.

Let us introduce the partial differential operators: The two operators XVt(X,Y) and YVt(X,Y) act on ZA as follows:

XVt(X,Y):=ddsVt(X+sZ,Y)|s=0,YVt(X,Y):=ddsVt(X,Y+sZ)|s=0.

The following formulas are given in [28, Lemma 1.2] (where they are deduced from the formula for the differential of the exponential map):

(A.6)ev(t)-1v(t)XVt(X,Y)=e-ty1-e-txtx,
ev(t)-1v(t)YVt(X,Y)=1-e-tyty.

Lemma A.4 (cf. [28, Proposition 1.3]).

We have

(A.7)tVt(X,Y)=XVt(X,Y)[X,Ft(X,Y)]+YVt(X,Y)[Y,Gt(X,Y)]

for small enough t.

Proof.

Throughout the proof we abbreviate Vt(X,Y), Ft(X,Y), and Gt(X,Y) to Vt, Ft and Gt. Taking t in the definition of Vt, we get

Vt=-1tVt+1t(XVt)X+1t(YVt)Y.

Thus, we must show that

-1tVt+1t(XVt)X+1t(YVt)Y=(XVt)[X,Ft]+(YVt)[Y,Gt].

Multiplying by ev(t)-1v(t) on both sides of this equation and using (A.6), we get

1te-ty+1tY-1tVt=1tety(1-e-tx)Ft+1t(1-e-ty)Gt.

Now multiplying by tety on both sides and using that tetyVt(X,Y)=Vt(Y,X), we get (A.5), which we know is valid. Working backwards, we get the desired result. (To have invertibility of 1v(t)ev(t)-1 we need v(t)<2π which can be arranged for small enough t.) ∎

For fA invertible and ZA we use the notation fZ=fZf-1.

Theorem A.5.

There exists ϵ>0 such that for any given Banach algebra A there exist continuous functions (X,Y)R(X,Y)A and (X,Y)S(X,Y)A defined for Xϵ and Yϵ such that R(0,0)=S(0,0)=0,

V(eR(X,Y)X,eS(X,Y)Y)=X+Y

for all Xϵ and Yϵ, and

R(X,Y)=Y4+[X,R(X,Y)]+[Y,R′′(X,Y)],
S(X,Y)=-X4+[X,S(X,Y)]+[Y,S′′(X,Y)]

for some R, R′′, S, and S′′. Moreover, if A is a C*-algebra and X and Y are skewadjoints, then R, R, R′′, S, S, and S′′ as above are all skewadjoints.

Proof.

We work in the setting of the free Banach algebra 𝒜(X,Y) and its Banach–Lie subalgebra (X,Y). We will construct Rt,St(X,Y) for |t|ϵ such that R0=S0=0 and

(A.8)Vt(eRtX,eStY)=X+Y

for all |t|ϵ. We will moreover show that λαRt=Rαt and λαSt=Sαt for all |α|1. To derive the theorem from this we will make the assignment ϵXX, ϵYY and then define R(X,Y)=Rϵ(1ϵX,1ϵY) and S(X,Y)=Sϵ(1ϵX,1ϵY).

Notice that (A.8) holds for t=0 once R0=S0=0. Thus, differentiating with respect to t, (A.8) is equivalent to

(A.9)tVt(Xt,Yt)+XVt(Xt,Yt)[teRte-Rt,Xt]
+YVt(Xt,Yt)[teSte-St,Yt]=0,

with R0=S0=0 (see [28, pp. 4–5]). Here we have set Xt=eRtX and Yt=eStY.

Let rt=adRt and st=adSt. Consider the initial value problem

(A.10)tRt=rt1-e-rtFt(X,e-rtestY),
tSt=st1-e-stGt(e-stertX,Y),

with initial conditions R0=S0=0. The function

(U,V,t)𝐻adU1-e-adUFt(X,e-adUeadVY)

has the form n=1tn-1Hn(U,V), where

Hn(U,V)=adU1-e-adUFn(X,e-adUeadVY)(X,Y)

for sufficiently small U. (Recall that Fn(X,Y) is a homogeneous polynomial in the noncommuting variables X and Y; here F(X,Y)=n=1Fn(X,Y)[[X,Y]].) Furthermore, using the simple estimate Fn(X,TY)(1+T)nFn(X,Y) for any operator T acting on (X,Y), we derive that n=1|t|n-1Hn(U,V)< for (U,V,t) in a sufficiently small neighborhood of (0,0,0). It follows that H has uniformly bounded Fréchet derivative for small enough t. The same can be said of the second equation in (A.10). This guarantees the existence and uniqueness of a solution to (A.10) in the Banach space (X,Y) for |t|<ϵ and some ϵ>0.

From (A.10) and the formula for the differential of the exponential map we deduce that

(A.11)teRt=Ft(ertX,estY)eRt,
teSt=Gt(ertX,estY)eSt.

With Rt and St that satisfy (A.11), equation (A.9) is equivalent to

tVt(Xt,Yt)+XVt(Xt,Yt)[Ft(Xt,Yt),Xt]+YVt(Xt,Yt)[Gt(Xt,Yt),Yt]=0.

But this is equation (A.7) from Lemma A.4, where X and Y haven been replaced by Xt and Yt. This proves (A.8).

Let 0<α1. It is straightforward to check that (λαRt/α,λαSt/α) is also a solution of (A.10) (cf. [28, p. 7] for the same verification for ft:=eRt and gt:=eSt). It follows that λαRt=Rαt and λαSt=Sαt.

Let us write

Rt=a(t)X+b(t)Y+n2Rt,n,

with Rt,nn(X,Y). From λαRt=Rαt we deduce that a(t) and b(t) are linear. But from (A.10) we see that tRt|t=0=Ft(X,Y)|t=0=X4. Thus,

Rt=tX4+n2Rt,n

for all |t|ϵ. Applying Lemma A.2 to Rt, we get

Rt=tX4+[X,Rt]+[Y,Rt′′]

for all |t|ϵ2 and some Rt,Rt′′(X,Y). We can derive similarly for St that

St=-tY4+[X,St]+[Y,St′′]

for |t|ϵ2 and suitable St,St′′(X,Y). Let us relabel ϵ2 as ϵ so that these representations of Rt and St are valid for |t|ϵ.

To prove the theorem let us make the assignment ϵXX, ϵYY, which, for X,Y with X,Yϵ, extends to a Banach algebra contractive homomorphism from 𝒜(X,Y) to A. Then R(X,Y)=Rϵ(Xϵ,Yϵ) and S(X,Y)=Sϵ(Xϵ,Yϵ) are as desired.

Finally, suppose that A is a C*-algebra and that X,Y are skewadjoints. Since σZ=-Z* is a continuous Lie algebra homomorphism and R(X,Y) is a convergent series of Lie brackets on X,Y we get that σR(X,Y)=R(σX,σY)=R(X,Y), i.e., R(X,Y) is skewadjoint. The same argument applies to S(X,Y) and to R,R′′,S,S′′. ∎

Acknowledgements

At an early stage of this research I benefited from discussions with Ping Wong Ng and Tracy Robins, for which I am grateful. This work was completed while visiting the Mittag-Leffler Institute during the 2016 program “Classification of Operator Algebras: Complexity, Rigidity, and Dynamics”. I am grateful to the Mittag-Leffler Institute and to the organizers of the program for their hospitality and support.

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Received: 2016-12-12
Revised: 2017-03-21
Published Online: 2017-04-20
Published in Print: 2019-11-01

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