# Abstract

We investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity of a

## 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

Let us endow *E* with the *E* (cf. [26]).
Given *A* and of norm at most 1, we can evaluate

Let

Let *n*-iterated commutators
in

Let

for

### Lemma A.1.

*Let *

*n*(i.e.,

### Proof.

We have *n*. So

### Lemma A.2.

*Let *

*If *

### Proof.

Let *n*.
For each

as desired.
Since

Let *A* be a Banach algebra and

Note that

### Theorem A.3.

*Let A be a Banach algebra. Let *

*If A is a *

### Proof.

We first work in the Banach algebra

Now let *A* be a Banach algebra. For

Applying the assignment

Suppose now that *A* is a *A*. Observe that the involution

### 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

Moreover, *X*
and *Y*.

Let

(Our notation matches that of [28, Sections 1.1–1.5], with the following exception: our

This is the first Kashiwara–Vergne equation.
Since

Let us define *X* and *Y*).
Define

Let
*t*. (We sometimes omit reference to *X* and *Y*
to simplify notation.) Let us also define

for small enough *t*.

Let us introduce the partial differential operators: The two operators

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

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

*We have*

*for small enough t.*

### Proof.

Throughout the proof we abbreviate

Thus, we must show that

Multiplying by

Now multiplying by *t*.)
∎

For

### Theorem A.5.

*There exists *

*for all *

*for some *

*A*is a

*X*and

*Y*are skewadjoints, then

*R*,

*S*,

### Proof.

We work in the setting of the free Banach algebra

for all

Notice that (A.8) holds for *t*, (A.8) is equivalent to

with

Let

with initial conditions

has the form

for sufficiently small *U*. (Recall that *T* acting on *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

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

With

But this is equation (A.7) from Lemma A.4, where *X* and *Y* haven been replaced by

Let

Let us write

with

for all

for all

for

To prove the theorem let us make the assignment *A*. Then

Finally, suppose that *A* is a

# 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.

### References

[1]
H. Bass,

[2]
M. Brešar, E. Kissin and V. S. Shulman,
Lie ideals: From pure algebra to

[3] E. B. Dynkin, Calculation of the coefficients in the Campbell–Hausdorff formula, Doklady Akad. Nauk SSSR (N.S.) 57 (1947), 323–326. Search in Google Scholar

[4]
G. A. Elliott and M. Rørdam,
The automorphism group of the irrational rotation

[5] P. de la Harpe, Simplicity of the projective unitary groups defined by simple factors, Comment. Math. Helv. 54 (1979), no. 2, 334–345. Search in Google Scholar

[6] P. de la Harpe and G. Skandalis, Déterminant associé à une trace sur une algébre de Banach, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 1, 241–260. Search in Google Scholar

[7]
P. de la Harpe and G. Skandalis,
Produits finis de commutateurs dans les

[8]
P. de la Harpe and G. Skandalis,
Sur la simplicité essentielle du groupe des inversibles et du groupe unitaire dans une

[9] I. N. Herstein, On the Lie and Jordan rings of a simple associative ring, Amer. J. Math. 77 (1955), 279–285. Search in Google Scholar

[10] I. N. Herstein, Topics in ring theory, The University of Chicago Press, Chicago 1969. Search in Google Scholar

[11] I. N. Herstein, On the Lie structure of an associative ring, J. Algebra 14 (1970), 561–571. Search in Google Scholar

[12] I. N. Herstein, On the multiplicative group of a Banach algebra, Symposia mathematica. Vol. VIII, Academic Press, London (1970), 227–232. Search in Google Scholar

[13] R. V. Kadison, Infinite unitary groups, Trans. Amer. Math. Soc. 72 (1952), 386–399. Search in Google Scholar

[14] R. V. Kadison, Infinite general linear groups, Trans. Amer. Math. Soc. 76 (1954), 66–91. Search in Google Scholar

[15] R. V. Kadison, On the general linear group of infinite factors, Duke Math. J. 22 (1955), 119–122. Search in Google Scholar

[16] I. Kaplansky, Rings of operators, W. A. Benjamin, New York 1968. Search in Google Scholar

[17]
E. Kirchberg and M. Rørdam,
Central sequence

[18] L. W. Marcoux, Sums of small number of commutators, J. Operator Theory 56 (2006), no. 1, 111–142. Search in Google Scholar

[19]
L. W. Marcoux and G. J. Murphy,
Unitarily-invariant linear spaces in

[20] C. R. Miers, Closed Lie ideals in operator algebras, Canad. J. Math. 33 (1981), no. 5, 1271–1278. Search in Google Scholar

[21]
P. W. Ng and L. Robert,
Sums of commutators in pure

[22]
P. W. Ng and E. Ruiz,
The structure of the unitary groups of certain simple

[23]
P. W. Ng and E. Ruiz,
Simplicity of the projective unitary group of the multiplier algebra of a simple stable nuclear

[24]
P. W. Ng and E. Ruiz,
On the structure of the projective unitary group of the multiplier algebra of a simple stable nuclear

[25]
P. W. Ng and E. Ruiz,
The automorphism group of a simple

[26] V. Pestov, Correction to: “Free Banach–Lie algebras, couniversal Banach–Lie groups, and more” [Pacific J. Math. 157 (1993), no. 1, 137–144], Pacific J. Math. 171 (1995), no. 2, 585–588. Search in Google Scholar

[27]
L. Robert,
On the Lie ideals of

[28] F. Rouvière, Symmetric spaces and the Kashiwara–Vergne method, Lecture Notes in Math. 2115, Springer, Berlin 2014. Search in Google Scholar

[29]
K. Thomsen,
Finite sums and products of commutators in inductive limit

[30] L. N. Vaserstein, Normal subgroups of the general linear groups over Banach algebras, J. Pure Appl. Algebra 41 (1986), no. 1, 99–112. Search in Google Scholar

**Received:**2016-12-12

**Revised:**2017-03-21

**Published Online:**2017-04-20

**Published in Print:**2019-11-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston