We investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity of a -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 -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 -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.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 . Let us denote by and the canonical basis of E. Let for . Let
is the free non-unital algebra in the variables and . We regard the elements of as polynomials in the noncommuting variables and . We regard the elements of as formal power series in and .
Let us endow E with the norm and then with the projective tensor product norm (i.e., the norm after identifying with ). Let denote the Banach algebra of convergent series in endowed with the norm. We have the inclusions . The Banach algebra is the free Banach algebra on the vector space E (cf. ). Given and elements in in some Banach algebra A and of norm at most 1, we can evaluate via the universal property of , i.e., by extending the assignment , to a contractive algebra homomorphism. The function is defined for and , and is given by a normally convergent power series in . In particular, it is continuous and has Fréchet derivatives of all orders for and .
Let . Define by for all and extend it to an algebra homomorphism on . Since maps to itself, we can further extend it to . We call the scaling automorphism. Notice that maps the Banach algebra to itself for .
Let denote the span of the n-iterated commutators in and , i.e., with . Define also
is the free Lie algebra on . Finally, let denote the closure of inside the Banach algebra .
Let denote the linear operator
for . The Dynkin–Specht–Wever theorem asserts that the map is the identity on . We can estimate that . From this we deduce that is a contractive map. Define by . Observe then extends to a contractive map from to .
Let be such that for all n (i.e., ). Then .
We have for all n. So . ∎
Let . Then there exist such that
If , then and are in .
Let with for all n. For each we can write for some . We remark that, from the definition of the norm on , we have that and . Applying on both sides of we get , where now and are in . Let us set and . Then
as desired. Since is a contraction, and for all . Hence, if , then . ∎
Let A be a Banach algebra and . Define by holomorphic functional calculus
Note that is well defined for .
Let A be a Banach algebra. Let . There exist continuous functions and defined for such that for all ,
If A is a -algebra and X and Y are skewadjoint, then so are and
Now let A be a Banach algebra. For such that define
Suppose now that A is a -algebra and that are skewadjoint elements of A. Observe that the involution is a continuous Lie algebra homomorphism; i.e., we have . Since is a convergent series of Lie brackets on we get that , i.e., is skewadjoint. The same argument applies to . ∎
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  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  is either formal or finite-dimensional one).
Let A be a Banach algebra. Let and . We have demonstrated in Theorem A.3 the existence of and for such that
Moreover, and are expressible as normally convergent series of Lie brackets in X and Y.
Let denote the map for all . Let and . Define
This is the first Kashiwara–Vergne equation. Since and are invertible operators for (hence, for ), and are defined for and are normally convergent series of Lie brackets on this domain. (More formally, we can first define and in the Banach–Lie algebra and then evaluate them at such that via the assignments and .)
Let us define for all and non-zero and small enough (depending on X and Y). Define
Let , which is defined also for small enough t. (We sometimes omit reference to X and Y to simplify notation.) Let us also define and . Then (A.4) implies that
for small enough t.
Let us introduce the partial differential operators: The two operators and act on as follows:
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]).
for small enough t.
Throughout the proof we abbreviate , , and to , and . Taking in the definition of , we get
Thus, we must show that
Multiplying by on both sides of this equation and using (A.6), we get
Now multiplying by on both sides and using that , we get (A.5), which we know is valid. Working backwards, we get the desired result. (To have invertibility of we need which can be arranged for small enough t.) ∎
For invertible and we use the notation .
There exists such that for any given Banach algebra A there exist continuous functions and defined for and such that ,
for all and , and
for some , , , and . Moreover, if A is a -algebra and X and Y are skewadjoints, then R, , , S, , and as above are all skewadjoints.
We work in the setting of the free Banach algebra and its Banach–Lie subalgebra . We will construct for such that and
for all . We will moreover show that and for all . To derive the theorem from this we will make the assignment , and then define and .
with (see [28, pp. 4–5]). Here we have set and .
Let and . Consider the initial value problem
with initial conditions . The function
has the form , where
for sufficiently small U. (Recall that is a homogeneous polynomial in the noncommuting variables and ; here .) Furthermore, using the simple estimate for any operator T acting on , we derive that for in a sufficiently small neighborhood of . 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 for and some .
From (A.10) and the formula for the differential of the exponential map we deduce that
Let us write
with . From we deduce that and are linear. But from (A.10) we see that . Thus,
for all . Applying Lemma A.2 to , we get
for all and some . We can derive similarly for that
for and suitable . Let us relabel as ϵ so that these representations of and are valid for .
To prove the theorem let us make the assignment , , which, for with , extends to a Banach algebra contractive homomorphism from to A. Then and are as desired.
Finally, suppose that A is a -algebra and that are skewadjoints. Since is a continuous Lie algebra homomorphism and is a convergent series of Lie brackets on we get that , i.e., is skewadjoint. The same argument applies to and to . ∎
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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