The continuity of derivations from group algebras: factorizable and connected groups

Abstract

A group is said to be factorizable if it has a finite number of abelian subgroups, H1, H2, … Hn, such that G = H1H2 … Hn. It is shown that, if G is a factorizable or connected locally compact group, then every derivation from L1 (G) to an arbitrary L1 (G)-bimodule X is continuous.

Factorization in finite-codimensional ideals of group algebras

“…Step 1 shows that a certain ideal, called the continuity ideal, has ®nite codimension in the algebra, and Step 2 uses an approximate identity argument to show that the restriction of the derivation to the continuity ideal is continuous. This approach does succeed in showing that all derivations from C Ã -algebras are continuous, [23], and is used in [37] to show that derivations from L 1 G are continuous for certain groups G. In the case of general locally compact groups, Corollary 6.2 would suf®ce to carry out Step 2 if Step 1 could be completed. However our current understanding of the structure of general group algebras seems inadequate for this.…”

“…A natural question to ask is as follows: which ideals in L 1 G may be decomposed as in the Main Theorem? It is shown in [37] that, if G is discrete or connected and N is a closed normal subgroup, then the ideal has such a decomposition. It can be shown that a necessary condition for I to have such a decomposition is that I should be weakly complemented.…”

“…Then J H has a right bounded approximate identity if and only if H is amenable; see [22, Theorems 10.1 and 10.2]. This fact may be used, in conjunction with a Lie groups argument, to show that, if G is connected, then ®nite-codimensional ideals in L 1 G are idempotent; see [37]. These methods in fact show that L 1 0 G is a sum of two left ideals each having a right bounded approximate identity, that is, that the approximate identities are on the same side.…”

Locally Compact Groups which have the Weakly Compact Homomorphism Property

“…Again using [Joh,Corollary 4.5], we may therefore assume that G is a connected Lie group. Now, by [Wil,Theorem 1.2], there are abelian subgroups Hx, ... ,Hn of G such that G = Hi ■ ■ ■ H" . Hence, as a consequence of [Joh,Theorem 4.2(a) If "irreducible representation" is supposed here to mean "topologically irreducible ""-representation on a Hubert space", as is usual in the group algebra context, this question asks if every G £ [Moore] is WCHP and is thus resolved by our theorem.…”

Examples of Factorization Without Bounded Approximate Units

“…( 2 )^( 3 )^ ( 7 ) (*) II V (2) is shown by an example of Paschke (see [19] or [9,Example 22.3]). Also, the algebras treated in [23], which have already been mentioned, do not satisfy (2) but they do satisfy (7). It is not known whether they satisfy (3).…”