By Leonard E Dickson

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3. If A is a unital Banach algebra, a Banach A-bimodule E is called unital if eA · x = x · eA = x (x ∈ E). 4 Let A be a Banach algebra. A Banach A-bimodule E is called pseudounital if E = {a · x · b : a, b ∈ A, x ∈ E}. Similarly, one defines pseudo-unital left and right Banach modules. 4 Let A be a Banach algebra, and let E be a Banach A-bimodule. We say that E is essential if the linear hull of {a·x·b : a, b ∈ A, x ∈} is dense in E. 10] to show: If A has a bounded approximate identity and if E is essential, then E is pseudo unital.

V) There is a left invariant mean on U C(G). Proof It is obvious that only (v) =⇒ (i) needs proof. Let m be a left invariant mean on U C(G). 8, m is also topologically left invariant. Let (eα )α be a bounded approximate identity for L1 (G) in P (G), and choose an ultrafilter U on the index set of (eα )α that dominates the order filter. Define m ˜ : L∞ (G) → C, φ → lim eα ∗ φ ∗ eα , m . 5(iii), this is well defined. It is easily seen that m ˜ is a mean on L∞ (G) (but check for yourself). e. m ˜ is topologically left invariant.

Let π : E ∗ → E0∗ be the restriction map. It is routinely checked that π is a module homomorphism, so that π ◦ D ∈ Z 1 (A, E0∗ ). Since H1 (A, E0∗ ) = {0}, there is φ0 ∈ E0∗ such that ˜ := D−adφ ∈ Z 1 (A, E ⊥ ). π◦D = adφ0 . Choose φ ∈ E ∗ such that φ|E0 = φ0 . It follows that D 0 ∗ We have E0⊥ ∼ (E/E ) (as Banach A-bimodules). 3. e. D = adφ−ψ . such that D It is often convenient to extend a derivation to a larger algebra. If a Banach algebra A is contained as a closed ideal in another Banach algebra B then the strict topology on B with respect to A is defined through the family of seminorms (pa )a∈A , where (b ∈ B).