2. CHAINS AND TRACES 11

Definition 2.6 (Catenation). Let x, y, z ∈ α ∩ β. The catenation of two

(α, β)-traces Λ = (x, y, w) and Λ = (y, z, w ) is defined by

Λ#Λ := (x, z, w + w ).

Let u ∈ D(x, y) and u ∈ D(y, z) and suppose that u and u are constant near the

ends ±1 ∈ D. For 0 λ 1 suﬃciently close to one the λ-catenation of u and

u is the map u#λu ∈ D(x, z) defined by

(u#λu )(ζ) :=

⎧

⎨

⎩

u

ζ+λ

1+λζ

, for Re ζ ≤ 0,

u

ζ−λ

1−λζ

, for Re ζ ≥ 0.

Lemma 2.7. If u ∈ D(x, y) and u ∈ D(y, z) are as in Definition 2.6 then

Λu#λu = Λu#Λu .

Thus the catenation of two (α, β)-traces is again an (α, β)-trace.

Proof. This follows directly from the definitions.