Zero Product Determined Algebras by Matej Brešar

Author:Matej Brešar
Language: eng
Format: epub
ISBN: 9783030802424
Publisher: Springer International Publishing

(6.12)

Now consider the map defined by

(compare Example 1.​11). Since xy = 0 implies yx ∈ U, Φ is a continuous bilinear map satisfying Φ(x, y) = 0 whenever xy = 0. As A is zpd, we have (by Proposition 4.​9 (iv)) Φ(x, y) = T(xy) for some linear map . In particular, Φ(xy, z) = Φ(x, yz) for all x, y, z ∈ A, which means that

for all x, y, z ∈ A. Since A 2 = A by Cohen’s factorization theorem, we thus have

and so (6.12) gives

As ρ is continuous and (6.11) holds, this implies that ρ([A, A]) = {0}; that is, φ = 0.

We have thus handled the case where φ is skew-symmetric. Let now φ be arbitrary. Then the continuous bilinear functional

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