ABSTRACT
A C* -algebra is a Banach algebra A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq654.tif"/> together with a mapping x ↦ x* on A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq655.tif"/> satisfying the following conditions:
(x*)* = x for all x ∈ A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq656.tif"/> .
(ax + by)* = āx* + b̄y* for all x, y ∈ A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq657.tif"/> and a, b ∈ C.
(xy)* = y*x* for all x, y ∈ A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq658.tif"/> .
‖x*x‖ = ‖x‖ 2 for all x ∈ A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq659.tif"/> .