ABSTRACT
Recall that the maximal ideal space M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq337.tif"/> A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq338.tif"/> of 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/eq339.tif"/> consists of all multiplicative linear functionals 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/eq340.tif"/> . The space M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq341.tif"/> A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq342.tif"/> is a compact Hausdorff space with the weak-star topology inherited from the Banach dual of A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq343.tif"/> . When A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq344.tif"/> is commutative, the set M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq345.tif"/> A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq346.tif"/> is in one-to-one correspondence with the set of (proper) maximal ideals in A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137292/676aa42d-79c8-4246-bc0f-e0fd47650f7d/content/eq347.tif"/> .
