uniform algebra

In functional analysis, a uniform algebra ''A'' on a compact Hausdorff topological space ''X'' is a closed (with respect to the uniform norm) subalgebra of the C*-algebra ''C(X)'' (the continuous complex-valued functions on ''X'') with the following properties:
:the constant functions are contained in ''A''
: for every ''x'', ''y'' $\in$ ''X'' there is ''f''$\in$''A'' with ''f''(''x'')$\ne$''f''(''y''). This is called separating the points of ''X''.

As a closed subalgebra of the commutative Banach algebra ''C(X)'' a uniform algebra is itself a unital commutative Banach algebra (when equipped with the uniform norm). Hence, it is, (by definition) a Banach function algebra.

A uniform algebra ''A'' on ''X'' is said to be natural if the maximal ideals of ''A'' are precisely the ideals $M_x$ of functions vanishing at a point ''x'' in ''X''.

Abstract characterization

If ''A'' is a unital commutative Banach algebra such that $, , a^2, , = , , a, , ^2$ for all ''a'' in ''A'', then there is a compact Hausdorff ''X'' such that ''A'' is isomorphic as a Banach algebra to a uniform algebra on ''X''. This result follows from the spectral radius formula and the Gelfand representation.

Notes

References

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Functional analysis
Banach algebras

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