#### Title

Pervasive Algebras and Maximal Subalgebras.

#### Publication Date

2011

#### Journal

Studia Mathematics

#### Volume

206

#### Issue

1

#### First Page

1

#### Last Page

24

#### Abstract

A uniform algebra *A *on its Shilov boundary *X *is *maximal* if *A *is not *C*(*X*) and no uniform algebra is strictly contained between *A *and *C*(*X*) . It is *essentially pervasive* if *A *is dense in *C*(*F*) whenever *F *is a proper closed subset of the essential set of *A*. If *A *is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show: (1) If *A *is pervasive and proper, and has a nonconstant unimodular element, then *A *contains an infinite descending chain of pervasive subalgebras on *X *. (2) It is possible to find a compact Hausdorff space *X *such that there is an isomorphic copy of the lattice of all subsets of N in the family of pervasive subalgebras of *C*(*X*). (3) In the other direction, if *A *is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word “strongly” is removed. We discuss examples involving Dirichlet algebras, *A*(*U*) algebras, Douglas algebras, and subalgebras of *H*∞(D), and develop new results that relate pervasiveness, maximality, and relative maximality to support sets of representing measures.

#### Link to Published Version

#### Recommended Citation

Gorkin, Pamela and O'Farrell, Anthony G.. "Pervasive Algebras and Maximal Subalgebras.." *Studia Mathematics* 206, no. 1 (2011)
: 1-24.

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