Primitive limit algebras and C\(^{*}\)-envelopes
We know that every operator algebra can be completely isometrically embedded in a unique minimal \(C^*\)-algebra which is called its \(C^*\)-envelope [cf. \textit{M. Hamana}, Publ. Res. Inst. Math. Sci. 15, 773-785 (1979; Zbl 0436.46046)], although we cannot in general compute this algebra. In this paper, however, an explicit presentation of the quotient of an NSAF algebra by any closed ideal is described as a subalgebra of an AF-algebra. Recall that an NSAF algebra is a limit of direct sums of nest algebras. A closed ideal \(I\) of a Banach algebra \(A\) is said to be prime if for any two (closed) ideals \(I_1\) and \(I_2\) so that \(I_1I_2 \subseteq I\), it follows that \(I_1 \subseteq I\) or \(I_2\subseteq I\). The kernel of an algebraically irreducible representation of \(A\) is called primitive. It is known that a primitive ideal of a Banach algebra \(A\) is closed and prime. The converse is valid for a separable \(C^*\)-algebra \(A\) [cf. \textit{J. Dixmier}, Bull. Soc. Math. Fr. 88, 95-112 (1960; Zbl 0124.32403)], but not generally true in the context of Banach algebras. The authors prove the converse for NSAF algebras. By a regular limit algebra we mean a norm-closed subalgebra of an AF \(C^*\)-algebra containing a canonical maximal abelian self-adjoint subalgebra. It is also shown that a regular limit algebra is \(C^*\)-primitive if (and only if) it is prime.
scientific article; zbMATH DE number 1845949
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