Morita equivalence of nest algebras (Q2857594)

This paper is mainly concerned with the specialisation of weak-\(*\) Morita equivalence and \(\Delta\)-equivalence of dual operator algebras to the setting of nest algebras. The notion of weak-\(*\) Morita equivalence is due to \textit{D. P. Blecher} and \textit{U. Kashyap} [J. Pure Appl. Algebra 212, No.~11, 2401--2412 (2008; Zbl 1143.47055)] and \(\Delta\)-equivalence has been introduced by the author [J. Funct. Anal. 256, No.~11, 3545--3567 (2009; Zbl 1188.47056)] and is equivalent to the concept of stable isomorphism of dual operator algebras.NEWLINENEWLINELet \({\mathcal N}, {\mathcal M}\) be nests, let \(A=\text{alg} {\mathcal N}\) and \( B=\text{alg} {\mathcal M}\) be the corresponding nest algebras, and let \(A_0, B_0\) be the subalgebras of compact operators in \(A\) and \(B\), respectively. The main result states that the nests \({\mathcal N}\) and \( {\mathcal M}\) are isomorphic if and only if \(A_0\) and \(B_0\) are strongly Morita equivalent if and only if \(A\) and \(B\) are weakly-\(*\) Morita equivalent. Strong and weak-\(*\) Morita equivalences can in this way be perceived as lattice related properties. Here, strong Morita equivalence refers to a Morita theory developed by \textit{D. P. Blecher} et al. [Mem. Am. Math. Soc. 681 (2000; Zbl 0966.46033)] for not necessarily self-adjoint operator algebras as a counterpart of that existing in the self-adjoint algebras context.NEWLINENEWLINEThe present paper goes further to address questions such as those concerning spatial Morita equivalence and stable isomorphism of nest algebras and presents, amongst several other results, a new proof showing that weak-\(*\) Morita equivalence is strictly weaker than \(\Delta\)-equivalence. The final section is dedicated to the presentation of a counterexample illustrating the fact that the second duals of strongly Morita equivalent algebras need not be stably isomorphic.