Preface

This boo k i s a slightly expanded an d revised version of th e lecture notes fro m

the NSF/CBMS Regiona l Conferenc e hel d in Tempe, Arizon a i n May , 1984, at

which th e thir d autho r wa s th e principal lecturer . I n th e boo k w e hav e trie d t o

summarize some of th e voluminous progress that has been made in the theory of

dual algebra s sinc e th e appearanc e i n 1978 o f Scot t Brown' s pioneerin g paper ,

which clearly showe d th e utility of thi s concept for studying th e structure theory

of bounde d linea r operators on Hilbert space. The aim of th e book i s t o present

an approach for studying (nonselfadjoint) dual algebras that allows one to obtain,

in particular, results on invariant subspaces and dilation theory.

The boo k i s pu t togethe r a s follows . Chapte r I consists o f preliminarie s o f a

general nature concerning dual algebras . Most of Chapter s II, III, IV, and X are

taken fro m [6a] , but Chapter s II I an d VI contai n som e ne w material , especiall y

Theorem 6.3 an d th e results leading up to it. Chapters IV and V are taken fro m

[10], and Chapter VIII comes from [7a] and [6b]. Chapter VIII is a rewrite of part

of [11] and [12a], and Chapter IX is taken from a version of [11a].

We wis h t o acknowledg e ou r indebtednes s t o Constanti n Aposto l an d Bel a

Sz.-Nagy, with whom we obtained many of the results to be found herein, and to

Frank Gilfeather and the other members of the Mathematical Sciences Section of

the Nationa l Scienc e Foundation , whos e suppor t an d encouragemen t hav e con-

tributed greatly to our efforts.

Hari Bercovici

Ann Arbor, Michigan Ciprian FoidS

December 1984 - . „

Carl Pearcy

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