On the picard group of involutive algebras and coalgebras

Abstract

We study how Hilbert bimodules correspond in the algebraic case to hermitian Morita equivalences and consequently we obtain a description of the hermitian Picard group of a commutative involutive algebra A as the semidirect product of the clansical hermitian Picard group of A and the automorphisms of A commuting with the involution. We also obtain similar decomposition results on hermitian Picard groups of involutive coalgebraa (C, w c ) , which show, at least in the cocommutative case, that this hermitian Picard group differs considerably from the non hermitian one. Contents 1 I n t r o d u c t i o n 3 H e r m i t i a n M o r i t a t h e o r y 3 A l g e b r a i c H i l b e r t b i m o d u l e s a n d t h e h e r m i t i a n P i c a r d g r o u p 4 T h e h e r m i t i a n P i c a r d g r o u p o f a n involutive c o a l g e b r a