Hilbert Modules over Operator Algebras

• Author(s): Paul S. Muhly, Baruch Solel,
• Publisher: American Mathematical Soc.
• Pages: 53
• ISBN_10: 0821803468
  ISBN_13: 9780821803462
  
• Language: en
• Categories: Mathematics / Algebra / General , Mathematics / Calculus ,

Description:... This memoir gives a general systematic analysis of the notions of "projectivity" and "injectivity" in the context of Hilbert modules over operator algebras. A Hilbert module over an operator algebra [italic capital]A is simply the Hilbert space of a (contractive) representation of [italic capital]A viewed as a module over [italic capital]A in the usual way. We introduce various notions of "projective" Hilbert modules and use them to investigate dilation and commutant lifting problems over certain infinite dimensional analogues of "incidence" algebras. We prove that commutant lifting holds for such an algebra if and only if the pattern indexing the algebra is a "tree" in the sense of computer directories.