Partial Isometries Between Hilbert Modules

Abstract: Hilbert modules are Banach spaces and share, of course, all their good properties. But geometrically they behave - as opposed with the very well-behaved Hilbert spaces - very much like pre-Hilbert spaces.

As a common root of most problems - if not all - one may highlight the fact that Hilbert modules need not be self-dual; one of the most striking consequences of missing self-duality is the fact that not all bounded modules maps need to possess an adjoint. (Intimately related: not all closed submodules are the range of a projection.) This raises the question how to define isometries, cosisometries, and partial isometries between Hilbert modules, without requiring explicitly in the definition that these maps are adjointable.

While the definition of isometries (as inner product preserving maps) is rather natural and well-known since long (they need not be adjointable), our definitions (proposed with Orr Shalit) of coisometries (they turn out to be adjointable) and partial isometries (they need not be adjointable) are rather recent.

As a specific problem, we will address the question how to find a (reasonable) composition law among partial isometries (making them the morphisms of a category). It turns out that for Hilbert spaces the problem can be solved, while for Hilbert modules we have to pass to the *partially defined* isometries. The proofs of some of the intermediate statements explore typical features of proofs in Hilbert module theory: Some are like those for Hilbert spaces; some reduce the proof (by means of a well-known technical tool) to that for Hilbert spaces; and some are simply ``different''. (Of course, the latter also for work Hilbert spaces; but they are ``different'' from what you would write down with all you arsenal of Hilbert space methods at your disposal.)

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | slides | video )

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