BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Michael Skeide (Università degli Studi del Molise) DTSTART:20231129T200000Z DTEND:20231129T210000Z DTSTAMP:20260420T053047Z UID:NYC-NCG/146 DESCRIPTION:Title: Partial Isometries Between Hilbert Modules\nby Michael Skeide (Unive rsità degli Studi del Molise) as part of Noncommutative geometry in NYC\n \n\nAbstract\nHilbert modules are Banach spaces and share\, of course\, al l their good properties. But geometrically they behave - as opposed with t he very well-behaved Hilbert spaces - very much like pre-Hilbert spaces.\n \nAs 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 mod ules 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 Hilber t modules\, without requiring explicitly in the definition that these maps are adjointable.\n\nWhile the definition of isometries (as inner product preserving maps) is rather natural and well-known since long (they need no t be adjointable)\, our definitions (proposed with Orr Shalit) of coisomet ries (they turn out to be adjointable) and partial isometries (they need n ot be adjointable) are rather recent.\n\nAs a specific problem\, we will a ddress the question how to find a (reasonable) composition law among parti al 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 Hilbe rt 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 space s\; and some are simply ``different''. (Of course\, the latter also for wo rk Hilbert spaces\; but they are ``different'' from what you would write d own with all you arsenal of Hilbert space methods at your disposal.)\n LOCATION:https://researchseminars.org/talk/NYC-NCG/146/ END:VEVENT END:VCALENDAR