Mathematical Sciences: C*-Algebras and the Geometry of      Hilbert Space
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{'SignBlockName': 'William Y. Velez', 'PO_EMAI': None, 'PO_PHON': None}
This is mathematical research in pursuit of two famous open                problems having to do with pure states on operator algebras. The                term state comes from quantum mechanics. In some formulations                   thereof, observables are represented by selfadjoint operators in                an appropriate algebra of operators, and states are certain                     linear functionals on the operator algebra; thus, specifying a                  state assigns a numerical value to each of the observables. The                 pure states are those not given as convex combinations of other                 states. In principal, they determine everything about the                       algebra, but pinning down the connection more precisely has led                 to daunting difficulties. One of the problems the present project               will address is that of extending the Stone-Weierstrass theorem,                which may be thought of as a result about commutative operator                  algebras, to the general, noncommutative situation. The other                   problem is to show the uniqueness of pure state extensions from                 maximal diagonal subalgebras of the algebra of all bounded                      operators on a Hilbert space.
{'FirstName': 'Charles', 'LastName': 'Akemann', 'PI_MID_INIT': 'A', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Charles A Akemann', 'EmailAddress': 'akemann@math.ucsb.edu', 'NSF_ID': '000298653', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of California-Santa Barbara', 'CityName': 'SANTA BARBARA', 'ZipCode': '931060001', 'PhoneNumber': '8058934188', 'StreetAddress': '3227 CHEADLE HALL', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'California', 'StateCode': 'CA', 'CONGRESSDISTRICT': '24', 'CONGRESS_DISTRICT_ORG': 'CA24', 'ORG_UEI_NUM': 'G9QBQDH39DF4', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF CALIFORNIA, SANTA BARBARA', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '126200', 'Text': 'MODERN ANALYSIS'}
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