Let E and F be Hilbert C*-modules over a C*-algebra A. New classes of (possibly unbounded) operators t: E->F are introduced and investigated - first of all graph regular operators. Instead of the density of the domain D(t) we only assume that t is essentially defined, that is, D(t) has an trivial ortogonal complement. Then t has a well-defined adjoint. We call an essentially defined operator t graph regular if its graph G(t) is orthogonally complemented and orthogonally closed if G(t) coincides with its biorthogonal complement. A theory of these operators and related concepts is developed: polar decomposition, functional calculus. Various characterizations of graph regular operators are given: (a, a_*, b)-transform and bounded transform. A number of examples of graph regular operators are presented (on commutative C*-algebras, a fraction algebra related to the Weyl algebra, Toeplitz algebra, C*-algebra of the Heisenberg group). A new characterization of operators affiliated to a C*-algebra in terms of resolvents is given as well as a Kato-Rellich theorem for affiliated operators. The association relation is introduced and studied as a counter part of graph regularity for concrete C*-algebras.:Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Sightings
1. Unitary *-module spaces
Algebraic essence of adjointability on Hilbert C*-modules . . . . . 13
a) Operators on Hilbert C*-modules - Notions. . . . . . . . . . . . . . 13
b) Essential submodules and adjointability . . . . . . . . . . . . . . . . 15
c) From Hilbert C*-modules to unitary *-module spaces . . . . . . 16
2. Operators on unitary *-module spaces
Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3. Graph regularity
Pragmatism between weak and (strong) regularity . . . . . . . . . 27
a) Types of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
b) The case C(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
c) Graph regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Transition. Orthogonal complementability and topology
Back to Hilbert C*-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Graph regular operators on Hilbert C*-modules
4. Commutative case: Operators on C_0(X)
Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Interjection. Unboundedness and graph regularity . . . . . . . . . . 55
5. Relation to adjointable operators
Sources of graph regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6. Concrete C*-algebras
Association relation and affiliation relation . . . . . . . . . . . . . . . . 61
7. Examples
Graph regular operators that are not regular . . . . . . . . . . . . . 67
a) Position and momentum operators as graph regular
operators on a fraction algebra related to the Weyl algebra . . 67
b) A graph regular but not regular operator on the
group C*-algebra of the Heisenberg group . . . . . . . . . . . . . . . 69
c) Unbounded Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . 70
8. Bounded transform
The canonical regular operator associated to a graph regular
operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9. Absolute value and polar decomposition . . . . . . . . . . . . . . . 79
10. Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
11. Special matrices of C*-algebras
Counter examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Abstract and open questions . . . . . . . . . . . . . . . . . . . . . . . . . 89
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Dank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Erklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:15095 |
| Date | 28 November 2016 |
| Creators | Gebhardt, René |
| Contributors | Schmüdgen, Konrad, Troitsky, Evgenij V., Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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