The Kadison-Singer PropertySpringer, 7 de nov. 2016 - 140 pàgines This book gives a complete classification of all algebras with the Kadison-Singer property, when restricting to separable Hilbert spaces. The Kadison-Singer property deals with the following question: given a Hilbert space H and an abelian unital C*-subalgebra A of B(H), does every pure state on A extend uniquely to a pure state on B(H)? This question has deep connections to fundamental aspects of quantum physics, as is explained in the foreword by Klaas Landsman. The book starts with an accessible introduction to the concept of states and continues with a detailed proof of the classification of maximal Abelian von Neumann algebras, a very explicit construction of the Stone-Cech compactification and an account of the recent proof of the Kadison-Singer problem. At the end accessible appendices provide the necessary background material. This elementary account of the Kadison-Singer conjecture is very well-suited for graduate students interested in operator algebras and states, researchers who are non-specialists of the field, and/or interested in fundamental quantum physics. |
Continguts
| 1 | |
| 3 | |
3 State Spaces and the KadisonSinger Property | 11 |
4 Maximal Abelian CSubalgebras | 23 |
5 Minimal Projections in Maximal Abelian von Neumann Algebras | 37 |
6 StoneČech Compactification | 59 |
7 The Continuous Subalgebra and the KadisonSinger Conjecture | 71 |
8 The KadisonSinger Problem | 85 |
Appendix A Preliminaries | 113 |
Appendix B Functional Analysis and Operator Algebras | 117 |
Appendix C Additional Material | 125 |
Appendix D Notes and Remarks | 132 |
Altres edicions - Mostra-ho tot
Frases i termes més freqüents
A1 B A2 abelian unital abelian von Neumann Ad(j arbitrary bijection C∗-algebra Cauchy–Schwarz inequality compact Hausdorff space continuous function continuous subalgebra converges convex convex set Corollary define Definition discrete subalgebra exactly one element Ext(f f e S(A filter finite Furthermore GB A2 Haar functions Hausdorff space Hence Hilbert space isomorphism Kadison-Singer conjecture Kadison-Singer property Lemma let x e linear linear subspace main text matrices maximal abelian subalgebras meet-semilattice minimal projections Mn(C n e j Neumann algebra operator algebras orthant Pn(A positive Proof Suppose Proposition prove real stable polynomial S(Ac self-adjoint separable Hilbert space set of projections space H SpringerBriefs in Mathematical Stone-ˇCech compactification strong topology subset Suppose H topological space total set totally ordered Tychonoff space Ultra(Z(X ultrafilter unique unitarily equivalent unitary von Neumann algebra whence zeroes