Dissertationes mathematicae. 482: Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces / P. Niemiec; Institute of Mathematics, Polish Academy of Sciences. - Warszawa: Instytut matematyczny PAN, 2012. - ii, 106 p. - Ref.: p.105-106. - ISSN 0012-3862  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

1 Introduction ................................................. 5 1.1 Preface ................................................. 5 1.2 Basic notation and terminology .......................... 7 2 General decomposition theorem ................................ 9 2.1 Preliminaries ........................................... 9 2.2 The b-transform ........................................ 11 2.3 Background on von Neumann algebras ..................... 12 2.4 Decompositions relative to ideals ...................... 13 3 Structural decomposition .................................... 17 3.1 Strong order ........................................... 17 3.2 Steering projections in W*-algebras .................... 18 3.2.1 Type II1 ........................................ I9 3.2.2 Types I and III ................................. 19 3.2.3 Type II∞ ........................................ 20 3.3 Decomposition relative to a steering projection ........ 21 3.4 Minimal and semiminimal tuples ......................... 24 3.5 Unities of ideals ...................................... 27 3.6 Decomposition relative to the unity .................... 28 4 Topological model ........................................... 32 4.1 Algebraic and order properties ......................... 32 4.2 Reconstructing infinite operations ..................... 36 4.3 Semigroup of semiminimal tuples ........................ 40 4.4 Model for the class .................................... 45 4.5 Types of tuples ........................................ 55 5 Prime decomposition ......................................... 58 5.1 Primes, semiprimes, atoms and fractals ................. 58 5.2 Strongly unitarily disjoint families ................... 62 5.3 Measure-theoretic preliminaries ........................ 65 5.4 Direct integrals and measurable domains ................ 72 5.5 'Continuous' direct sums ............................... 81 5.6 Prime decomposition .................................... 88 6 Classification of ideals ................................... 100 6.1 Types of isomorphisms ................................. 100 6.2 Classification of ideals up to isomorphism ............ 102 6.3 Concluding remarks .................................... 103 6.3.1 Finite-dimensional tuples ...................... 103 6.3.2 Problem of axiomatization ...................... 104 6.3.3 'Continuous' ideals ............................ 104 6.3.4 Length of tuples ............................... 105 References .................................................... 105