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SectionReferences

[1]
  
W. Amrein and K. B. Sinha On pairs of projections in a Hilbert space, Linear Algebra and its Applications, 208/209, pp. 425–435, (1994), doi: 10.1016/0024-3795(94)90454-5.
[2]
  
M. Argerami, Majorisation and the Carpenter’s Theorem, Integral Equations and Operator Theory, 82, no. 1, pp. 33–49, (2015), doi: 10.1007/s00020-014-2180-7.
[3]
  
W. Arveson, Diagonals of normal operators with finite spectrum, Proceedings of the National Academy of Sciences of the United States of America, 104, no. 4, pp. 1152–1158, (2007), doi: 10.1073/pnas.0605367104.
[4]
  
J. E. Avron, R. Seiler and B. Simon The index of a pair of projections, Journal of Functional Analysis, 120, no. 1, pp. 220–237, (1994), doi: 10.1006/jfan.1994.1031.
[5]
  
M.-T. Benameur, A. L. Carey, J. Phillips, A. Rennie, F. A. Sukochev and K. P. Wojciechowski An analytic approach to spectral flow in von Neumann algebras, Analysis, geometry and topology of elliptic operators, pp. 297–352, (2006).
[6]
  
M. Bownik and J. Jasper The Schur–Horn theorem for operators with finite spectrum, Transactions of the American Mathematical Society, 367, no. 7, pp. 5099–5140, (2013), doi: 10.1090/S0002-9947-2015-06317-X.
[7]
  
L. B. Brown, Ext of certain free product \(C^{\ast}\)-algebras, Journal of Operator Theory, 6, no. 1, pp. 135–141, (1981).
[8]
  
L. G. Brown, R. G. Douglas and P. A. Fillmore Unitary equivalence modulo the compact operators and extensions of \(C^{\ast}\)-algebras, Proceedings of a Conference on Operator Theory in Lecture Notes in Mathematics, 345, no. 7, pp. 58–128, (1973).
[9]
  
L. G. Brown and H. H. Lee Homotopy classification of projections in the corona algebra of a non-simple \(C^*\)-algebra, Canadian Journal of Mathematics, 64, no. 4, pp. 755–777, (2012), doi: 10.4153/CJM-2011-092-x.
[10]
  
A. Carey and J. Phillips Spectral flow in Fredholm modules, eta invariants and the JLO cocycle, \(K\)-Theory, 31, no. 2, pp. 135–194, (2004), doi: 10.1023/B:KTHE.0000022922.68170.61.
[11]
  
A. Carey, J. Phillips, A. Rennie and F. A. Sukochev, The local index formula in semifinite von Neumann algebras. II. The even case, Advances in Mathematics, 202, no. 2, pp. 517–554, (2006), doi: 10.1016/j.aim.2005.03.010.
[12]
  
A. Carey, J. Phillips and F. A. Sukochev, Spectral flow and Dixmier traces, Advances in Mathematics, 173, no. 1, pp. 68–113, (2003), doi: 10.1016/S0001-8708(02)00015-4.
[13]
  
C. Davis, Separation of two linear subspaces, Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum, 192, pp. 172–187, (1958).
[14]
  
J. Dixmier, Position relative de deux variétés linéaires fermées dans un espace de Hilbert, Revue Scientifique, 86, pp. 387–399, (1948).
[15]
  
E. G. Effros, Why the circle is connected: an introduction to quantized topology, The Mathematical Intelligencer, 11, no. 1, pp. 27–34, (1989), doi: 10.1007/BF03023772.
[16]
  
P. R. Halmos, Two subspaces, Transactions of the American Mathematical Society, 144, pp. 381–389, (1969).
[17]
  
R. V. Kadison, The Pythagorean Theorem I: the finite case, Proceedings of the National Academy of Sciences of the United States of America, 99, no. 7, pp. 4178–4184, (2002), doi: http://www.pnas.org/content/99/7/4178.
[18]
  
R. V. Kadison, The Pythagorean Theorem II: the infinite discrete case, Proceedings of the National Academy of Sciences of the United States of America, 99, no. 8, pp. 5217–5222, (2002), doi: http://www.pnas.org/cgi/reprintframed/99/8/5217.
[19]
  
V. Kaftal, P. W. Ng and S. Zhang, Strong sums of projections in von Neumann factors, Journal of Functional Analysis, 257, no. 8, pp. 2497–2529, (2009), doi: 10.1016/j.jfa.2009.05.027.
[20]
  
V. Kaftal and G. Weiss, An infinite dimensional Schur–Horn Theorem and majorization theory, Journal of Functional Analysis, 259, no. 12, pp. 3115–3162, (2010), doi: 10.1016/j.jfa.2010.08.018.
[21]
  
T. Kato, Perturbation theory for linear operators, (1966), pp. xix+592, Springer–Verlag New York, Inc., New York.
[22]
  
M. G. Krein, M. A. Kranoselskiĭ and D. P. Milman, Defect numbers of linear operators in Banach space and some geometrical problems, Sobor. Trudov. Insst. Mat. Akad. Nauk SSSR, 11, pp. 97–112, (1948).
[23]
  
S. Strătilă, Modular theory in operator algebras, pp. 492, (1981), Editura Academiei Republicii Socialiste România, Bucharest; Abacus Press, Tunbridge Wells.
[24]
  
M. Takesaki, Theory of operator algebras. I, pp. xx+415, (2002), in Encyclopaedia of Mathematical Sciences, 124, Springer-Verlag, Berlin.