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Dilation Theory: A Guided Tour

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Operator Theory, Functional Analysis and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 282))

Abstract

Dilation theory is a paradigm for studying operators by way of exhibiting an operator as a compression of another operator which is in some sense well behaved. For example, every contraction can be dilated to (i.e., is a compression of) a unitary operator, and on this simple fact a penetrating theory of non-normal operators has been developed. In the first part of this survey, I will leisurely review key classical results on dilation theory for a single operator or for several commuting operators, and sample applications of dilation theory in operator theory and in function theory. Then, in the second part, I will give a rapid account of a plethora of variants of dilation theory and their applications. In particular, I will discuss dilation theory of completely positive maps and semigroups, as well as the operator algebraic approach to dilation theory. In the last part, I will present relatively new dilation problems in the noncommutative setting which are related to the study of matrix convex sets and operator systems, and are motivated by applications in control theory. These problems include dilating tuples of noncommuting operators to tuples of commuting normal operators with a specified joint spectrum. I will also describe the recently studied problem of determining the optimal constant \(c = c_{\theta ,\theta '}\), such that every pair of unitaries U, V  satisfying V U = e iθ UV  can be dilated to a pair of cU′, cV ′, where U′, V ′ are unitaries that satisfy the commutation relation \(V'U' = e^{i\theta '} U'V'\). The solution of this problem gives rise to a new and surprising application of dilation theory to the continuity of the spectrum of the almost Mathieu operator from mathematical physics.

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Notes

  1. 1.

    The reader should be aware that Theorem 7.4 is sometimes referred to as Arveson’s dilation theorem, whereas I used this name already for the more specific Theorem 5.1.

  2. 2.

    The existence of the C*-envelope was obtained much earlier, without making use of boundary representations; see [118].

References

  1. J. Agler, Rational dilation on an annulus. Ann. Math. 121, 537–563 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Agler, J.E. McCarthy, Pick Interpolation and Hilbert Function Spaces. Graduate Studies in Mathematics, vol. 44 (American Mathematical Society, Providence, 2002)

    Google Scholar 

  3. J. Agler, J.E. McCarthy, Distinguished varieties. Acta Math. 194, 133–153 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Agler, N.J. Young, Operators having the symmetrized bidisc as a spectral set. Proc. Edinb. Math. Soc. 43, 195–210 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Agler, J. Harland, B.J. Raphael, Classical Function Theory, Operator Dilation Theory and Machine Computation on Multiply-Connected Domains. Memoirs of the American Mathematical Society (American Mathematical Society, Providence, 2008)

    Google Scholar 

  6. M.A. Akcoglu, L. Sucheston, Dilations of positive contractions on L p spaces. Can. Math. Bull. 20, 285–292 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. C.G. Ambrozie, A. Gheondea, An interpolation problem for completely positive maps on matrix algebras: solvability and parametrization. Linear Multilinear Algebra 63, 826–851 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Ambrozie, V. Muller, Commutative dilation theory, in Operator Theory, ed. by D. Alpay (Springer, Berlin, 2014)

    MATH  Google Scholar 

  9. T. Andô, On a pair of commutative contractions. Acta Sci. Math. 24, 88–90 (1963)

    MathSciNet  MATH  Google Scholar 

  10. W.B. Arveson, Subalgebras of C ∗-algebras. Acta Math. 123, 141–224 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  11. W.B. Arveson, Subalgebras of C∗-algebras II. Acta Math. 128, 271–308 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. W.B. Arveson, Subalgebras of C ∗-algebras III: multivariable operator theory. Acta Math. 181, 159–228 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. W.B. Arveson, Dilation theory yesterday and today, in A Glimpse of Hilbert Space Operators: Paul R. Halmos in Memoriam, ed. by S. Axler, P. Rosenthal, D. Sarason (Birkhäuser, Basel, 2010)

    Google Scholar 

  14. W.B. Arveson, Non-commutative Dynamics and E-semigroups. Springer Monographs in Mathematics (Springer, Berlin, 2003)

    Google Scholar 

  15. W.B. Arveson, The noncommutative Choquet boundary. J. Am. Math. Soc. 21, 1065–1084 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Aubrun, L. Lami, C. Palazuelos, M. Plavala, Entangleability of cones (2020). ar**v:1911.09663

    Google Scholar 

  17. A. Avila, S. Jitomirskaya, The ten martini problem. Ann. Math. 170, 303–342 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Avron, P.H.M.v. Mouche, B. Simon, On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys. 132, 103–118 (1990)

    Google Scholar 

  19. C. Badea, B. Beckermann, Spectral sets, in Handbook of Linear Algebra, ed. by L. Hogben (Chapman and Hall/CRC, Boca Raton, 2014)

    Google Scholar 

  20. J. Bellisard, Lipshitz continuity of gap boundaries for Hofstadter-like spectra. Commun. Math. Phys. 160, 599–613 (1994)

    Article  MathSciNet  Google Scholar 

  21. A. Ben-Tal, A. Nemirovski, On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty. SIAM J. Optim. 12, 811–833 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Bercovici, D. Timotin, The numerical range of a contraction with finite defect numbers. J. Math. Anal. Appl. 417, 42–56 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Bercovici, C. Foias, L. Kerchy, B. Sz.-Nagy, Harmonic Analysis of Operators on Hilbert Space. Universitext (Springer, Berlin, 2010)

    Google Scholar 

  24. B.V.R. Bhat, An index theory for quantum dynamical semigroups. Trans. Am. Math. Soc. 348, 561–583 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. B.V.R. Bhat, A generalized intertwining lifting theorem, in Operator Algebras and Applications, II, Waterloo, ON, 1994-1995, Fields Institute Communications, vol. 20 (American Mathemtical Society, Providence, 1998), pp. 1–10

    Google Scholar 

  26. B.V.R. Bhat, T. Bhattacharyya, Dilations, completely positive maps and geometry (forthcoming book)

    Google Scholar 

  27. B.V.R. Bhat, M. Mukherjee, Inclusion systems and amalgamated products of product systems. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 13, 1–26 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. B.V.R. Bhat, M. Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 3, 519–575 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. T. Bhattacharyya, S. Pal, S. Shyam Roy, Dilations of - contractions by solving operator equations. Adv. Math. 230, 577–606 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Bluhm, I. Nechita, Joint measurability of quantum effects and the matrix diamond. J. Math. Phys. 59, 112202 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. F.P. Boca, Rotation C*-Algebras and Almost Mathieu Operators (The Theta Foundation, Bucharest, 2001)

    Google Scholar 

  32. F.P. Boca, A. Zaharescu, Norm estimates of almost Mathieu operators. J. Funct. Anal. 220, 76–96 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. J.W. Bunce, Models for n-tuples of noncommuting operators. J. Funct. Anal. 57, 21–30 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  34. D.R. Buske, J.R. Peters, Semicrossed products of the disk algebra: contractive representations and maximal ideals. Pac. J. Math. 185, 97–113 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. M.D. Choi, K.R. Davidson, A 3 × 3 dilation counterexample. Bull. Lond. Math. Soc. 45, 511–519 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. M.D. Choi, C.K. Li, Constrained unitary dilations and numerical ranges. J. Operator Theory 46, 435–447 (2001)

    MathSciNet  MATH  Google Scholar 

  37. M.-D. Choi, G.A. Elliott, N. Yui, Gauss polynomials and the rotation algebra. Invent. Math. 99, 225–246 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  38. D. Cohen, Dilations of matrices. Thesis (M.Sc.), Ben-Gurion University (2015). ar**v:1503.07334

    Google Scholar 

  39. M.J. Crabb, A.M. Davie, Von Neumann’s inequality for hilbert space operators. Bull. Lond. Math. Soc. 7, 49–50 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Crouzeix, Numerical range and functional calculus in Hilbert space. J. Funct. Anal. 244, 668–990 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. K.R. Davidson, E.G. Katsoulis, Dilation theory, commutant lifting and semicrossed products. Doc. Math. 16, 781–868 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. K.R. Davidson, M. Kennedy, The Choquet boundary of an operator system. Duke Math. J. 164, 2989–3004 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. K.R. Davidson, M. Kennedy, Noncommutative Choquet theory. ar**v:1905.08436

    Google Scholar 

  44. K.R. Davidson, D.R. Pitts, Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras. Integr. Equ. Operator Theory 31, 321–337 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  45. K.R. Davidson, D.R. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras. Math. Ann. 311, 275–303 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  46. K.R. Davidson, A. Dor-On, O.M. Shalit, B. Solel, Dilations, inclusions of matrix convex sets, and completely positive maps. Int. Math. Res. Not. 2017, 4069–4130 (2017)

    MathSciNet  MATH  Google Scholar 

  47. K.R. Davidson, A. Dor-On, O.M. Shalit, B. Solel, Dilations, inclusions of matrix convex sets, and completely positive maps (2018). ar**v:1601.07993v3 [math.OA]

    Google Scholar 

  48. K.R. Davidson, A.H. Fuller, E.T.A. Kakariadis, Semicrossed products of operator algebras: a survey. New York J. Math. 24a, 56–86 (2018)

    Google Scholar 

  49. E.B. Davies, Quantum Theory of Open Systems (Academic, Cambridge, 1976)

    MATH  Google Scholar 

  50. A. Dor-On, Techniques in operator algebras: classification, dilation and non-commutative boundary theory. Thesis (Ph.D.) University of Waterloo (2017)

    Google Scholar 

  51. A. Dor-On, G. Salomon, Full Cuntz-Krieger dilations via non-commutative boundaries. J. Lond. Math. Soc. 98, 416–438 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  52. R. Douglas, On extending commutative ssemigroupsemigroups of isometries. Bull. Lond. Math. Soc. 1, 157–159 (1969)

    Article  MATH  Google Scholar 

  53. M. Dritschel, S. McCullough, Boundary representations for families of representations of operator algebras and spaces. J. Operator Theory 53, 159–167 (2005)

    MathSciNet  MATH  Google Scholar 

  54. M.A. Dritschel, S. McCullough, The failure of rational dilation on a triply connected domain. J. Am. Math. Soc. 18, 873–918 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  55. S.W. Drury, A generalization of von Neumann’s inequality to the complex ball. Proc. Am. Math. Soc. 68, 300–304 (1978)

    MathSciNet  MATH  Google Scholar 

  56. S.W. Drury, Remarks on von Neumann’s Inequality. Lecture Notes in Mathematics, vol. 995 (Springer, Berlin, 1983)

    Google Scholar 

  57. E.G. Effros, S. Winkler, Matrix convexity: operator analogues of the bipolar and Hahn-Banach theorems. J. Funct. Anal. 144, 117–152 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  58. E. Egerváry, On the contractive linear transformations of n-dimensional vector space. Acta Sci. Math. Szeged 15, 178–182 (1954)

    MathSciNet  MATH  Google Scholar 

  59. G.A. Elliott, Gaps in the spectrum of an almost periodic Schrödinger operator. C.R. Math. Rep. Acad. Sci. Can. 4, 255–299 (1982)

    Google Scholar 

  60. D.E. Evans, J.T. Lewis, Dilations of dynamical semi-groups. Commun. Math. Phys. 50, 219–227 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  61. E. Evert, J.W. Helton, I. Klep, S. McCullough, Extreme points of matrix convex sets, free spectrahedra, and dilation theory. J. Geom. Anal. 28, 1373–1408 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  62. S. Fackler, J. Glück, A toolkit for constructing dilations on Banach spaces. Proc. Lond. Math. Soc. 118, 416–440 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  63. C. Foias, A.E. Frazho, The Commutant Lifting Approach to Interpolation Problems (Birkhäuser, Basel, 1990)

    Google Scholar 

  64. A.E. Frazho, Models for noncommuting operators. J. Funct. Anal. 48, 1–11 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  65. T. Fritz, T. Netzer, A. Thom, Spectrahedral containment and operator systems with finite-dimensional realization. SIAM J. Appl. Algebra Geom. 1, 556–574 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  66. A.H. Fuller, Finitely correlated representations of product systems of C*-correspondences over \(\mathbb {N}^k\). J. Funct. Anal. 260, 574–611 (2011)

    Google Scholar 

  67. D.J. Gaebler, Continuous unital dilations of completely positive semigroups. J. Funct. Anal. 269, 998–1027 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  68. T.W. Gamelin, Uniform Algebras, vol. 311 (American Mathematical Society, Providence, 2005)

    MATH  Google Scholar 

  69. M. Gerhold, O.M. Shalit, Dilations of q-commuting unitaries (2019). ar**v:1902.10362

    Google Scholar 

  70. M. Gerhold, O.M. Shalit, On the matrix range of random matrices (2020). ar**v:1911.12102

    Google Scholar 

  71. U. Haagerup, M. Rørdam, Perturbations of the rotation C*-algebras and of the Heisenberg commutation relations. Duke Math. J. 77, 227–256 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  72. P.R. Halmos, Normal dilations and extensions of operators. Summa Brasil. Math. 2, 125–134 (1950)

    MathSciNet  Google Scholar 

  73. M. Hartz, M. Lupini, Dilation theory in finite dimensions and matrix convexity (2019). ar**v:1910.03549

    Google Scholar 

  74. J.W. Helton, I. Klep, S. McCullough, The matricial relaxation of a linear matrix inequality. Math. Program. 138, 401–445 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  75. J.W. Helton, I. Klep, S. McCullough, Matrix convex hulls of free semialgebraic sets. Trans. Am. Math. Soc. 368, 3105–3139 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  76. J.W. Helton, I. Klep, S. McCullough, M. Schweighofer, Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions. Memoirs of the American Mathematical Society (American Mathematical Society, Providence, 2019)

    Google Scholar 

  77. D.R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B. 14, 2239–2249 (1976)

    Article  Google Scholar 

  78. J.A. Holbrook, Schur norms and the multivariate von Neumann inequality, in Operator Theory: Advances and Applications, vol. 127 (Birkhäuser, Basel, 2001)

    Google Scholar 

  79. Z. Hu, R. **a, S. Kais, A quantum algorithm for evolving open quantum dynamics on quantum computing devices (2019). ar**v:1904.00910

    Google Scholar 

  80. B. Huber, T. Netzer, A note on non-commutative polytopes and polyhedra (2019). ar**v:1809.00476

    Google Scholar 

  81. M. Izumi, E0-semigroups: around and beyond Arveson’s work. J. Oper. Theory 68, 335–363 (2012)

    MATH  Google Scholar 

  82. E.T.A. Kakariadis, O.M. Shalit, Operator algebras of monomial ideals in noncommuting variables. J. Math. Anal. Appl. 472, 738–813 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  83. I. Kaplansky, Modules over operator algebras. Am. J. Math. 75, 839–858 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  84. E. Katsoulis, C. Ramsey, Crossed Products of Operator Algebras. Memoirs of the American Mathematical Society (American Mathematical Society, Providence, 2019)

    Google Scholar 

  85. M. Kennedy, O.M. Shalit, Essential normality, essential norms and hyperrigidity. J. Funct. Anal. 270, 2812–2815 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  86. D. Keshari, N. Mallick, q-commuting dilation. Proc. Am. Math. Soc. 147, 655–669 (2019)

    Google Scholar 

  87. G. Knese, The von Neumann inequality for 3 × 3 matrices. Bull. Lond. Math. Soc. 48, 53–57 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  88. Ł. Kosiński, Three-point Nevanlinna-Pick problem in the polydisc. Proc. Lond. Math. Soc. 111, 887–910 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  89. K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory. Lecture Notes in Physics, vol. 190 (Springer, Berlin, 1983)

    Google Scholar 

  90. T.-L. Kriel, An introduction to matrix convex sets and free spectrahedra (2016). ar**v:1611.03103v6

    Google Scholar 

  91. B. Kümmerer, Markov dilations and W*-algberas. J. Funct. Anal. 63, 139–177 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  92. M. Laca, From endomorphisms to automorphisms and back: dilations and full corners. J. Lond. Math. Soc. 61, 893–904 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  93. E.C. Lance, Hilbert C*-modules: A Toolkit for Operator Algebraists, vol. 210 (Cambridge University Press, Cambridge, 1995)

    Book  MATH  Google Scholar 

  94. J. Levick, R.T.W. Martin, Matrix N-dilations of quantum channels. Oper. Matrices 12, 977–995 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  95. E. Levy, O.M. Shalit, Dilation theory in finite dimensions: the possible, the impossible and the unknown. Rocky Mountain J. Math. 44, 203–221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  96. C.K. Li, Y.T. Poon, Convexity of the joint numerical range. SIAM J. Matrix Anal. Appl. 21, 668–678 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  97. C.K. Li, Y.T. Poon, Interpolation by completely positive maps. Linear Multilinear Algebra 59, 1159–1170 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  98. D. Markiewicz, O.M. Shalit, Continuity of CP-semigroups in the point-strong topology. J. Operator Theory 64, 149–154 (2010)

    MathSciNet  MATH  Google Scholar 

  99. J.E. McCarthy, O.M. Shalit, Unitary N-dilations for tuples of commuting matrices. Proc. Am. Math. Soc. 14, 563–571 (2013)

    MathSciNet  MATH  Google Scholar 

  100. P.S. Muhly, B. Solel, Tensor algebras over C ∗-correspondences: representations, dilations, and C ∗-envelopes. J. Funct. Anal. 158, 389–457 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  101. P.S. Muhly, B. Solel, An algebraic characterization of boundary representations, in Operator Theory: Advances and Applications, vol. 104 (Birkhäuser, Basel, 1998), pp. 189–196

    MATH  Google Scholar 

  102. P.S. Muhly, B. Solel, Quantum Markov processes (correspondences and dilations). Int. J. Math. 13, 863–906 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  103. P.S. Muhly, B. Solel, Hardy algebras, W*-correspondences and interpolation theory. Math. Ann. 330, 353–415 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  104. P. Muhly, B. Solel, Quantum Markov semigroups (product systems and subordination). Int. J. Math. 18, 633–669 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  105. V. Müller, F-H. Vasilescu, Standard models for some commuting multioperators. Proc. Am. Math. Soc. 117, 979–989 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  106. B. Nagy, On contractions in Hilbert space. Acta Sci. Math 15, 87–92 (2013)

    MATH  Google Scholar 

  107. I.L. Nielsen, M.A. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)

    MATH  Google Scholar 

  108. D. Opela, A generalization of Andô’s theorem and Parrott’s example. Proc. Am Math. Soc. 134, 2703–2710 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  109. S. Pal, O.M. Shalit, Spectral sets and distinguished varieties in the symmetrized bidisc. J. Funct. Anal. 266, 5779–5800 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  110. S. Parrott, Unitary dilations for commuting contractions. Pac. J. Math. 34, 481–490 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  111. K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics, vol. 85 (Birkhäuser, Basel, 2012)

    Google Scholar 

  112. W.L. Paschke, Inner product modules over B ∗-algebras. Trans. Am. Math. Soc. 182, 443–468 (1973)

    MathSciNet  MATH  Google Scholar 

  113. B. Passer, Shape, scale, and minimality of matrix ranges. Trans. Am. Math. Soc. 372, 1451–1484 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  114. B. Passer, V.I. Paulsen, Matrix range characterization of operator system properties (preprint). ar**v:1912.06279

    Google Scholar 

  115. B. Passer, O.M. Shalit, Compressions of compact tuples. Linear Algebra Appl. 564, 264–283 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  116. B. Passer, O.M. Shalit, B. Solel, Minimal and maximal matrix convex sets. J. Funct. Anal. 274, 3197–3253 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  117. V.I. Paulsen, Representations of function algebras, abstract operator spaces, and Banach space geometry. J. Funct. Anal. 109(1), 113–129 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  118. V.I. Paulsen, Completely Bounded Maps and Operator Algebras (Cambridge University Press, Cambridge, 2002)

    MATH  Google Scholar 

  119. V.I. Paulsen, M. Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Cambridge Studies in Advanced Mathematics, vol. 152 (Cambridge University Press, Cambridge, 2016)

    Google Scholar 

  120. V.I. Paulsen, I.G. Todorov, M. Tomforde, Operator system structures on ordered spaces. Proc. Lond. Math. Soc. 102, 25–49 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  121. G. Pisier, Similarity Problems and Completely Bounded Maps. Lecture Notes of Mathematics, vol. 1618 (Springer, Berlin, 1996)

    Google Scholar 

  122. G. Pisier, Introduction to Operator Space Theory, vol. 294 (Cambridge University Press, Cambridge, 2003)

    Book  MATH  Google Scholar 

  123. G. Popescu, Isometric dilations for infinite sequences of noncommuting operators. Trans. Am. Math. Soc. 316, 523–536 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  124. G. Popescu, Von Neumann inequality for \((B(\mathcal {H})^n)_1\). Math. Scand. 68, 292–304 (1991)

    Google Scholar 

  125. G. Popescu, Poisson transforms on some C*-algebras generated by isometries. J. Funct. Anal. 161, 27–61 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  126. G. Popescu, Free holomorphic functions on the unit ball of \(B(\mathcal {H})^n\). J. Funct. Anal. 241, 268–333 (2006)

    Google Scholar 

  127. G. Popescu, Operator theory on noncommutative varieties. Ind. Univ. Math. J. 56, 389–442 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  128. G. Popescu, Berezin transforms on noncommutative polydomains. Trans. Am. Math. Soc. 368, 4357–4416 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  129. B. Prunaru, Lifting fixed points of completely positive semigroups. Integr. Equ. Oper. Theory 72, 219–222 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  130. M. Ptak, Unitary dilations of multiparameter semigroups of operators. Ann. Polon. Math. 45, 237–243 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  131. M. Rieffel, Induced representations of C*-algebras. Adv. Math. 13, 176–257 (1974)

    Article  MATH  Google Scholar 

  132. M. Rieffel, C*-algebras associated with irrational rotations. Pac. J. Math. 93, 415–429 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  133. F. Riesz, B. Sz.-Nagy, Functional Analysis (Dover, New York, 1990) (first published in 1955)

    Google Scholar 

  134. G. Salomon, O.M. Shalit, E. Shamovich, Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball. Trans. Am. Math. Soc. 370, 8639–8690 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  135. G. Salomon, O.M. Shalit, E. Shamovich, Algebras of noncommutative functions on subvarieties of the noncommutative ball: the bounded and completely bounded isomorphism problem. J. Funct. Anal. (2019). ar**v:1806.00410

    Google Scholar 

  136. D. Sarason, On spectral sets having connected complement. Acta Sci. Math. 26, 289–299 (1965)

    MathSciNet  MATH  Google Scholar 

  137. D. Sarason, Generalized interpolation in H ∞. Trans. Am. Math. Soc. 127, 179–203 (1967)

    MathSciNet  MATH  Google Scholar 

  138. J. Sarkar, Applications of Hilbert module approach to multivariable operator theory, in Operator Theory, ed. by D. Alpay (Springer, Berlin, 2014)

    Google Scholar 

  139. J. Sarkar, Operator theory on symmetrized bidisc. Ind. Univ. Math. J. 64, 847–873 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  140. D. SeLegue, Minimal Dilations of CP maps and C*-extension of the Szegö limit theorem. Ph.D Dissertation, University of California, Berkeley (1997)

    Google Scholar 

  141. O.M. Shalit, E0-dilation of strongly commuting CP0-semigroups. J. Funct. Anal. 255, 46–89 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  142. O.M. Shalit, What type of dynamics arise in E0-dilations of commuting Quantum Markov Semigroups?. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 11, 393–403 (2008)

    Article  MATH  Google Scholar 

  143. O.M. Shalit, Representing a product system representation as a contractive semigroup and applications to regular isometric dilations. Can. Math. Bull. 53, 550–563 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  144. O.M. Shalit, E-dilations of strongly commuting CP-semigroups (the nonunital case). Houston J. Math. 35(1), 203–232 (2011)

    MathSciNet  MATH  Google Scholar 

  145. O.M. Shalit, A sneaky proof of the maximum modulus principle. Am. Math. Month. 120(4), 359–362 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  146. O.M. Shalit, Operator theory and function theory in Drury-Arveson space and its quotients, in Operator Theory, ed. by D. Alpay (Springer, Berlin, 2014)

    Google Scholar 

  147. O.M. Shalit, A First Course in Functional Analysis (Chapman and Hall/CRC, Boca Raton, 2017)

    Book  MATH  Google Scholar 

  148. O.M. Shalit, M. Skeide, Three commuting, unital, completely positive maps that have no minimal dilation. Integr. Equ. Oper. Theory 71, 55–63 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  149. O.M. Shalit, M. Skeide, CP-semigroups and dilations, subproduct systems and superproduct systems: the multi-parameter case and beyond** (in preparation)

    Google Scholar 

  150. O.M. Shalit, B. Solel, Subproduct systems. Doc. Math. 14, 801–868 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  151. M. Słociński, Unitary dilation of two-parameter semi-groups of contractions. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 22, 1011–1014 (1974)

    MathSciNet  MATH  Google Scholar 

  152. A. Skalski, On isometric dilations of product systems of C*-correspondences and applications to families of contractions associated to higher-rank graphs. Ind. Univ. Math. J. 58, 2227–2252 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  153. A. Skalski, J. Zacharias, Wold decomposition for representations of product systems of C*-correspondences. Int. J. Math. 19, 455–479 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  154. M. Skeide, Hilbert modules and applications in quantum probability. Habilitationsschrift, Cottbus (2001). http://web.unimol.it/skeide/

  155. M. Skeide, Isometric dilations of representations of product systems via commutants. Int. J. Math. 19, 521–539 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  156. M. Skeide, Classification of E 0-semigroups by product systems. Mem. Am. Math. Soc. 240, 1137 (2016)

    MathSciNet  Google Scholar 

  157. B. Solel, Representations of product systems over semigroups an dilations of commuting CP maps. J. Funct. Anal. 235, 593–618 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  158. B. Solel, Regular dilations of representations of product systems. Math. Proc. R. Ir. Acad. 108, 89–110 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  159. W.F. Stinespring, Positive functions on C*-algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)

    MathSciNet  MATH  Google Scholar 

  160. J. Stochel, F.H. Szafraniec, Unitary dilation of several contractions, in Operator Theory: Advances and Applications, vol. 127 (Birkhäuser, Basel, 2001), pp. 585–598

    MATH  Google Scholar 

  161. E. Stroescu, Isometric dilations of contractions on Banach spaces. Pac. J. Math. 47, 257–262 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  162. F.H. Szafraniec, Murphy’s Positive definite kernels and Hilbert C*-modules sreorganized. Banach Center Publ. 89, 275–295 (2010)

    Article  MATH  Google Scholar 

  163. B. Sz.-Nagy, Sur les contractions de l’espace de Hilbert. Acta Sci. Math. 15, 87–92 (1953)

    Google Scholar 

  164. B. Sz.-Nagy, Extensions of linear transformations in Hilbert space which extend beyond this space. Appendix to F. Riesz and B. Sz.-Nagy, Functional Analysis, Ungar, New York, 1960. Translation of “Prolongements des transformations de l’espace de Hilbert qui sortent de cet espace”, Budapest (1955)

    Google Scholar 

  165. N.T. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory. J. Funct. Anal. 16, 83–100 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  166. A. Vernik, Dilations of CP-maps commuting according to a graph. Houston J. Math. 42, 1291–1329 (2016)

    MathSciNet  MATH  Google Scholar 

  167. F. vom Ende, G. Dirr, Unitary dilations of discrete-time quantum-dynamical semigroups. J. Math. Phys. 60, 1–17 (2019)

    Google Scholar 

  168. J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr. 4, 258–281 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  169. B. Xhabli, Universal operator system structures on ordered spaces and their applications. Thesis (Ph.D.) University of Houston (2009)

    Google Scholar 

  170. B. Xhabli, The super operator system structures and their applications in quantum entanglement theory. J. Funct. Anal. 262, 1466–1497 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  171. A. Zalar, Operator positivestellensatze for noncommutative polynomials positive on matrix convex sets. J. Math. Anal. Appl. 445, 32–80 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This survey paper grew out of the talk that I gave at the International Workshop on Operator Theory and its Applications (IWOTA) that took place in the Instituto Superior Técnico, Lisbon, Portugal, in July 2019. I am grateful to the organizers of IWOTA 2019 for inviting me to speak in this incredibly successful workshop, and especially to Amélia Bastos, for inviting me to contribute to these proceedings. I used a preliminary version of this survey as lecture notes for a mini-course that I gave in the workshop Noncommutative Geometry and its Applications, which took place in January 2020, in NISER, Bhubaneswar, India. I am grateful to the organizers Bata Krishna Das, Sutanu Roy and Jaydeb Sarkar, for the wonderful hospitality and the opportunity to speak and organize my thoughts on dilation theory. I also owe thanks to Michael Skeide and to Fanciszek Szafraniec, for helpful feedback on preliminary versions. Finally, I wish to thank an anonymous referee for several useful comments and corrections.

This project was partially supported by ISF Grant no. 195/16.

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  1. Faculty of Mathematics, Technion - Israel Institute of Technology, Haifa, Israel

    Orr Moshe Shalit

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  1. Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    M. Amélia Bastos

  2. Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal

    Luís Castro

  3. Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Lisboa, Portugal

    Alexei Yu. Karlovich

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Shalit, O.M. (2021). Dilation Theory: A Guided Tour. In: Bastos, M.A., Castro, L., Karlovich, A.Y. (eds) Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications, vol 282. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-51945-2_28

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