Additive units of product systems

Rajarama Bhat, B.; Lindsay, J. Martin; Mukherjee, Mithun

doi:10.1090/tran/7092

Additive units of product systems
HTML articles powered by AMS MathViewer

by B. V. Rajarama Bhat, J. Martin Lindsay and Mithun Mukherjee PDF

Trans. Amer. Math. Soc. 370 (2018), 2605-2637 Request permission

Abstract:

We introduce the notion of additive units, or “addits”, of a pointed Arveson system and demonstrate their usefulness through several applications. By a pointed Arveson system we mean a spatial Arveson system with a fixed normalised reference unit. We show that the addits form a Hilbert space whose codimension-one subspace of “roots” is isomorphic to the index space of the Arveson system and that the addits generate the type I part of the Arveson system. Consequently the isomorphism class of the Hilbert space of addits is independent of the reference unit. The addits of a pointed inclusion system are shown to be in natural correspondence with the addits of the generated pointed product system. The theory of amalgamated products is developed using addits and roots, and an explicit formula for the amalgamation of pointed Arveson systems is given, providing a new proof of its independence of the particular reference units. (This independence justifies the terminology “spatial product” of spatial Arveson systems.) Finally a cluster construction for inclusion subsystems of an Arveson system is introduced, and we demonstrate its correspondence with the action of the Cantor–Bendixson derivative in the context of the random closed set approach to product systems due to Tsirelson and Liebscher.

References

William Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc. 80 (1989), no. 409, iv+66. MR 987590, DOI 10.1090/memo/0409
William Arveson, Continuous analogues of Fock space. IV. Essential states, Acta Math. 164 (1990), no. 3-4, 265–300. MR 1049159, DOI 10.1007/BF02392756
William Arveson, Noncommutative dynamics and $E$-semigroups, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1978577, DOI 10.1007/978-0-387-21524-2
B. V. Rajarama Bhat, Minimal dilations of quantum dynamical semigroups to semigroups of endomorphisms of $C^\ast$-algebras, J. Ramanujan Math. Soc. 14 (1999), no. 2, 109–124. MR 1727708
B. V. Rajarama Bhat, Volkmar Liebscher, Mithun Mukherjee, and Michael Skeide, The spatial product of Arveson systems is intrinsic, J. Funct. Anal. 260 (2011), no. 2, 566–573. MR 2737413, DOI 10.1016/j.jfa.2010.09.001
B. V. R. Bhat, V. Liebscher, and M. Skeide, A problem of powers and the product of spatial product systems, Quantum probability and related topics, QP–PQ: Quantum Probab. White Noise Anal., vol. 23, World Sci. Publ., Hackensack, NJ, 2008, pp. 93–106. MR 2590656, DOI 10.1142/9789812835277_{0}007
B. V. Rajarama Bhat, Volkmar Liebscher, and Michael Skeide, Subsystems of Fock need not be Fock: spatial CP-semigroups, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2443–2456. MR 2607874, DOI 10.1090/S0002-9939-10-10260-3
B. V. Rajarama Bhat and Mithun Mukherjee, Inclusion systems and amalgamated products of product systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), no. 1, 1–26. MR 2646788, DOI 10.1142/S0219025710003924
B. V. Rajarama Bhat and Michael Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), no. 4, 519–575. MR 1805844, DOI 10.1142/S0219025700000261
J. L. B. Cooper, One-parameter semigroups of isometric operators in Hilbert space, Ann. of Math. (2) 48 (1947), 827–842. MR 27129, DOI 10.2307/1969382
Alain Guichardet, Symmetric Hilbert spaces and related topics, Lecture Notes in Mathematics, Vol. 261, Springer-Verlag, Berlin-New York, 1972. Infinitely divisible positive definite functions. Continuous products and tensor products. Gaussian and Poissonian stochastic processes. MR 0493402, DOI 10.1007/BFb0070306
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
K. Kuratowski, Sur les décompositions semi-continus d’espaces métriques compacts, Fund. Math. 11 (1928), 169–183.
Volkmar Liebscher, Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces, Mem. Amer. Math. Soc. 199 (2009), no. 930, xiv+101. MR 2507929, DOI 10.1090/memo/0930
Volkmar Liebscher, The relation of spatial product and tensor product of arveson systems-the random set point of view, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18 (2015), no. 4, 1550029, 22 pp.
J. Martin Lindsay, Quantum stochastic analysis—an introduction, Quantum independent increment processes. I, Lecture Notes in Math., vol. 1865, Springer, Berlin, 2005, pp. 181–271. MR 2132095, DOI 10.1007/11376569_{3}
Oliver T. Margetts and R. Srinivasan, Invariants for $E_0$-semigroups on $II_1$ factors, Commun. Math. Phys.323 (2013), 1155–1184, DOI 10.1007/S00220-013-1790-2.
Oliver T. Margetts and R. Srinivasan, Cohomology for spatial super-product systems, arXiv:OA 1608.00610v1.
Daniel Markiewicz, On the product system of a completely positive semigroup, J. Funct. Anal. 200 (2003), no. 1, 237–280. MR 1974096, DOI 10.1016/S0022-1236(02)00168-4
Paul S. Muhly and Baruch Solel, Quantum Markov processes (correspondences and dilations), Internat. J. Math. 13 (2002), no. 8, 863–906. MR 1928802, DOI 10.1142/S0129167X02001514
Mithun Mukherjee, Index computation for amalgamated products of product systems, Banach J. Math. Anal. 5 (2011), no. 1, 148–166. MR 2738527, DOI 10.15352/bjma/1313362987
K. R. Parthasarathy, An introduction to quantum stochastic calculus, Monographs in Mathematics, vol. 85, Birkhäuser Verlag, Basel, 1992. MR 1164866, DOI 10.1007/978-3-0348-8641-3
Robert T. Powers, A nonspatial continuous semigroup of $*$-endomorphisms of ${\mathfrak {B}}({\mathfrak {H}})$, Publ. Res. Inst. Math. Sci. 23 (1987), no. 6, 1053–1069. MR 935715, DOI 10.2977/prims/1195175872
Robert T. Powers, New examples of continuous spatial semigroups of $\ast$-endomorphisms of $\mathfrak {B}(\mathfrak {H})$, Internat. J. Math. 10 (1999), no. 2, 215–288. MR 1687149, DOI 10.1142/S0129167X99000094
Robert T. Powers, Addition of spatial $E_0$-semigroups, Operator algebras, quantization, and noncommutative geometry, Contemp. Math., vol. 365, Amer. Math. Soc., Providence, RI, 2004, pp. 281–298. MR 2106824, DOI 10.1090/conm/365/06707
Orr Moshe Shalit and Baruch Solel, Subproduct systems, Doc. Math. 14 (2009), 801–868. MR 2608451
Michael Skeide, Classification of $E_0$-semigroups by product systems, Mem. Amer. Math. Soc. 240 (2016), no. 1137, vi+126. MR 3460113, DOI 10.1090/memo/1137
Michael Skeide, Commutants of von Neumann modules, representations of $\scr B^a(E)$ and other topics related to product systems of Hilbert modules, Advances in quantum dynamics (South Hadley, MA, 2002) Contemp. Math., vol. 335, Amer. Math. Soc., Providence, RI, 2003, pp. 253–262. MR 2029630, DOI 10.1090/conm/335/06015
Michael Skeide, A simple proof of the fundamental theorem about Arveson systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), no. 2, 305–314. MR 2235551, DOI 10.1142/S0219025706002378
Michael Skeide, The index of (white) noises and their product systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), no. 4, 617–655. MR 2282723, DOI 10.1142/S0219025706002573
Michael Skeide, The Powers sum of spatial CPD-semigroups and CP-semigroups, Noncommutative harmonic analysis with applications to probability II, Banach Center Publ., vol. 89, Polish Acad. Sci. Inst. Math., Warsaw, 2010, pp. 247–263. MR 2730893, DOI 10.4064/bc89-0-17
Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici, and László Kérchy, Harmonic analysis of operators on Hilbert space, Revised and enlarged edition, Universitext, Springer, New York, 2010. MR 2760647, DOI 10.1007/978-1-4419-6094-8
W. A. Sutherland, Introduction to metric and topological spaces, Clarendon Press, Oxford, 1975. MR 0442869
B. Tsirelson, Automorphisms of the type $II_1$ Arveson system of Warren’s noise, arXiv:math/0612303v1.
Boris Tsirelson, From random sets to continuous tensor products: answers to three questions of W. Arveson, arXiv:math/0001070v1.
Boris Tsirelson, From slightly coloured noises to unitless product systems, arXiv:FA 0006165 vl.
Boris Tsirelson, Non-isomorphic product systems, Advances in quantum dynamics (South Hadley, MA, 2002) Contemp. Math., vol. 335, Amer. Math. Soc., Providence, RI, 2003, pp. 273–328. MR 2029632, DOI 10.1090/conm/335/06017
Boris Tsirelson, On automorphisms of type II Arveson systems (probabilistic approach), New York J. Math. 14 (2008), 539–576. MR 2448659