Abstract
After completing my Ph.D. on some problems in the theory of uniform distribution in Vienna in early 1968 I felt in need of a change of mathematical direction and started reading K.R. Parthasarathy’s book Probability measures on metric spaces ([20]), which impressed me so much that I decided to go to Manchester and work with Parthasarathy. When I eventually arrived there in 1969 I discovered that Partha—as I learned to call him—had moved on to the mathematical foundations of quantum mechanics and quantum field theory, and was working on certain connections between continuous tensor products and classical central limit theorems. The fact that he was starting on a new mathematical venture, combined with his extraordinary willingness to share problems and ideas, allowed me to begin working with him on these problems immediately after my arrival. It was a happy time for me: his and Shyama’s kindness, warmth and hospitality (as well as Shyama’s fiery Indian cooking) all contributed to the fond memories I still have of my stay in Manchester. Much to my regret Partha and his family went back to India in 1970, and I went on to Bedford College in London. In 1972 Partha came back to England to spend some time at Warwick and I joined him there for a while continuing our earlier collaboration on topics related to continuous tensor products and noncommutative versions of the central limit theorem (cf. [26]).
Dedicated to K.R. Parthasarathy on the occasion of his 60th birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Araki, Factorisable representations of current algebra, Publ. Res. Inst. Math. Sci. 5 (1970), 361–422.
H. Araki and E.J. Woods, Complete Boolean algebras of type I factors, Publ. Res. Inst. Math. Sci. 2 (1966), 157–242.
A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1(1981), 431–450.
A. Connes and B. Weiss, Property T and asymptotically invariant sequences, Israel J. Math. 37 (1980), 209–210.
H.A. Dye, On groups of measure preserving transformations I, Amer. J. Math. 81 (1959), 119–159.
———, On groups of measure preserving transformations II, Amer. J. Math. 85 (1963), 551–576.
———, On the ergodic mixing theorem, Trans. Amer. Math. Soc. 118 (1965), 123–130.
W. Geller and J. Propp, The projective fundamental group of a ℤ 2 –shift, Ergod. Th. & Dynam. Sys. 15 (1995), 1091–1118.
B.V. Gnedenko and A.N. Kolmogorov, Limit distributions for sums of independent random variables, Addison–Wesley, Reading, Massachusetts, 1954.
H. Helson and W. Parry, Cocycles and spectra, Ark. Mat. 16 (1978), 195–206.
T. Hida, Stationary stochastic processes, Mathematical Notes, Princeton University Press, Princeton, N.J., 1970.
J.W. Kammeyer, A complete classification of two–point extensions of a multidimensional Bernoulli shift, Doctoral Dissertation, University of Maryland, 1988.
J.W. Kammeyer, A classification of the finite extensions of a multidimensional Bernoulli shift, Trans. Amer. Math. Soc. 335 (1993), 443–475.
P.W. Kasteleyn, The statistics of dimers on a lattice. I, Phys. D 27 (1961), 1209–1225.
A. Katok, Constructions in ergodic theory.
A. Katok and R.J. Spatzier, Subeliiptic estimates of polynomall differential operators and applications to rigidity of abelian actions, Math. Res. Lett. 1 (1994), 193–202.
———, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 131–156.
A. Livshitz, Cohomology of dynamical system, Izv. Akad. Nauk SSSR Ser. Mat. 6 (1972), 1278–1301.
W. Parry, A note on cocycles in ergodic theory, Compositio Math. 28 (1966), 303–330.
K.R. Parthasarathy, Probability measures on metric spaces, Academic Press, New York-London, 1967.
———, Multipliers on locally compact groups, Lecture Notes in Mathematics, vol. 93, Springer Verlag, Berlin-Heidelberg-New York, 1969.
———, Infinitely divisible representations and positive definite functions on a compact group, Comm. Math. Phys. 16 (1970), 148–156.
K.R. Parthasarathy and K. Schmidt, Infinitely divisible projective representations, cocycles and Levy-Khnnchine formula on locally compact groups, Preprint (1970).
———, Factorisabee representations of current groups and the Arak-Woods imbedding theorem, Acta Math. 128 (1972), 53–71.
———, Positive definite kernels, continuous tensor products and central limit theorems of probability theory, Lecture Notes in Mathematics, vol. 272, Springer Verlag, Berlin-Heidelberg-New York, 1972.
———, Stable positive definite functions, Trans. Amer. Math. Soc. 203 (1975), 161–174.
———, A new method for constructing factorisable representations of current groups and current algebras, Comm. Math. Phys. 50 (1976), 167–175.
———, On the cohomology of a hyperfinite action, Monatsh. Math. 84 (1977), 37–48.
K. Schmidt, Cocycles on ergodic transformation groups, MacMillan (India), Delhi, 1977.
———, Amenability, Kazhdan’ property T, strong ergodicity and invariant means for ergodic group actions, Ergod. Th. & Dynam. Sys. 1 (1981), 223–236.
K. Schmidt, Dynamical systems of algebraic origin, Birkhauser Verlag, Basel-Berlin, 1995.
———, The cohomology of higher-dimensional shifts of finite type, Pacific J. Math. 170 (1995), 237–270.
———, On the cohomology of algebraic Zd-actions with values in compact Lie groups, Preprint (1995).
———, Tilings, fundamental cocycles and fundamental groups of symbolic Zd-actions. Preprint (1996).
G. Segal, Unitary representations of some infinite dimensional groups, Comm. Math. Phys. 80 (1981), 301–342.
R.F. Streater, Current commutation relations, continuous tensor products and infinitely divisible group representations, Rend. Sc. Intern. E. Fermi 45 (1969), 247–263.
V.S. Varadarajan, Geometry of Quantum Theory (2nd edition), Springer Verlag, Berlin-Heidelberg-New York, 1985.
R.J. Zimmer, Ergodic theory and semisimple Lie groups, Birkhauser Verlag, Basel-Berlin, 1984.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Hindustan Book Agency
About this chapter
Cite this chapter
Schmidt, K. (1996). From infinitely divisible representations to cohomological rigidity. In: Bhatia, R. (eds) Analysis, Geometry and Probability. Texts and Readings in Mathematics, vol 10. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-80250-87-8_10
Download citation
DOI: https://doi.org/10.1007/978-93-80250-87-8_10
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-12-8
Online ISBN: 978-93-80250-87-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)