DOI QR코드

DOI QR Code

DIRECT PRODUCTED W*-PROBABILITY SPACES AND CORRESPONDING AMALGAMATED FREE STOCHASTIC INTEGRATION

  • Cho, Il-Woo (Saint Ambrose University Department of Mathematics)
  • Published : 2007.02.28

Abstract

In this paper, we will define direct producted $W^*-porobability$ spaces over their diagonal subalgebras and observe the amalgamated free-ness on them. Also, we will consider the amalgamated free stochastic calculus on such free probabilistic structure. Let ($A_{j},\;{\varphi}_{j}$) be a tracial $W^*-porobability$ spaces, for j = 1,..., N. Then we can define the corresponding direct producted $W^*-porobability$ space (A, E) over its N-th diagonal subalgebra $D_{N}\;{\equiv}\;\mathbb{C}^{{\bigoplus}N}$, where $A={\bigoplus}^{N}_{j=1}\;A_{j}\;and\;E={\bigoplus}^{N}_{j=1}\;{\varphi}_{j}$. In Chapter 1, we show that $D_{N}-valued$ cumulants are direct sum of scalar-valued cumulants. This says that, roughly speaking, the $D_{N}-freeness$ is characterized by the direct sum of scalar-valued freeness. As application, the $D_{N}-semicircularityrity$ and the $D_{N}-valued$ infinitely divisibility are characterized by the direct sum of semicircularity and the direct sum of infinitely divisibility, respectively. In Chapter 2, we will define the $D_{N}-valued$ stochastic integral of $D_{N}-valued$ simple adapted biprocesses with respect to a fixed $D_{N}-valued$ infinitely divisible element which is a $D_{N}-free$ stochastic process. We can see that the free stochastic Ito's formula is naturally extended to the $D_{N}-valued$ case.

Keywords

direct producted $W^*-probability$ spaces over their diagonal subalgebras;$D_N$-freeness;$D_N$-semicircularity;$D_N$-valued infinitely divisibility;$D_N$-valued simple adapted biprocesses;$D_N$-valued free stochastic integrals;$It\^{o}'s$ formula;$D_N$-free brownian motions

References

  1. M. Anshelevich, Free Stochastic Measures via Noncrossing Partitions, Preprint, 1999
  2. I. Cho, Toeplitz Noncommutative Probability Spaces over Toeplitz Matricial Algebras, Preprint, 2002
  3. I. Cho, The Moment Series of the Generating Operator of $L(F_2)*_{L(K)}L(F_2) $ Preprint, 2003
  4. I. Cho, Graph von Neumann Algebras, ACTA Applied Math, (2007) To be appeared
  5. I. Cho, Group Freeness and Certain Amalgamated Freeness, (2007) Submitted to J. of KMS https://doi.org/10.4134/JKMS.2008.45.3.597
  6. I. Cho, The Characterization of Amalgamated Free Blocks of a Graph von Neumann Algebra, (2007) Submitted to JAMC
  7. K. J. Horadam, The word problem and related results for graph product groups, Proc. Amer. Math. Soc. 82 (1981), no. 2, 157-164
  8. A. Nica, R-transform in Free Probability, Lectures in the special semester 'Free probability theory and operator spaces', IHP, Paris, 1999
  9. A. Nica, R-transforms of free joint distributions and non-crossing partitions, J. Funct. Anal. 135 (1996), no. 2, 271-296 https://doi.org/10.1006/jfan.1996.0011
  10. R. Speicher, Combinatorics of Free Probability Theory, IHP course note
  11. R. Speicher, Free Calculus, Lecture Note for Summer School on Quantum Probability, Grenoble, 1998
  12. R. Speicher, Combinatorial theory of the free product with amalgamation and operator- valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627, 1-88 pp
  13. F. Radulescu, Singularity of the radial subalgebra of $L(F_N)$ and the Pukanszky invariant, Pacific J. Math. 151 (1991), no. 2, 297-306 https://doi.org/10.2140/pjm.1991.151.297
  14. D. Voiculescu, K. Dykemma, and A. Nica, Free random variables, A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups, CRM Monograph Series, 1. American Mathematical Society, Providence, RI, 1992

Cited by

  1. C *-Subalgebras Generated by a Single Operator in B(H) vol.108, pp.3, 2009, https://doi.org/10.1007/s10440-009-9478-5
  2. Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I vol.107, pp.1-3, 2009, https://doi.org/10.1007/s10440-008-9380-6
  3. CLASSIFICATION ON ARITHMETIC FUNCTIONS AND CORRESPONDING FREE-MOMENT L-FUNCTIONS vol.52, pp.3, 2015, https://doi.org/10.4134/BKMS.2015.52.3.717
  4. ℂ-VALUED FREE PROBABILITY ON A GRAPH VON NEUMANN ALGEBRA vol.47, pp.3, 2010, https://doi.org/10.4134/JKMS.2010.47.3.601
  5. Applications of automata and graphs: Labeling operators in Hilbert space. II. vol.50, pp.6, 2009, https://doi.org/10.1063/1.3141524
  6. C *-algebras generated by partial isometries vol.26, pp.1-2, 2008, https://doi.org/10.1007/s12190-007-0009-0