Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ℂ-VALUED FREE PROBABILITY ON A GRAPH VON NEUMANN ALGEBRA

  • Cho, Il-Woo (DEPARTMENT OF MATHEMATICS SAINT AMBROSE UNIVERSITY)
  • Published : 2010.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In [6] and [7], we introduced graph von Neumann algebras which are the (groupoid) crossed product algebras of von Neumann algebras and graph groupoids via groupoid actions. We showed that such crossed product algebras have the graph-depending amalgamated reduced free probabilistic properties. In this paper, we will consider a scalar-valued $W^*$-probability on a given graph von Neumann algebra. We show that a diagonal graph $W^*$-probability space (as a scalar-valued $W^*$-probability space) and a graph W¤-probability space (as an amalgamated $W^*$-probability space) are compatible. By this compatibility, we can find the relation between amalgamated free distributions and scalar-valued free distributions on a graph von Neumann algebra. Under this compatibility, we observe the scalar-valued freeness on a graph von Neumann algebra.

Keywords

References

  1. B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics, 184. Springer-Verlag, New York, 1998.
  2. I. Cho, Graph-matrices over additive graph groupoids, Submitted to JAMC.
  3. I. Cho, Group-freeness and certain amalgamated freeness, J. Korean Math. Soc. 45 (2008), no. 3, 597-609. https://doi.org/10.4134/JKMS.2008.45.3.597
  4. I. Cho, Direct producted $W^{*}$-probability spaces and corresponding amalgamated free stochastic integration, Bull. Korean Math. Soc. 44 (2007), no. 1, 131-150. https://doi.org/10.4134/BKMS.2007.44.1.131
  5. I. Cho, Measures on graphs and groupoid measures, Complex Anal. Oper. Theory 2 (2008), no. 1, 1-28. https://doi.org/10.1007/s11785-007-0031-0
  6. I. Cho, Graph von Neumann algebras, Acta Appl. Math. 95 (2007), no. 2, 95-134. https://doi.org/10.1007/s10440-006-9081-y
  7. I. Cho, Characterization of amalgamated free blocks of a graph von Neumann algebra, Complex Anal. Oper. Theory 1 (2007), no. 3, 367-398. https://doi.org/10.1007/s11785-007-0017-y
  8. I. Cho, Vertex-compressed algebras of a graph von Neumann algebra, Acta Appl. Math. (2008), To Appear.
  9. I. Cho, Operator Algebraic Quotient Structures of Graph von Neumann Algebras,CAOT, (2008), To Appear.
  10. I. Cho, and P. E. T. Jorgensen, $C^{*}$-algebras generated by partial isometries, JAMC, (2008), To Appear.
  11. I. Cho, and P. E. T. Jorgensen, $C^{*}$-subalgebras generated by partial isometries, JMP, (2008), To Appear.
  12. I. Cho, and P. E. T. Jorgensen, Applications in automata and graphs: Labeling operators in Hilbert space I, (2007) Submitted to Acta Appl. Math: Special Issues.
  13. I. Cho, and P. E. T. Jorgensen, Applications in automata and graphs: Labeling operators in Hilbert space II, (2008) Submitted to JMP.
  14. I. Cho, and P. E. T. Jorgensen, $C^{*}$-subalgebras generated by a single operator in B(H), (2008) Submitted to Acta Appl. Math: Special Issues.
  15. $C^{*}$-dynamical systems induced by partial isometries, (2008) Preprint.
  16. I. Cho, and P. E. T. Jorgensen, R. Diestel, Graph Theory: 3-rd edition, Graduate Texts in Mathematics, 173. Springer-Verlag, Berlin, 2005.
  17. R. Exel, Interaction, (2004) Preprint.
  18. R. Exel, A new look at the crossed-product of a $C^{*}$-algebra by an endomorphism, Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1733-1750. https://doi.org/10.1017/S0143385702001797
  19. R. Gliman, V. Shpilrain, and A. G. Myasnikov, Computational and Statistical Group Theory, Contemporary Mathematics, 298. American Mathematical Society, Providence, RI, 2002.
  20. V. F. R. Jones, Subfactors and Knots, CBMS Regional Conference Series in Mathematics, 80. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991.
  21. M. T. Jury and D. W. Kribs, Ideal structure in free semigroupoid algebras from directed graphs, J. Operator Theory 53 (2005), no. 2, 273-302.
  22. A. G. Myasnikov and V. Shapilrain, Group Theory, Statistics and Cryptography, Contemporary Mathematics, 360. American Mathematical Society, Providence, RI, 2004.
  23. A. Nica, R-transform in Free Probability, IHP course note, available at www.math.uwaterloo.ca/˜anica.
  24. A. Nica, D. Shlyakhtenko, and R. Speicher, R-cyclic families of matrices in free probability, J. Funct. Anal. 188 (2002), no. 1, 227-271. https://doi.org/10.1006/jfan.2001.3814
  25. A. Nica and R. Speicher, R-diagonal Pair–A Common Approach to Haar Unitaries and Circular Elements, www.mast.queensu.ca/˜speicher.
  26. F. Radulescu, Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index, Invent. Math. 115 (1994), no. 2, 347-389. https://doi.org/10.1007/BF01231764
  27. I. Raeburn, Graph Algebras, CBMS Regional Conference Series in Mathematics, 103. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005.
  28. B. Solel, You can see the arrows in a quiver operator algebra, J. Aust. Math. Soc. 77 (2004), no. 1, 111-122. https://doi.org/10.1017/S1446788700010181
  29. R. Speicher, Combinatorial theory of the free product with amalgamation and operatorvalued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627, x+88 pp.
  30. R. Speicher, Combinatorics of free probability theory IHP course note, available at www.mast.queensu.ca/˜speicher.
  31. D. Voiculescu, Operations on certain non-commutative operator-valued random variables, Asterisque No. 232 (1995), 243-275.
  32. D. Voiculescu, K. Dykemma, and A. Nica, Free Random Variables, CRM Monograph Series, 1. American Mathematical Society, Providence, RI, 1992.