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http://dx.doi.org/10.4134/JKMS.2010.47.3.601

ℂ-VALUED FREE PROBABILITY ON A GRAPH VON NEUMANN ALGEBRA  

Cho, Il-Woo (DEPARTMENT OF MATHEMATICS SAINT AMBROSE UNIVERSITY)
Publication Information
Journal of the Korean Mathematical Society / v.47, no.3, 2010 , pp. 601-631 More about this Journal
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
graph groupoids; crossed products; graph von Neumann algebras; $w^*$-probability spaces;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 0
연도 인용수 순위
1 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.
2 I. Cho, and P. E. T. Jorgensen, Applications in automata and graphs: Labeling operators in Hilbert space II, (2008) Submitted to JMP.
3 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.
4 $C^{*}$-dynamical systems induced by partial isometries, (2008) Preprint.
5 I. Cho, and P. E. T. Jorgensen, R. Diestel, Graph Theory: 3-rd edition, Graduate Texts in Mathematics, 173. Springer-Verlag, Berlin, 2005.
6 R. Exel, Interaction, (2004) Preprint.
7 A. Nica, R-transform in Free Probability, IHP course note, available at www.math.uwaterloo.ca/˜anica.
8 A. Nica, D. Shlyakhtenko, and R. Speicher, R-cyclic families of matrices in free probability, J. Funct. Anal. 188 (2002), no. 1, 227-271.   DOI   ScienceOn
9 A. Nica and R. Speicher, R-diagonal Pair–A Common Approach to Haar Unitaries and Circular Elements, www.mast.queensu.ca/˜speicher.
10 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.   DOI
11 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.
12 I. Cho, Direct producted $W^{*}$-probability spaces and corresponding amalgamated free stochastic integration, Bull. Korean Math. Soc. 44 (2007), no. 1, 131-150.   DOI   ScienceOn
13 I. Cho, Measures on graphs and groupoid measures, Complex Anal. Oper. Theory 2 (2008), no. 1, 1-28.   DOI
14 I. Cho, Graph von Neumann algebras, Acta Appl. Math. 95 (2007), no. 2, 95-134.   DOI
15 I. Cho, Characterization of amalgamated free blocks of a graph von Neumann algebra, Complex Anal. Oper. Theory 1 (2007), no. 3, 367-398.   DOI
16 I. Cho, Vertex-compressed algebras of a graph von Neumann algebra, Acta Appl. Math. (2008), To Appear.
17 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.
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.   DOI   ScienceOn
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 A. G. Myasnikov and V. Shapilrain, Group Theory, Statistics and Cryptography, Contemporary Mathematics, 360. American Mathematical Society, Providence, RI, 2004.
22 I. Cho, Group-freeness and certain amalgamated freeness, J. Korean Math. Soc. 45 (2008), no. 3, 597-609.   DOI   ScienceOn
23 B. Solel, You can see the arrows in a quiver operator algebra, J. Aust. Math. Soc. 77 (2004), no. 1, 111-122.   DOI
24 D. Voiculescu, Operations on certain non-commutative operator-valued random variables, Asterisque No. 232 (1995), 243-275.
25 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.
26 R. Speicher, Combinatorics of free probability theory IHP course note, available at www.mast.queensu.ca/˜speicher.
27 D. Voiculescu, K. Dykemma, and A. Nica, Free Random Variables, CRM Monograph Series, 1. American Mathematical Society, Providence, RI, 1992.
28 B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics, 184. Springer-Verlag, New York, 1998.
29 I. Cho, Graph-matrices over additive graph groupoids, Submitted to JAMC.
30 I. Cho, Operator Algebraic Quotient Structures of Graph von Neumann Algebras,CAOT, (2008), To Appear.
31 I. Cho, and P. E. T. Jorgensen, $C^{*}$-algebras generated by partial isometries, JAMC, (2008), To Appear.
32 I. Cho, and P. E. T. Jorgensen, $C^{*}$-subalgebras generated by partial isometries, JMP, (2008), To Appear.