Group-freeness and certain amalgamated freeness
1
조회
0
다운로드
2
인용
0
참고문헌
키워드
초록
In this paper, we will consider certain amalgamated free product structure in crossed product algebras. Let M be a von Neumann algebra acting on a Hilbert space H and G, a group and let α : G → AutM be an action of G on M, where AutM is the group of all automorphisms on M: Then the crossed product M = M ×α G of M and G with respect to ® is a von Neumann algebra acting on H l²(G), generated by M and {ug}g∈G, where ug is the unitary representation of g on l²(G), We show that M ×α(G₁* G₂) = (M ×α G₁) *M (M ×α G₂). We compute moments and cumulants of operators in M. By doing that, we can verify that there is a close relation between Group Freeness and Amalgamated Freeness under the crossed product. As an application, we can show that if FN is the free group with N-generators, then the crossed product algebra LM(Fn) ≡ M ×α Fn satisfies that LM(Fn) = LM(Fk₁ ) *M LM(Fk₂ ), whenever n = k₁ + k₂ for n, k₁, k₂∈ N.
Abstract
In this paper, we will consider certain amalgamated free product structure in crossed product algebras. Let M be a von Neumann algebra acting on a Hilbert space H and G, a group and let α : G → AutM be an action of G on M, where AutM is the group of all automorphisms on M: Then the crossed product M = M ×α G of M and G with respect to ® is a von Neumann algebra acting on H l²(G), generated by M and {ug}g∈G, where ug is the unitary representation of g on l²(G), We show that M ×α(G₁* G₂) = (M ×α G₁) *M (M ×α G₂). We compute moments and cumulants of operators in M. By doing that, we can verify that there is a close relation between Group Freeness and Amalgamated Freeness under the crossed product. As an application, we can show that if FN is the free group with N-generators, then the crossed product algebra LM(Fn) ≡ M ×α Fn satisfies that LM(Fn) = LM(Fk₁ ) *M LM(Fk₂ ), whenever n = k₁ + k₂ for n, k₁, k₂∈ N.
참고문헌 (22)
- G. C. Bell, "Growth of the asymptotic dimension function for groups", ~ (2005)
- I. Cho, "Graph von Neumann algebras", Acta Applicandae Mathematicae, 95, pp.95~134 (2007)
- I. Cho, "Characterization of amalgamated free blocks of a graph von Neumann algebra", Complex Analysis and Operator Theory, 1, pp.367~398 (2007)
- Ilwoo Cho, "Direct producted $W^{*}$-probability spaces and corresponding amalgamated free stochastic integration", 대한수학회지, 44(1), pp.131~150 (2007), 대한수학회
- R. Gliman, "Computational and Statistical Group Theory, Contemporary Mathematics", pp.298~ (2002), American Mathematical Society
- V. Jones, "Subfactors and Knots, CBMS Regional Conference Series in Mathematics", pp.80~ (1991), Published for the Conference Board of the Mathematical Sciences
- M. T. Jury, "Ideal structure in free semigroupoid algebras from directed graphs", Journal of Operator Theory, 53(2), pp.273~302 (2005)
- D. W. Kribs, "Free semigroupoid algebras", Journal of The Ramanujan Mathematical Society, 19(2), pp.117~159 (2004)
- A. G. Myasnikov, "Group Theory, Statistics, and Cryptography, Contemporary Mathematics", pp.360~ (2004), American Mathematical Society
- A. Nica, "R-cyclic families of matrices in free probability", Journal of Functional Analysis, 188(1), pp.227~271 (2002)
- F. Radulescu, "Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index", Inventiones mathematicae, 115(2), pp.347~389 (1994)
- D. Shlyakhtenko, "Some applications of freeness with amalgamation", Journal für die reine und angewandte Mathematik, 500, pp.191~212 (1998)
- D. Shlyakhtenko, "A-valued semicircular systems", Journal of Functional Analysis, 166(1), pp.1~47 (1999)
- P. Sniady, "Continuous family of invariant subspaces for R-diagonal operators", Inventiones mathematicae, 146(2), pp.329~363 (2001)
- B. Solel, "You can see the arrows in a quiver operator algebra", Journal of the Australian Mathematical Society, 77(1), pp.111~122 (2004)
- R. Speicher, "Combinatorial theory of the free product with amalgamation and operatorvalued free probability theory", Memoirs of the American Mathematical Society, 132(627), pp.88~ (1998)
- J. Stallings, "Centerless groups–an algebraic formulation of Gottlieb’s theorem", Topology, 4, pp.129~134 (1965)
- D. Voiculescu, "Operations on certain non-commutative operator-valued random variables", Asterisque, (232), pp.243~275 (1995)
- D. Voiculescu, "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" (1992), American Mathematical Society