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On the equational theory of projection lattices of finite von neumann factors

Published online by Cambridge University Press:  12 March 2014

Christian Herrmann
Christian Herrmann*
Affiliation:
Technische Universität Darmstadt, Fb Mathematik, Schlossgartenstr. 7, D 64289 Darmstadt, Germany. E-mail: herrmann@mathematik.tu-darmstadt.de
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Abstract

For a finite von Neumann algebra factor M, the projections form a modular ortholattice L(M). We show that the equational theory of L(M) coincides with that of some resp. all L(ℂn×n ) and is decidable. In contrast, the uniform word problem for the variety generated by all L(ℂn×n) is shown to be undecidable.

Keywords

von Neumann algebraprojection latticecontinuous geometryequational theory
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 1102 - 1110
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1]Bennett, M. K., Biatomic lattices, Algebra Universalis, vol. 24 (1987), pp. 60–73.CrossRefGoogle Scholar
[2]Berberian, Sterling K., The regular ring of a finite A W* -algebra, Annals of Mathematics, vol. 65 (1957), pp. 224–240.CrossRefGoogle Scholar
[3]Berberian, Sterling K., The maximal ring of quotients of a finite von Neumann algebra. Rocky Mountain Journal of Mathematics, vol. 12 (1982), pp. 149–164.CrossRefGoogle Scholar
[4]Birkhoff, Garret, Lattice theory, American Mathematical Society, Providence, 1967.Google Scholar
[5]Birkhoff, Garret and von Neumann, John, The logic of quantum mechanics. Annals of Mathematics, vol. 37 (1936), pp. 823–843.CrossRefGoogle Scholar
[6]Burris, Stanley and Sankappanavar, H. P., A course in universal algebra, Springer, Berlin, 1981.CrossRefGoogle Scholar
[7]Dunn, J. Michael, Hagge, Tobias J., Moss, Lawrence S., and Wang, Zhenghan, Quantum logic as motivated by quantum computing, this Journal, vol. 70 (2005), pp. 353–359.Google Scholar
[8]Giudici, Luca, Dintorni del teorema di coordinatizatione di von Neumann, Ph.D. thesis. Universita di Milano, 1995.Google Scholar
[9]Goodearl, Kenneth R., Von Neumann regular rings, Krieger, Malabar, 1991.Google Scholar
[10]Gross, Herbert, Quadratic forms in infinite-dimensional vector spaces, Birkäuser, Basel, 1979.Google Scholar
[11]Gurevich, Yuri, The word problem for certain classes of semigroups, Algebra and Logic, vol. 5 (1966), pp. 25–35.Google Scholar
[12]Gurevich, Yuri and Lewis, Harry R., The word problem for cancellation semigroups with zero. this Journal, vol. 49 (1984), pp. 184–191.Google Scholar
[13]Hagge, Tobias J., QL(ℂn) determines n, this Journal, vol. 72 (2007), pp. 1194–1196.Google Scholar
[14]Herrmann, Christian, Complemented modular lattices with involution and orthogonal geometry. Algebra Universalis, vol. 61 (2009), p. 26.CrossRefGoogle Scholar
[15]Herrmann, Christian, Generators for complemented modular lattices and the von Neumann-Jonsson coordinatization theorems, to appear in Algebra Universalis, 2009.Google Scholar
[16]Herrmann, Christian, Micol, Florence, and Roddy, Michael S., On n-distributive modular ortholattices, Algebra Universalis, vol. 53 (2005), pp. 143–147.CrossRefGoogle Scholar
[17]Holland, Samuel S., Current interest in orthomodular lattices, Trends in lattice theory (Abbott, J. C., editor), van Nostrand Reinhold, Cincinatti, 1970, pp. 41–116.Google Scholar
[18]Huhn, András P., Schwach distributive Verbände I, Acta Scientiarum Mathematicarum, vol. 33 (1972), pp. 297–305.Google Scholar
[19]Hutchinson, George, Recursively unsolvable word problems of modular lattices and diagram chasing, Journal of Algebra, vol. 26 (1973), pp. 385–399.CrossRefGoogle Scholar
[20]Kaplansky, Irving, Von Neumann's characterization of factors of type II1, Collected works of John von Neumann, vol. III, Pergamon Press, Oxford, 1963, pp. 562–563.Google Scholar
[21]Lipshttz, Leonid L., The undecidablility of the wordproblems for projective geometries and modular lattices, Transactions of the American Mathematical Society, vol. 193 (1974), pp. 171–180.CrossRefGoogle Scholar
[22]Maeda, Fumitomo, Kontinuierliche Geometrien, Springer, Berlin, 1958.CrossRefGoogle Scholar
[23]Murray, Francis J. and von Neumann, John, On rings of operators, Annals of Mathematics, vol. 37 (1936), pp. 116–229.CrossRefGoogle Scholar
[24]Murray, Francis J. and von Neumann, John, On rings of operators IV, Annals of Mathematics, vol. 44 (1943), pp. 761–808.CrossRefGoogle Scholar
[25]Rédei, Miklós, Quantum logic in algebraic approach, Kluwer, Dordrecht, 1998.CrossRefGoogle Scholar
[26]Roddy, Michael S., On the word problem for orthocomplemented modular lattices, Canadian Journal of Mathematics, vol. 61 (1989), pp. 961–1004.CrossRefGoogle Scholar
[27]Tarski, Alfred, A decision method for elementary algebra and geometry, RAND Corporation, Santa Monica, 1948.Google Scholar
[28]von Neumann, John, Continuous geometry, Princeton University Press, Princeton, 1960.Google Scholar
[29]von Neumann, John, Continuous geometries with a transition probability, American Mathematical Society, Providence, 1981.CrossRefGoogle Scholar