Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$ K\sb 0$ of certain subdiagonal subalgebras of von Neumann algebras


Author: Richard Baker
Journal: Proc. Amer. Math. Soc. 116 (1992), 13-19
MSC: Primary 46L80; Secondary 19A49
DOI: https://doi.org/10.1090/S0002-9939-1992-1093591-7
MathSciNet review: 1093591
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that $ {K_0}$ of any finite maximal subdiagonal subalgebra of a separably acting finite von Neumann algebra is isomorphic to $ {K_0}$ of the diagonal of the subalgebra. It results that $ {K_0}$ of any finite, $ \sigma $-weakly closed, maximal triangular subalgebra of a separably acting finite von Neumann algebra is isomorphic to $ {K_0}$ of the diagonal of the subalgebra, provided that the diagonal of the subalgebra is a Cartan subalgebra of the von Neumann algebra. In addition, given any separably acting type $ {\text{II}_1}$ factor $ \mathcal{M}$, we explicitly compute $ {K_0}$ of those triangular subalgebras $ \mathcal{T}$ of $ \mathcal{M}$ that have the property that there exists a UHF subalgebra $ \mathcal{A}$ of $ \mathcal{M}$ and a standard triangular UHF algebra $ \mathcal{S}$ in $ \mathcal{A}$ such that $ \mathcal{A}$ is $ \sigma $-weakly dense in $ \mathcal{M}$ and $ \mathcal{T}$ is the $ \sigma $-weak closure of $ \mathcal{S}$.


References [Enhancements On Off] (What's this?)

  • [1] W. B. Arveson, Analyticity in operator algebras, Amer. J. Math. 89 (1967), 578-642. MR 0223899 (36:6946)
  • [2] R. L. Baker, A certain class of triangular algebras in type $ {\text{II}_1}$ hyperfinite factors, Proc. Amer. Math. Soc. 112 (1991), 163-169. MR 1049840 (91h:46106)
  • [3] J. Dixmier, Von Neumann algebras, North Holland, Amsterdam, New York, and Oxford, 1981. MR 641217 (83a:46004)
  • [4] P. S. Muhly, K.-S. Saito, and B. Solel, Coordinates for triangular operators algebras, Ann. of Math. (2) 127 (1988), 245-278. MR 932297 (89h:46088)
  • [5] F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2) 44 (1943), 716-808. MR 0009096 (5:101a)
  • [6] D. R. Pitts, On the $ {K_0}$ groups of nest algebras, preprint.
  • [7] -, Factorization problems for nests: factorization methods and characterizations of the universal factorization property, J. Funct. Anal. 79 (1988), 57-90. MR 950084 (90a:46160)
  • [8] J. R. Peters and B. H. Wagner, Triangular AF algebras and nest subalgebras of UHF algebras, preprint. MR 1191255 (94c:46116)
  • [9] C. Qiu, $ K$-theory of analytic crossed products, Rocky Mountain J. Math. (to appear). MR 1201109 (94d:46069)
  • [10] J. Tomiyama, On the projections of norm one in the direct product operator algebras, Tôhoku Math. J. (2) 11 (1959), 305-313. MR 0108739 (21:7453)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46L80, 19A49

Retrieve articles in all journals with MSC: 46L80, 19A49


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1093591-7
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society