Banach Journal of Mathematical Analysis

Nearly relatively compact projections in operator algebras

Lawrence G. Brown

Abstract

Let A be a C-algebra, and let A be its enveloping von Neumann algebra. Akemann suggested a kind of noncommutative topology in which certain projections in A play the role of open sets, and he used two operator inequalities in connection with compactness. Both of these inequalities are equivalent to compactness for a closed projection in A, but only one is equivalent to relative compactness for a general projection. A third operator inequality, also related to compactness, was used by the author. The study of all three inequalities can be unified by considering a numerical invariant which is equivalent to the distance of a projection from the set of relatively compact projections. Tomita’s concept of regularity of projections seems relevant, and so we give some results and examples on regularity. We also include a few related results on semicontinuity.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 2 (2018), 259-293.

Dates
Received: 8 December 2016
Accepted: 17 January 2017
First available in Project Euclid: 21 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1505959314

Digital Object Identifier
doi:10.1215/17358787-2017-0033

Mathematical Reviews number (MathSciNet)
MR3779714

Zentralblatt MATH identifier
06873501

Subjects
Primary: 46L05: General theory of $C^*$-algebras
Secondary: 47C15: Operators in $C^*$- or von Neumann algebras

Keywords
compact projection regular projection semicontinuous operator

Citation

Brown, Lawrence G. Nearly relatively compact projections in operator algebras. Banach J. Math. Anal. 12 (2018), no. 2, 259--293. doi:10.1215/17358787-2017-0033. https://projecteuclid.org/euclid.bjma/1505959314


Export citation

References

  • [1] C. A. Akemann, The general Stone–Weierstrass problem, J. Funct. Anal. 4 (1969), 277–294.
  • [2] C. A. Akemann, Approximate units and maximal abelian $C^{*}$-subalgebras, Pacific J. Math. 33 (1970), 543–550.
  • [3] C. A. Akemann, Left ideal structure of $C^{*}$-algebras, J. Funct. Anal. 6 (1970), 305–317.
  • [4] C. A. Akemann, A Gelfand representation theory for $C^{*}$-algebras, Pacific J. Math. 39 (1971), 1–11.
  • [5] C. A. Akemann and G. K. Pedersen, Complications of semicontinuity in $C^{*}$-algebra theory, Duke Math. J. 40 (1973), 785–795.
  • [6] C. A. Akemann, G. K. Pedersen, and J. Tomiyama, Multipliers of $C^{*}$-algebras, J. Funct. Anal. 13 (1973), 277–301.
  • [7] L. G. Brown, Semicontinuity and multipliers of $C^{*}$-algebras, Canad. J. Math. 40 (1988), no. 4, 865–988.
  • [8] L. G. Brown, Determination of $A$ from $M(A)$ and related matters, C. R. Math. Acad. Sci. Soc. R. Can. 10 (1988), no. 6, 273–278.
  • [9] L. G. Brown, The rectifiable metric on the set of closed subspaces of Hilbert spaces, Trans. Amer. Math. Soc. 337 (1993), no. 1, 279–289.
  • [10] R. C. Busby, Double centralizers and extensions of $C^{*}$-algebras, Trans. Amer. Math. Soc. 132 (1968), 79–99.
  • [11] F. Combes, Sur les faces d’une $C^{*}$-algèbre, Bull. Sci. Math. (2) 93 (1969), 37–62.
  • [12] J. Dixmier, Position relative de deux variétés linéaires fermées dans un espace de Hilbert, Revue Sci. 86 (1948), 387–399.
  • [13] E. G. Effros, Order ideals in a $C^{*}$-algebra and its dual, Duke Math. J. 30 (1963), 391–411.
  • [14] P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389.
  • [15] R. V. Kadison, Irreducible operator algebras, Proc. Natl. Acad. Sci. USA 43 (1957), 273–276.
  • [16] J. L. Kelley, General Topology, Van Nostrand, Toronto, 1955.
  • [17] G. K. Pedersen, Measure theory for $C^{*}$-algebras, II, Math. Scand. 22 (1968), 63–74.
  • [18] G. K. Pedersen, $SAW^{*}$-algebras and corona $C^{*}$-algebras, contributions to noncommutative topology, J. Operator Theory 15 (1986), no. 1, 15–32.
  • [19] I. Raeburn and A. M. Sinclair, The $C^{*}$-algebra generated by two projections, Math. Scand. 65 (1989), no. 2, 278–290.
  • [20] S. Sakai, On linear functionals of $W^{*}$-algebras, Proc. Japan Acad. 34 (1958), 571–574.
  • [21] M. Tomita, Spectral theory of operator algebras, I, Math. J. Okayama Univ. 9 (1959/1960), 63–98.