## Advances in Operator Theory

### $C^*$-algebra distance filters

#### Abstract

‎We use nonsymmetric distances to give a self-contained account of $C^*$-algebra filters and their corresponding compact projections‎, ‎simultaneously simplifying and extending their general theory‎.

#### Article information

Source
Adv. Oper. Theory, Volume 3, Number 3 (2018), 655-681.

Dates
Received: 10 October 2017
Accepted: 4 March 2018
First available in Project Euclid: 4 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1522807283

Digital Object Identifier
doi:10.15352/aot.1710-1241

Mathematical Reviews number (MathSciNet)
MR3795107

Zentralblatt MATH identifier
06902459

Subjects
Primary: 46L05: General theory of $C^*$-algebras
Secondary: 06A75: Generalizations of ordered sets ‎46L85‎ ‎54E99

#### Citation

Bice, Tristan; Vignati, Alessandro. $C^*$-algebra distance filters. Adv. Oper. Theory 3 (2018), no. 3, 655--681. doi:10.15352/aot.1710-1241. https://projecteuclid.org/euclid.aot/1522807283

#### References

• If two projections are close, then they are unitarily equivalent, MathOverflow, 2013.
• Extending akemann's non-commutative Urysohn lemma, MathOverflow, 2015.
• C. A. Akemann, A Gelfand representation theory for $C^*$-algebras, Pacific J. Math. 39 (1971), 1–11.
• C. A. Akemann, Approximate units and maximal abelian $C^*$-subalgebras, Pacific J. Math. 33 (1970), 543–550.
• C. A. Akemann, J. Anderson, and G. K. Pedersen, Approaching infinity in $C^*$-algebras, J. Operator Theory 21 (1989), no. 2, 255–271.
• C. A. Akemann, J. Anderson, and G. K. Pedersen, Excising states of $C^*$-algebras, Canad. J. Math. 38 (1986), no. 5, 1239–1260.
• C. A. Akemann and G. K. Pedersen, Facial structure in operator algebra theory, Proc. London Math. Soc. (3) 64 (1992), no. 2, 418–448.
• T. Bice, Filters in $C^*$-algebras, Canad. J. Math. (2013), no. 3, 485–509.
• T. Bice, The projection calculus, Münster J. Math. 6 (2013), 557–581.
• T. Bice, Semicontinuity in ordered Banach spaces, preprint, arXiv:1604.03154.
• T. Bice, Distance domains, preprint, arXiv:1704.01024.
• T. Bice and P. Koszmider, $C^*$-algebras with and without $\ll$-increasing approximate units, 2017, arXiv:1707.09287.
• B. Blackadar, Operator algebras: Theory of $C^*$-algebras and von neumann algebras, Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry, III. Springer-Verlag, Berlin, 2017.
• D. P. Blecher and N. Weaver, Quantum measurable cardinals, J. Funct. Anal. 273 (2017), no. 5, 1870–1890.
• L. G. Brown, Semicontinuity and multipliers of $C^*$-algebras, Canad. J. Math. 40 (1988), no. 4, 865–988.
• I. Farah and E. Wofsey, Set theory and operator algebras, Appalachian Set Theory: 2006-2012, London Mathematical Society Lecture Note Series, Cambridge University Press, 2013.
• F. Kittaneh, Inequalities for the Schatten $p$-norm. IV, Comm. Math. Phys. 106 (1986), no. 4, 581–585.
• A. W. Marcus, D. A. Spielman, and N. Srivastava, Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem, Ann. of Math. (2) 182 (2015), no. 1, 327–350.
• E. Ortega, M. Rørdam, and H. Thiel, The Cuntz semigroup and comparison of open projections, J. Funct. Anal. 260 (2011), no. 12, 3474–3493.
• G. K. Pedersen, $C^*$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1979.
• I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyatsov, Model theory for metric structures, Volume 2 of Model Theory with Applications to Algebra and Analysis (Z. Chatzidakis, D. Macpherson, A. Pillay, and A. Wilkie, eds.), London Mathematical Society Lecture Note Series, no. 350, Cambridge University Press, 2008, pp. 315–427.