Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-02-05T10:52:44.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Elementary operators on prime C*-algebras II†

Published online by Cambridge University Press:  18 May 2009

Martin Mathieu
Martin Mathieu
Affiliation:
Mathematisches Institut der, Universitat Tübingen, 7400 Tübingen, West Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Compact elementary operators acting on the algebra ℒ(H) of all bounded operators on some Hilbert space H were characterised by Fong and Sourour in [9]. Akemann and Wright investigated compact and weakly compact derivations on C*-algebras [1], and also studied compactness properties of the sum and the product of the right and the left regular representation of an element in a C*-algebra [2]. They used the concept of a compact Banach algebra element due to Vala [17]: an element a in a Banach algebra A is called compact if the mapping x → axa is compact on A. This notion has been further investigated by Ylinen [18, 19, 20], who showed in particular that a is a compact element of the C*-algebra A if x ↦ axa is weakly compact on A [19].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 3 , September 1988 , pp. 275 - 284
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

1.Akemann, C. A. and Wright, S., Compact and weakly compact derivations of C*-algebras, Pacific J. Math. 85 (1979), 253–259.CrossRefGoogle Scholar
2.Akemann, C. A. and Wright, S., Compact actions on C*-algebras, Glasgow Math. J. 21 (1980), 143–149.CrossRefGoogle Scholar
3.Apostol, C. and Fialkow, L. A., Structural properties of elementary operators, Canad. J. Math. 38 (1986), 1485–1524.CrossRefGoogle Scholar
4.Archbold, R. J., On factorial states of operator algebras, J. Functional Analysis 55 (1984), 25–38.CrossRefGoogle Scholar
5.Barnes, B. A., Murphy, G. J., Smyth, M. R. F. and West, T. T., Riesz and Fredholm theory in Banach algebras, Pitman Research Notes in Mathematics 67 (1982).Google Scholar
6.Breuer, M., Fredholm theories in von Neumann algebras I, Math. Ann. 178 (1968), 243–254.CrossRefGoogle Scholar
7.Dixmier, J., Les C*-algébres et leurs représentations, (Gauthier-Villars, 1969).Google Scholar
8.Dunford, N. and Schwartz, J. T., Linear operators Part I, (Interscience, New York, 1958).Google Scholar
9.Fong, C. K. and Sourour, A. R., On the operator identity σAkXBk = 0, Canad. J. Math. 31 (1979), 845–857.CrossRefGoogle Scholar
10.Ho, Y., A note on derivations, Bull. Inst. Math. Acad. Sinica 5 (1977), 1–5.Google Scholar
11.Magajna, B., A system of operator equations, Canad. Math. Bull. 30 (1987), 200–209.CrossRefGoogle Scholar
12.Martindale, W. S., Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584.CrossRefGoogle Scholar
13.Mathieu, M., Elementary operators on prime C*-algebras, I, to appear.Google Scholar
14.Mathieu, M., Applications of ultraprime Banach algebras in the theory of elementary operators, Thesis, (Tübingen, 1986).Google Scholar
15.Pedersen, G. K., C*-atgebras and their automorphism groups, (Academic Press, 1979).Google Scholar
16.Vala, K., On compact sets of compact operators, Ann. Acad. Sci. Fenn. Ser. A I 351 (1964).Google Scholar
17.Vala, K., Sur les éléments compacts d'une algèbre normée, Ann. Acad. Sci. Fenn. Ser. A I 407 (1967).Google Scholar
18.Ylinen, K., Compact and finite-dimensional elements of normed algebras, Ann. Acad. Sci. Fenn. Ser. A I 428 (1968).Google Scholar
19.Ylinen, K., Dual C*-algebras, weakly semi-completely continuous elements, and the extreme rays of the positive cone, Ann. Acad. Sci. Fenn. Ser. A 1 599 (1975).Google Scholar
20.Ylinen, K., Weakly completely continuous elements of C*-algebras, Proc. Amer. Math. Soc. 52 (1975), 323–326.Google Scholar