Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Unitary equivalences for essential extensions of -algebras

Author(s): Huaxin Lin
Journal: Proc. Amer. Math. Soc. 137 (2009), 3407-3420.
MSC (2000): Primary 46L05
Posted: May 15, 2009
MathSciNet review: 2515410
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Let be a unital separable -algebra and $B=C\otimes {\mathcal K},$ where is a unital -algebra. Let $\tau: A\to M(B)/B$ be a unital full essential extension of by We show that there is a bijection between elements in a quotient group of onto the strong unitary equivalence classes of unital full essential extensions $\sigma$ for which $[\sigma]=[\tau]$ in Consequently, when this group is zero, unitarily equivalent full essential extensions are strongly unitarily equivalent. When is a non-unital but $\sigma$ -unital simple -algebra with continuous scale, we also study the problem when two approximately unitarily equivalent essential extensions are strongly approximately unitarily equivalent. A group is used to compute the strongly approximate unitary equivalence classes in the same approximate unitary equivalent class of essential extensions.

References:

1.: B. Blackadar, -theory for Operator Algebras, 2nd ed., Mathematical Sciences Research Institute Publications, 5. Cambridge University Press, Cambridge, 1998. MR 1656031 (99g:46104)
2.: L. G. Brown, The universal coefficient theorem for and quasidiagonality, Operator Algebras and Group Representations, vol. 17, Pitman Press, Boston, 1983, pp. 60-64. MR 731763 (85m:46066)
3.: L. G. Brown, R. G. Douglas and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of $C\sp{*}$ -algebras, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973), pp. 58-128. Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973. MR 0380478 (52:1378)
4.: L. G. Brown, R. G. Douglas and P. A. Fillmore, Extensions of $C\sp*$ -algebras and -homology, Ann. of Math. (2) 105 (1977), 265-324. MR 0458196 (56:16399)
5.: G. A. Elliott and D. Kucerovsky, An abstract Voiculescu-Brown-Douglas-Fillmore absorption theorem, Pacific J. Math. 198 (2001), 385-409. MR 1835515 (2002i:46052)
6.: G. Gong and H. Lin, Almost multiplicative morphisms and -theory, Internat. J. Math. 11 (2000), 983-1000. MR 1797674 (2001j:46081)
7.: H. Lin, Simple $C\sp *$ -algebras with continuous scales and simple corona algebras, Proc. Amer. Math. Soc. 112 (1991), no. 3, 871-880. MR 1079711 (92e:46118)
8.: H. Lin, -algebra extensions of , Memoirs Amer. Math. Soc. 115 (1995), no. 550. MR 1257081 (96b:46098)
9.: H. Lin, Extensions by -algebras of real rank zero. II, Proc. London Math. Soc. (3) 71 (1995), 641-674. MR 1347408 (96j:46072)
10.: H. Lin, Extensions of by simple -algebras of real rank zero, Amer. J. Math. 119 (1997), 1263-1289. MR 1481815 (98m:46088)
11.: H. Lin, An Introduction to the Classification of Amenable -algebras, World Scientific, River Edge, NJ, 2001. MR 1884366 (2002k:46141)
12.: H. Lin, A separable Brown-Douglas-Fillmore theorem and weak stability, Trans. Amer. Math. Soc. 356 (2004), 2889-2925.MR 2052601 (2005d:46116)
13.: H. Lin, Simple corona -algebras, Proc. Amer. Math. Soc. 132 (2004), 3215-3224. MR 2073295 (2005e:46100)
14.: H. Lin, Extensions by simple -algebras: quasidiagonal extensions, Canad. J. Math. 57 (2005), 351-399. MR 2124922 (2005m:46094)
15.: H. Lin, Full extensions and approximate unitary equivalence, Pacific J. Math. 229 (2007), 389-428. MR 2276517 (2008k:4u6165)
16.: G. K. Pedersen, $SAW\sp *$ -algebras and corona $C\sp *$ -algebras, contributions to noncommutative topology, J. Operator Theory 15 (1986), 15-32. MR 816232 (87a:46095)
17.: M. Pimsner, S. Popa and D. Voiculescu, Homogeneous $C\sp{*}$ -extensions of $C(X)\otimes K(H)$ . I, J. Operator Theory 1 (1979), 55-108. MR 526291 (82e:46093a)
18.: N. C. Phillips, A survey of exponential rank. $C\sp *$ -algebras: 1943-1993 (San Antonio, TX, 1993), 352-399, Contemp. Math., 167, Amer. Math. Soc., Providence, RI, 1994. MR 1292021 (95j:46069)
19.: J. Rosenberg and C. Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov's generalized -functor, Duke Math. J. 55 (1987), 431-474. MR 894590 (88i:46091)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|