P. Baum, A. Connes, and N. Higson :
“Classifying space for proper actions and \( K \) -theory of group \( C^* \) -algebras 241–291
in
\( C^* \) -algebras: 1943–1993San Antonio, TX, 13–14 January 1993 ).
Edited by R. S. Doran .
Contemporary Mathematics 167 .
American Mathematical Society (Providence, RI ),
1994 .
MR
1292018 Zbl
0830.46061 incollection
Abstract
People
BibTeX
We announce a reformulation of the conjecture in [1982; 1988]. The advantage of the new version is that it is simpler and applies more generally than the earlier statement. A key point is to use the universal example for proper actions introduced in [1988]. There, the universal example seemed somewhat peripheral to the main issue. Here, however, it will play a central role.
@incollection {key1292018m,
AUTHOR = {Baum, Paul and Connes, Alain and Higson,
Nigel},
TITLE = {Classifying space for proper actions
and \$K\$-theory of group \$C^*\$-algebras},
BOOKTITLE = {\$C^*\$-algebras: 1943--1993},
EDITOR = {Doran, Robert S.},
SERIES = {Contemporary Mathematics},
NUMBER = {167},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1994},
PAGES = {241--291},
DOI = {10.1090/conm/167/1292018},
NOTE = {(San Antonio, TX, 13--14 January 1993).
MR:1292018. Zbl:0830.46061.},
ISSN = {0271-4132},
ISBN = {9780821851753},
}
P. F. Baum, N. Higson, and R. J. Plymen :
“Representation theory of \( p \) -adic groups: A view from operator algebras 111–149
in
The mathematical legacy of Harish-Chandra: A celebration of representation theory and harmonic analysis
(Baltimore, MD, 9–10 January 1998 ).
Edited by R. Doran and V. Varadarajan .
Proceedings of Symposia in Pure Mathematics 68 .
American Mathematical Society (Providence, RI ),
2000 .
MR
1767895 Zbl
0982.19006 incollection
Abstract
People
BibTeX
@incollection {key1767895m,
AUTHOR = {Baum, P. F. and Higson, N. and Plymen,
R. J.},
TITLE = {Representation theory of \$p\$-adic groups:
{A} view from operator algebras},
BOOKTITLE = {The mathematical legacy of {H}arish-{C}handra:
{A} celebration of representation theory
and harmonic analysis},
EDITOR = {Doran, R. and Varadarajan, V.},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {68},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2000},
PAGES = {111--149},
DOI = {10.1090/pspum/068/1767895},
NOTE = {(Baltimore, MD, 9--10 January 1998).
MR:1767895. Zbl:0982.19006.},
ISSN = {0082-0717},
ISBN = {9780821811979},
}
P. Baum :
“On the index of equivariant elliptic operators 41–49
in
Operator algebras, quantization, and noncommutative geometry: A centennial celebration honoring John von Neumann and Marshall H. Stone .
Edited by R. Doran and R. V. Kadison .
Contemporary Mathematics 365 .
American Mathematical Society (Providence, RI ),
2004 .
MR
2106816 Zbl
1081.46047 incollection
Abstract
People
BibTeX
For a countable discrete group \( \Gamma \) , the BC (Baum–Connes) conjecture can be stated in terms of \( \operatorname{Spin}^c \) -manifolds and Dirac operators. The BC map sends an appropriate equivariant Dirac operator to its index. Somewhat related to this is the theorem of W. Lück on the range of the trace at the identity.
@incollection {key2106816m,
AUTHOR = {Baum, Paul},
TITLE = {On the index of equivariant elliptic
operators},
BOOKTITLE = {Operator algebras, quantization, and
noncommutative geometry: {A} centennial
celebration honoring {J}ohn von {N}eumann
and {M}arshall {H}. {S}tone},
EDITOR = {Doran, R. and Kadison, R. V.},
SERIES = {Contemporary Mathematics},
NUMBER = {365},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2004},
PAGES = {41--49},
DOI = {10.1090/conm/365/06699},
NOTE = {MR:2106816. Zbl:1081.46047.},
ISSN = {0271-4132},
}
P. Baum :
“\( K \) -homology and D-branes81–94
in
Superstrings, geometry, topology, and \( C^* \) -algebras
(Fort Worth, TX, 18–22 May 2009 ).
Edited by R. S. Doran, G. Friedman, and J. Rosenberg .
Proceedings of Symposia in Pure Mathematics 81 .
American Mathematical Society (Providence, RI ),
2010 .
MR
2681759 Zbl
1210.81079 incollection
Abstract
People
BibTeX
\( K \) -homology is the dual theory to \( K \) -theory. In algebraic geometry [14], [7], the \( K \) -homology of a (possibly singular) projective variety \( X \) is the Grothendieck group of coherent algebraic sheaves on \( X \) . In topology there are three ways to define \( K \) -homology. First \( K \) -homology is the homology theory determined by the Bott spectrum. Second, \( K \) -homology is the group of geometric \( K \) -cycles introduced by Baum–Douglas [6]. Third, using funtional analysis, \( K \) -homology is the group of abstract elliptic operators as in the work of M. .F. Atiyah [1], Brown–Douglas–Filmore [15], and G. Kasparov [21].
The D-branes of string theory [31] are twisted geometric \( K \) -cycles which are endowed with some additional structure. The charge of a D-brane is the element in the twisted \( K \) -homology of spacetime determined by the underlying twisted \( K \) -cycle of the D-brane. Essentially, the Baum–Douglas theory [6] was rediscovered in terms of constraints on open strings. The aim of this expository note is to briefly describe this development.
@incollection {key2681759m,
AUTHOR = {Baum, Paul},
TITLE = {\$K\$-homology and {D}-branes},
BOOKTITLE = {Superstrings, geometry, topology, and
\$C^*\$-algebras},
EDITOR = {Doran, R. S. and Friedman, G. and Rosenberg,
J.},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {81},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2010},
PAGES = {81--94},
DOI = {10.1090/pspum/081/2681759},
NOTE = {(Fort Worth, TX, 18--22 May 2009). MR:2681759.
Zbl:1210.81079.},
ISSN = {0082-0717},
ISBN = {9780821848876},
}
A.-M. Aubert, P. Baum, and R. Plymen :
“Geometric structure in the representation theory of reductive \( p \) -adic groups, II 71–90
in
Harmonic analysis on reductive, \( p \) -adic groups
(San Francisco, 16 January 2010 ).
Edited by R. S. Doran, P. J. Sally, Jr., and L. Spice .
Contemporary Mathematics 543 .
American Mathematical Society (Providence, RI ),
2011 .
Part I was published in C. R. Math. Acad. Sci. Paris 345 :10 (2007)
MR
2798423 Zbl
1246.22019 incollection
Abstract
People
BibTeX
@incollection {key2798423m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger},
TITLE = {Geometric structure in the representation
theory of reductive \$p\$-adic groups,
{II}},
BOOKTITLE = {Harmonic analysis on reductive, \$p\$-adic
groups},
EDITOR = {Doran, Robert S. and Sally, Jr., Paul
J. and Spice, Loren},
SERIES = {Contemporary Mathematics},
NUMBER = {543},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2011},
PAGES = {71--90},
DOI = {10.1090/conm/543/10730},
NOTE = {(San Francisco, 16 January 2010). Part
I was published in \textit{C. R. Math.
Acad. Sci. Paris} \textbf{345}:10 (2007),
doesn't include ``reductive'' in title.
MR:2798423. Zbl:1246.22019.},
ISSN = {0271-4132},
ISBN = {9780821849859},
}
P. Baum, E. Guentner, and R. Willett :
“Exactness and the Kadison–Kaplansky conjecture 1–33
in
Operator algebras and their applications: A tribute to Richard V. Kadison
(San Antonio, TX, 10–11 January 2015 ).
Edited by R. S. Doran and E. Park .
Contemporary Mathematics 671 .
American Mathematical Society (Providence, RI ),
2016 .
Dedicated to Richard Kadison on the occasion of his ninetieth birthday with affection and admiration.
MR
3546676 Zbl
1366.46045 incollection
Abstract
People
BibTeX
We survey results connecting exactness in the sense of \( C^* \) -algebra theory, coarse geometry, geometric group theory, and expander graphs. We summarize the construction of the (in)famous non-exact monster groups whose Cayley graphs contain expanders, following Gromov, Arzhantseva, Delzant, Sapir, and Osajda. We explain how failures of exactness for expanders and these monsters lead to counterexamples to Baum–Connes type conjectures: the recent work of Osajda allows us to give a more streamlined approach than currently exists elsewhere in the literature.
We then summarize our work on reformulating the Baum–Connes conjecture using exotic crossed products, and show that many counterexamples to the old conjecture give confirming examples to the reformulated one; our results in this direction are a little stronger than those in our earlier work. Finally, we give an application of the reformulated Baum–Connes conjecture to a version of the Kadison–Kaplansky conjecture on idempotents in group algebras.
@incollection {key3546676m,
AUTHOR = {Baum, Paul and Guentner, Erik and Willett,
Rufus},
TITLE = {Exactness and the {K}adison--{K}aplansky
conjecture},
BOOKTITLE = {Operator algebras and their applications:
{A} tribute to {R}ichard {V}. {K}adison},
EDITOR = {Doran, Robert S. and Park, Efton},
SERIES = {Contemporary Mathematics},
NUMBER = {671},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2016},
PAGES = {1--33},
DOI = {10.1090/conm/671/13501},
NOTE = {(San Antonio, TX, 10--11 January 2015).
Dedicated to Richard Kadison on the
occasion of his ninetieth birthday with
affection and admiration. MR:3546676.
Zbl:1366.46045.},
ISSN = {0271-4132},
ISBN = {9781470419486},
}
