04312nam a22004095a 450000100100000000300120001000500170002200600190003900700150005800800410007302000180011402400210013204000140015307200170016708400220018424501480020626000820035430000340043633600260047033700260049633800360052234700240055849000540058250510900063650600660172652017640179265000310355665000200358765000480360770000350365570000290369070000270371970000280374670000330377485600320380785600630383989-091109CH-001817-320091109150325.0a fot ||| 0|cr nn mmmmamaa091109e20081006sz fot ||| 0|eng d a978303719560470a10.4171/0602doi ach0018173 7aPBMW2bicssc a19-xxa58-xx2msc10aK-Theory and Noncommutative Geometryh[electronic resource] /cGuillermo Cortiñas, Joachim Cuntz, Max Karoubi, Ryszard Nest, Charles A. Weibel3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2008 a1 online resource (454 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aEMS Series of Congress Reports (ECR) ;x2523-515X00tCategorical aspects of bivariant K-theory /rRalf Meyer --tInheritance of isomorphism conjectures under colimits /rArthur Bartels, Siegfried Echterhoff, Wolfgang Lück --tCoarse and equivariant co-assembly maps /rHeath Emerson, Ralf Meyer --tOn K1 of a Waldhausen category /rFernando Muro, Andrew Tonks --tTwisted K-theory – old and new /rMax Karoubi --tEquivariant cyclic homology for quantum groups /rChristian Voigt --tA Schwartz type algebra for the tangent groupoid /rPaulo Carrillo Rouse --tC*-algebras associated with the ax + b-semigroup over ℕ /rJoachim Cuntz --tOn a class of Hilbert C*-manifolds /rWend Werner --tDuality for topological abelian group stacks and T-duality /rUlrich Bunke, Thomas Schick, Markus Spitzweck, Andreas Thom --tDeformations of gerbes on smooth manifolds /rPaul Bressler, Alexander Gorokhovsky, Ryszard Nest, Boris Tsygan --tTorsion classes of finite type and spectra /rGrigory Garkusha, Mike Prest --tParshin’s conjecture revisited /rThomas Geisser --tAxioms for the norm residue isomorphism /rCharles A. Weibel.1 aRestricted to subscribers:uhttps://www.ems-ph.org/ebooks.php aSince its inception 50 years ago, K-theory has been a tool for
understanding a wide-ranging family of mathematical structures and their
invariants: topological spaces, rings, algebraic varieties and operator
algebras are the dominant examples. The invariants range from
characteristic classes in cohomology, determinants of matrices, Chow
groups of varieties, as well as traces and indices of elliptic operators.
Thus K-theory is notable for its connections with other branches of
mathematics.
Noncommutative geometry develops tools which allow
one to think of noncommutative algebras in the same footing as commutative
ones: as algebras of functions on (noncommutative) spaces. The algebras
in question come from problems in various areas of mathematics and mathematical
physics; typical examples include algebras of pseudodifferential operators, group algebras,
and other algebras arising from quantum field theory.
To study noncommutative geometric problems one considers invariants of the relevant noncommutative
algebras. These invariants include algebraic and topological K-theory, and also cyclic homology,
discovered independently by Alain Connes and Boris Tsygan, which can be regarded both as a noncommutative
version of de Rham cohomology and as an additive version of K-theory.
There are primary and secondary Chern characters which pass from
K-theory to cyclic homology. These characters are relevant both to noncommutative and commutative
problems, and have applications ranging from index theorems to the detection of singularities of commutative
algebraic varieties.
The contributions to this volume represent
this range of connections between K-theory, noncommmutative geometry, and other branches of mathematics.07aAlgebraic geometry2bicssc07a$K$-theory2msc07aGlobal analysis, analysis on manifolds2msc1 aCortiñas, Guillermo,eeditor.1 aCuntz, Joachim,eeditor.1 aKaroubi, Max,eeditor.1 aNest, Ryszard,eeditor.1 aWeibel, Charles A.,eeditor.40uhttps://doi.org/10.4171/060423cover imageuhttps://www.ems-ph.org/img/books/cortinas.jpg