Equivariant K-Theory of Compact Connected Lie Groups Dedicated to Daniel Quillen on the Occasion of his Sixtieth Birthday

We compute the equivariant K-theory K _G ^* (G)for a compact connected Lie group Gsuch that ₁ (G)is torsion free (where Gacts on itself by conjugation). We prove that K _G ^* (G)is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a compact connected Lie group Gwith ₁ (G)torsion, namely PSU(3), and compute the corresponding equivariant K-theory.     

On the First Galois Cohomology Group of the Algebraic Group SL1(D)

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL₁(A). For a field extension K of F, we study the first Galois Cohomology group H ¹(K,SL₁(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.     

Un théorème du point fixe de Lefschetz en géométrie d'Arakelov

We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic K₀-theory for these varieties. We then state a Riemann-Roch theorem for the natural transformation of equivariant arithmetic K₀ -theory induced by the restriction to the fixed point scheme and we show that it implies a version of Bismut's conjecture of an equivariant arithmetic Riemann-Roch theorem.     

Equivariant homology and cohomology of groups

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.     

A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof

We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic K ₀-theory for these varieties. We use the equivariant analytic torsion to define direct image maps in this context and we prove a Riemann-Roch theorem for the natural transformation of equivariant arithmetic K ₀-theory induced by the restriction to the fixed point scheme; this theorem can be viewed as an analog, in the context of Arakelov geometry, of the regular case of the theorem proved by P. Baum, W. Fulton and G. Quart in [BaFQ]. We show that it implies an equivariant refinement of the arithmetic Riemann-Roch theorem, in a form conjectured by J.-M. Bismut (cf. [B2, Par. (l), p. 353] and also Ch. Soulé’s question in [SABK, 1.5, p. 162]). Oblatum 22-I-1999 & 20-II-2001?Published online: 4 May 2001     

K-theory of braided tensor ring categories with higher commutativity

We show that theK-theory of a Waldhausen categoryC with anA-ring structure is anA ring spectrum. If theA structure onC supports anE _n structure, so thatBC group completes to ann-fold loop space, thenK (C) is anE _n ring spectrum. In particular, theK-theory of the category of crossedG-sets,G a finite group, is anE ₂ ring spectrum.     

Characters of the infinite symplectic group—A Riesz ring approach

We classify both the finite and infinite characters of the inductive limit symplectic group G. An important feature of our technique is the systematic use of a multiplicative structure on an “ordered completion” of the K₀-group for the group C^*-algebra A of G. We also give explicit examples of the K-theory for certain primitive quotients of A.     

K-theory for finite covariant systems

LetA be a real or complex Banach algebra and assume that is an action of a finite groupG onA by means of continuous automorphisms. To such a finite covariant system (A, G, ), we associate an Abelian groupK(A, G, ). We obtain some classical exact sequences for an algebraA and a closed invariant idealI. We also compute the group in a few important special cases. Doing so, we relate our new invariant to the classicalK ₀ andK ₁ of a Banach algebra and to theK-theory of ₂-graded Banach algebras. Finally, we obtain a result that gives a close relationship of our groupK(A, G, ) with theK-theory of the crossed productA G. In particular, we prove a six-term exact sequence involving our groupK(A, G, ) and theK-groups ofA G. In this way, we hope to contribute to the well-known problem of finding theK-theory of the crossed productA G in the case of an action of a finite group.     

Hopf Algebra Equivariant Cyclic Cohomology, K-theory and Index Formulas

For an algebra with an action of a Hopf algebra we establish the pairing between equivariant cyclic cohomology and equivariant K-theory for . We then extend this formalism to compact quantum group actions and show that equivariant cyclic cohomology is a target space for the equivariant Chern character of equivariant summable Fredholm modules. We prove an analogue of Julg's theorem relating equivariant K-theory to ordinary K-theory of the C^*-algebra crossed product, and characterize equivariant vector bundles on quantum homogeneous spaces.     

Equivariant Kasparov Theory and Generalized Homomorphisms

Let G be a locally compact group. We describe elements of KK ^G (A, B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal property of KK ^G : It is the universal split exact stable homotopy functor. To describe a Kasparov triple (, , F) for A, B by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form ; and more generally if the group action on A is proper in the sense of Exel and Rieffel.     

A Note on K-Theory of Azumaya Algebras

For an Azumaya algebra A which is free over its centre R, we prove that K-theory of A is isomorphic to K-theory of R up to its rank torsions. We conclude that K _i (A, ?/m) = K _i (R, ?/m) for any m relatively prime to the rank and i ≥ 0. This covers, for example, K-theory of division algebras, K-theory of Azumaya algebras over semilocal rings, and K-theory of graded central simple algebras indexed by a totally ordered abelian group.     

A K-theoretic note on geometric quantization

We give a purely K-theoretic proof of a case of the “quantization commutes with reduction” result, conjectured by Guillemin and Sternberg and proved by Meinrenken and Vergne. We show that the quantization is simply a pushforward in K-theory, and use Lerman's symplectic cutting and the localization theorem in equivariant K-theory to prove that quantization commutes with reduction. The case where G=S ¹ and the action is free on the zero level set of the moment map is addressed. Received: 9 March 1999     

The K-Theory of C *-Algebras with Finite Dimensional Irreducible Representations

We study the K-theory of unital C^*-algebras A satisfying the condition that all irreducible representations are finite and of some bounded dimension. We construct computational tools, but show that K-theory is far from being able to distinguish between various interesting examples. For example, when the algebra A is n-homogeneous, i.e., all irreducible representations are exactly of dimension n, then K_*(A) is the topological K-theory of a related compact Hausdorff space, this generalises the classical Gelfand-Naimark theorem, but there are many inequivalent homogeneous algebras with the same related topological space. For general A we give a spectral sequence computing K_*(A) from a sequence of topological K-theories of related spaces. For A generated by two idempotents, this becomes a 6-term long exact sequence.     

Mackey structures on equivariant algebraic K-theory

We study the Mackey structure of the G-spectrum K _G C associated to a monoidal G-category C. It is proved that the coefficient system of K _G C coincides, as a (graded) Mackey functor, with the system of equivariant K-groups in the sense of Fröhlich and Wall. It is also shown that for any exact category U, there exists a G-spectrum Q _G U representing the equivariant K-theory of U in the sense of Dress and Kuku, and that Q _G U is naturally G-homotopy equivalent to K _G IsoU if every short exact sequence in U splits.     

Equivariant Cyclic Cohomology of H-Algebras

We define an equivariant K ₀-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.     

Lambda-structure on Grothendieck groups of Hermitian vector bundles

We define a “compactification” of the representation ring of the linear group scheme over Specℤ, in the spirit of Arakelov geometry. We show that it is a λ-ring which is canonically isomorphic to a localized polynomial ring and that it plays a universal role with respect to natural operations on theK ₀-theory of hermitian bundles defined by Gillet-Soulé. As a byproduct, we prove that the natural pre-λ-ring structure of theK ₀-theory of hermitian bundles is a λ-ring structure. This last result plays a key role in the proof of the main results of [18] and [12].     

Equivariant pretheories and invariants of torsors

In the present paper we introduce and study the notion of an equivariant pretheory (basic examples are equivariant Chow groups of Edidin and Graham, Thomason??s equivariant K-theory and equivariant algebraic cobordism). Using the language of equivariant pretheories we generalize the theorem of Karpenko and Merkurjev on G-torsors and rational cycles. As an application, to every G-torsor E and a G-equivariant pretheory we associate a ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information about the motivic J-invariant of E, in the case of Grothendieck??s K ₀ indexes of the respective Tits algebras and in the case of algebraic cobordism ?? it gives a quotient of the cobordism ring of G.     

Index Theorem for Equivariant Dirac Operators on Noncompact Manifolds

Let D be a (generalized) Dirac operator on a noncompact complete Riemannian manifold M acted on by a compact Lie group G. Let v: M g = Lie G be an equivariant map, such that the corresponding vector field on M does not vanish outside of a compact subset. These data define an element of K-theory of the transversal cotangent bundle to M. Hence, by embedding of M into a compact manifold, one can define a topological index of the pair (D,v) as an element of the completed ring of characters of G. We define an analytic index of (D,v) as an index space of certain deformation of D and we prove that the analytic and topological indexes coincide. As a main step of the proof, we show that index is an invariant of a certain class of cobordisms, similar to the one considered by Ginzburg, Guillemin and Karshon. In particular, this means that the topological index of Atiyah is also invariant under this class of noncompact cobordisms. As an application, we extend the Atiyah–Segal–Singer equivariant index theorem to our noncompact setting. In particular, we obtain a new proof of this theorem for compact manifolds.     

Algebras almost commuting with Clifford algebras in a II∞ factor

We are concerned here with certain Banach algebras of operators contained within a fixed II factor N. These algebras may be thought of as noncommutative classifying spaces for the functor Ext _* ^N The basic objects of study are the algebras A _kN (for n=1, 2,...). Here, we are given an essentially unique representation of the complex Clifford algebra C _k N and the elements of A _k are those operators in N which exactly commute with the first k–1 generators of C _k and also commute with the kth generator modulo a symmetric ideal N. Up to isomorphism, these algebras are periodic with period 2.We determine completely the homotopy types of A ₁ ^–1 and A ₂ ^–1 Here, A ₁ ^–1 is homotopy equivalent to the space of (Breuer) Fredholm operators in N, while A ₂ ^–1 is homotopy equivalent to the group K _N ^–1 ={x N^–1¦ x=1+k, k K_N}. We use these results to compute the K-theory of A ₁ and A ₂.For a fixed C ^*-algebra A, we define abelian groups G _k,N(A) of equivalence classes of homomorphisms AA _k. Letting N = M (H) for a II₁ factor M we define similar abelian groups G _k(A, M) where we replace N by L(E) for countably generated right Hilbert M-modules E with (left) actions C _k L(E). Using ideas of Skandalis, we show that G _k,NG_k(A, M) so that the G _k,N are stable half-exact homotopy functors because the G _k(·, M) are such.In general, we show that G _k(A, M)KK ^k(A, M) and so our theory fits neatly into Kasparov KK-theory. We investigate many interesting examples from our point of view.