Filtrations of Simplicial Functors and the Novikov Conjecture

We show that the Strong Novikov Conjecture for the maximal C^*-algebra C^*(π) of a discrete group π is equivalent to a statement in topological K-theory for which the corresponding statement in algebraic K-theory is always true. We also show that for any group π, rational injectivity of the full assembly map for K_*^t(C^*(π)) follows from rational injectivity of the restricted assembly map. (Received: February 2006)     

Classification of extensions of classifiable -algebras

For a certain class of extensions of C^*-algebras in which B and A belong to classifiable classes of C^*-algebras, we show that the functor which sends to its associated six term exact sequence in K-theory and the positive cones of K₀(B) and K₀(A) is a classification functor. We give two independent applications addressing the classification of a class of C^*-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first named author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph C^*-algebras.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

Algebraic K-theory of non-linear projective spaces

In the spirit of “The Fundamental Theorem for the algebraic K-theory of spaces: I” (J. Pure Appl. Algebra 160 (2001) 21–52) we introduce a category of sheaves of topological spaces on n-dimensional projective space and present a calculation of its K-theory, a “non-linear” analogue of Quillen's isomorphism K_i(P_Rⁿ)₀ⁿK_i(R).     

Morava K-Theories of p-Groups with Cyclic Maximal Subgroups and Other Related Groups

Let P be a non-Abelian finite p-group, p odd, with cyclic maximal subgroups, and let K(n)^*(–) denote the nth Morava K-theory at p. In this paper we determine the algebras K(n)^*(BP) and K(n)^*(BG) for all groups G with Sylow p-subgroups isomorphic to P, giving further evidence for the fact that Morava K-theory as an invariant of finite groups, is finer than ordinary modp cohomology. Mathematics Subject Classifications (2000): 55N20, 55N22.     

Generic representations of the finite general linear groups and the Steenrod algebra: II

The category of generic representations over the finite fieldF _q , used in PartI to study modules over the Steenrod algebra, is here related to the modular representation theory of the groups GL _n (F _q ). This leads to a simple and elegant approach to the classic objects of study: irreducible representations, extensions of modules, homology stability, etc. Connections to current research in algebraicK-theory involving stableK-theory and Topological Hochschild Homology are also explained.Partially funded by the NSF.     

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.     

Real C*-Algebras, United K-Theory, and the Künneth Formula

We define united K-theory for real C^*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors – real K-theory, complex K-theory, and self-conjugate K-theory – and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Künneth-type, nonsplitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C^*-algebras A and B which holds as long as the complexification of A is in the bootstrap category . Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C^*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product is not determined solely by the greatest common divisor of K and l. Hence, we have examples of nonisomorphic, simple, purely infinite, real C^*-algebras whose complexifications are isomorphic.     

Semi-Topological K-Homology and Thomason''s Theorem

In this paper, we introduce the 'semi-topological K-homology' of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasi-projective complex variety Y coincides with the connective topological K-homology of the associated analytic space Y ^an. From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the 'Bott inverted' semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of X ^an. In combination with a result of Friedlander and the author, this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason's celebrated theorem that 'Bott inverted' algebraic K-theory with /n coefficients coincides with topological K-theory with /n coefficients.     

Algebraic geometry of topological spaces I

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case . We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C ^*-algebras, and for a homology theory of commutative algebras to vanish on C ^*-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil K-theory implies that commutative C ^*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.     

Analysis on discrete cocompact subgroups of the generic filiform Lie groups

Summary Let F_n, n≧ 1, denote the sequence of generic filiform (connected, simply connected) Lie groups. Here we study, for each F_n, the infinite dimensional simple quotients of the group C*-algebra of (the most obvious) one of its discrete cocompact subgroups D_n. For D_n, the most attractive concrete faithful representations are given in terms of Anzai flows, in analogy with the representations of the discrete Heisenberg group H₃ ⊆G₃ on L²(T) that result from the irrational rotation flows on T; the representations of D_n generate infinite-dimensional simple quotients A_n,θ of the group C*-algebra C*(D_n). For n>1, there are other infinite-dimensional simple quotients of C*(D_n) arising from non-faithful representations of D_n. Flows for these are determined, and they are also characterized and represented as matrix algebras over simple affine Furstenberg transformation group C*-algebras of the lower dimensional tori.     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

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.     

Gersten conjecture for equivariant K-theory and applications

For a reductive group scheme G over a regular semi-local ring A, we prove the Gersten conjecture for the equivariant K-theory. As a consequence, we show that if F is the field of fractions of A, then K^G₀(A) @ K^G₀(F){K^G_0(A) \cong K^G_0(F)}, generalizing the analogous result for a dvr by Serre (Inst Hautes études Sci Publ Math 34:37–52, 1968). We also show the rigidity for the K-theory with finite coefficients of a Henselian local ring in the equivariant setting. We use this rigidity theorem to compute the equivariant K-theory of algebraically closed fields.     

Embedding an AH-Algebra into a Simple C*-Algebra with Prescribed KK-Data

Let X be a connected finite CW complex. We show that, given a positive homomorphism Hom(K _*(C(X)), K _*(A)) with [1 _C(X)][1 _A ], where A is a unital separable simple C ^*-algebra with real rank zero, stable rank one and weakly unperforated K ₀(A), there exists a homomorphism h: C(X)A such that h induces . We also prove a structure result for unital separable simple C ^*-algebras A with real rank zero, stable rank one and weakly unperforated K ₀(A), namely, there exists a simple AH-algebra of real rank zero contained in A which determines the K-theory of A.     

The Volodin model for hermitian algebraic K-theory

We define the Volodin hermitian algebraic K-theory for a (discrete) ring with an involution and show that it is isomorphic to Karoubi's hermitian algebraic K-theory. We also construct the Volodin model X(R _*) of hermitian algebraic K-theory for a simplicial ring R _* and show that it is a homotopy fiber of the map B Ô(R _*)B Ô(R _*)⁺. We also prove the general linear version of this result, which has been claimed in the existing literature, but whose proof was overlooked.     

On A(X) → TC(X) for some X

Let X be a connected based space and p be a two-regular prime number. If the fundamental group of X has order p, we compute the two-primary homotopy groups of the homotopy fiber of the trace map A(X) → TC(X) relating algebraic K-theory of spaces to topological cyclic homology. The proof uses a theorem of Dundas and an explicit calculation of the cyclotomic trace map K(ℤ[C_p])→ TC(ℤ[C_p]).     

Real C*-Algebras, United KK-Theory, and the Universal Coefficient Theorem

We define united KK-theory for real C*-algebras A and B such that A is separable and B is -unital, extending united K-theory in the sense that KK^CRT( , B) = K^CRT(B). United KK-theory combines real, complex, and self-conjugate KK-theory; but unlike unaugmented KK-theory for real C*-algebras, it admits a Universal Coefficient Theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK^CRT(A,B) appears as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classifies KK-equivalence for real C*-algebras whose complexification is in the bootstrap category.Mathematics Subject Classification (2000): 19K35, 46L80.     

Topological Representation of Sheaf Cohomology of Sites

For a site & (with enough points), we construct a topological space X(&) and a full embedding ^* of the category of sheaves on & into those on X _(&) (i.e., a morphism of toposes :Sh (X_(&)) Sh(&)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms H^*(&,A) = H^*(X_(&), ^*A) for any Abelian sheaf A on &. As a particular case, this will give for any scheme Y a topological space X _(Y) and a functorial isomorphism between the étale cohomology H^*(Y _ét,A) and the ordinary sheaf cohomology H^*(X(_(Y),),^*A), for any sheaf A for the étale topology on Y.     

On the Homotopy Type of the Unitary Group and the Grassmann Space of Purely Infinite Simple C*-Algebras

We completely determine the homotopy groups _n (.) of the unitary group and the space of projections of purely infinite simple C ^*-algebras in terms of K-theory. We also prove that the unitary group of a purely infinite simple C ^*-algebra A is a contractible topological space if and only if K0(A) = K1(A) = {0}, and again if and only if the unitary group of the associated generalized Calkin algebra L(HA) / K(HA) is contractible. The well-known Kuiper's theorem is extended to a new class of C ^*-algebras.