The real spectrum of an ideal and KO-theory exact sequences

A construction in abstract real algebra is used to define invariants S _n(A) of commutative rings, with or without identity. If A=C(X) is the ring of continuous real functions on a compact space, then S _n(A) = k0^–n(X), and, for any A, S _n(A) Z[1/2]-W _n(A) Z[1/2], where the W _n(A) are the Witt groups of A. In addition, a short exact sequence of rings yields a long exact sequence of the groups S _n. The functors S _n(A) thus provide a solution of a problem proposed by Karoubi. This paper primarily deals with the exact sequences involving a ring A and an ideal I A. Work supported in part by NSF Grant DMS85-06816.     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

The algebraic theory of torsion. II: Products

The algebraic K-theory product K ₀(A) K ₁ B K ₁(A B) for rings A, B is given a chain complex interpretation, using the absolute torsion invariant introduced in Part I. Given a finitely dominated A-module chain complex C and a round finite B-module chain complex D, it is shown that the A B-module chain complex C D has a round finite chain homotopy structure. Thus, if X is a finitely dominated CW complex and Y is a round finite CW complex, the product X × Y is a CW complex with a round finite homotopy structure.     

Abstract Bott periodicity inKK-theory

We show that the universalC*-algebras KqA and K²A are homotopy equivalent and define abstract analogues of the Bott elements inKK-theory.     

Pluriadjoint bundles of polarized surfaces

LetX be a complex connected projective smooth algebraic surface and letL be an ample line bundle onX. The maps associated with the pluriadjoint bundles (K _X L) ¹,t2, are studied by combining an ampleness result forK _X L with a very recent result by Reider. It turns out that apart from some exceptions and up to reductions, 1) (K _X L)³ is very ample; 2) (K _X L) ² is ample and spanned by global sections and is very ample unless eitherg (L)=2 (arithmetic genus ofL) orX contains an elliptic curveE withE ²=0,E·L=1;3) when (K _X L) ² is not very ample, the associated map has degree 4, equality implying thatg (L)=2 and .     

Noncommutative Chern Characters of Compact Lie Group C*-Algebras

For compact Lie groups, the Chern characters K^*(G) Q H^* _DR(G;Q) have been already constructed. In this paper, we construct and study the corresponding noncommutative Chern characters. They are homomorphisms ch_C*: K_*(C^*(G)) from quantum K-groups into entire current periodic cyclic homology groups of group C^*-algebras. We also obtain the corresponding algebraic version ch_alg: K_*(C^*(G)) HP_*(C^*(G)), which can be identified with the classical Chern character K_* (C(T)) HP_* (C(T)), where T is the maximal torus of G.     

Tame and wild subspace problems

Assume thatB is a finite-dimensional algebra over an algebraically closed fieldk, B _d =Spec k[(B _d ] is the affine algebraic scheme whoseR-points are theB _k k[B_d]-module structures onR ^d, and M_d is a canonical B_k k[B_d]-module supported by k[B_d]^d. Further, say that an affine subscheme of B_d isclass true if the functor F_gn X M_{d k[B]} X induces an injection between the sets of isomorphism classes of indecomposable finite-dimensional modules over k[] andB. If B_d contains a class-true plane for somed, then the schemes B_e contain class-true subschemes of arbitrary dimensions. Otherwise, each B_d contains a finite number of classtrue puncture straight linesL(d, i) such that for eachn, almost each indecomposableB-module of dimensionn is isomorphic to someF _{L(d, i)} (X); furthermore,F _{L(d, i)} (X) is not isomorphic toF _{L(l, j)} (Y) if(d, i) (l, j) andX 0. The proof uses a reduction to subspace problems, for which an inductive algorithm permits us to prove corresponding statements.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 313–352, March, 1993.     

The obstruction to excision in K-theory and in cyclic homology

Let f:A→B be a ring homomorphism of not necessarily unital rings and an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in K-theory is the failure of the map between relative K-groups K_*(A:I)→K_*(B:f(I)) to be an isomorphism; it is measured by the birelative groups K_*(A,B:I). Similarly the groups HN_*(A,B:I) measure the obstruction to excision in negative cyclic homology. We show that the rational Jones-Goodwillie Chern character induces an isomorphism

Operator space tensor products of <Emphasis Type="Italic">C</Emphasis>*-algebras

For C*-algebras A and B, the identity map from into A _λ B is shown to be injective. Next, we deduce that the center of the completion of the tensor product A⊗B of two C*-algebras A and B with centers Z(A) and Z(B) under operator space projective norm is equal to . A characterization of isometric automorphisms of and A _h B is also obtained. Dedicated to Ajit Iqbal Singh on her 65th birthday.     

On commuting compact self-adjoint operators on a Pontryagin space

Suppose thatA ₁,A ₂, ...,A _n are compact commuting self-adjoint linear maps on a Pontryagin spaceK of indexk and that their joint root subspaceM ₀ at the zero eigenvalue in ⁿ is a nondegenerate subspace. Then there exist joint invariant subspacesH andF inK such thatK=FH,H is a Hilbert space andF is finite-dimensional space withkdimF(n+2)k. We also consider the structure of restrictionsA _j|F in the casek=1.     

The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.     

Adjoints of semigroups acting on vector-valued function spaces

LetT(t) be the translation group onY=C ₀(ℝ×K)=C ₀(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY ^⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y ^⊙=L ¹(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT ₀(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT ₀(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ ^⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C ₀(Ω;X) andL ^p (μ;X). This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful to the CWI and the Dutch National Science Foundation NWO for financial support.     

Extremal signatures for bivariate Chebyshev approximation problems

The problem of finding a Chebyshev solution of the real matrix equationAX+YB=C, whereC is anm×n matrix, is considered. This equation is equivalent to a linear system [I _n A,B ^T I _m ]z=d. The characterization and the computation of best linear Chebyshev approximations are connected with the notion of extremal signature. The purpose of this paper is to analyze the extremal signatures of this problem.     

NonstableK-Theory for operator algebras

We show that the homotopy groups of the group of quasi-unitaries inC ^*-algebras form a homology theory on the category of allC ^*-algebras which becomes topologicalK-theory when stabilized. We then show how this functorial setting, in particular the half-exactness of the involved functors, helps to calculate the homotopy groups of the group of unitaries in a series ofC ^*-algebras. The calculations include the case of all AbelianC ^*-algebras and allC ^*-algebras of the formAB, whereA is one of the Cuntz algebras O_n n=2, 3, ..., an infinite dimensional simpleAF-algebra, the stable multiplier or corona algebra of a-unitalC ^*-algebra, a properly infinite von Neumann algebra, or one of the projectionless simpleC ^*-algebras constructed by Blackadar.     

The Hilbert Scheme Parameterizing Finite Length Subschemes of the Line with Support at the Origin

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that A _k k[x]_(x)/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x]_(x).     

Caractérisation par l'uniformité des fibrés universels sur la Grassmanienne

Conclusion SoitE un fibré uniforme surG(d,n). Nous avons étudié exhaustivement les cas où rangEd. montrant queE est soit somme directe de fibrés ou droites, soit, sir=d, indécomposable et isomorphe àH _d O _G () où àH _d ^* O _G ().Si rangE=(n-d), nous avons étudié les cas oùE est décomposable en fibrés en droites ou indécomposable et isomorphe àQ _{n d} O _G () où àQ _n ^* _d O _G (). L'étude des fibrés indécomposables de rangd, et (n-d) nous a montré que les quatre fibrés universels surG(d,n), sont caractérisés par leur rang, leur polynôme de Chern et al propriété d'uniformité.     

A classifying space forK 1 (X, R) and extensions to Hilbert modules

In this paper we show that the self-adjoint Fredholm operators in a type II factor form a classifying space forK ¹ (X) R for X a compact Hausdorff space. We also extend this result to the standard Hilbert module over a simple, purely infinite C^*-algebra which is either unital or has a countable approximate identity consisting of projections.     

Lifting-Probleme für Vektorfunktionen und ⊗-Sequenzen

In this article the topologically exact sequences of locally convex spaces are characterized for which for every locally convex space F the map id : FE F Q is a homomorphism, or equivalently, the map id L : FK F E is a topological injection. This is motivated by the problem of lifting Q-valued functions with certain given properties to E-valued functions with the same or slightly weaker properties, which may also be considered as the investigation of parameter dependences of solutions of linear (differential) equations. Applications to partial differential equations and to Fredholm functions are given.