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.     

Localizations of Transfors

Let be n-dimensional teisi, i.e., higher-dimensional Gray-categorical structures. The following questions can be asked. Does a left q-transfor , i.e., a functor 2 _q , induce a right q-transfor , i.e., a functor More generally, does a functor induce a functor For k-arrows c and whose (k – 1)-sources and targets agree, does a q-transfor induce a q-transfor , for appropriate k-arrows For k-arrows c and whose (k – 1)-sources and targets agree, does a q-transfor induce a (q + k + 1)-transfor , for appropriate k-arrows I give answers to these questions in the cases where n-dimensional teisi and their tensor product have been defined, i.e., for n 3, and for n up to 5 in some cases that do not need all data and axioms of n-dimensional teisi.I apply the above to compositions in teisi, in particular to braidings and syllepses. One of the results is that a braiding on a monoidal 2-category induces a pseudo-natural transformation , where is the reverse of ? –, which is almost, but not quite, equal to – ?. However, in higher dimensions need not be reversible, so a braiding on a higher-dimensional tas can not be seen as a transfor A B B A.     

Nonuniqueness Sequences for Weighted Algebras of Holomorphic Functions in the Unit Circle

Suppose that is a system of continuous subharmonic functions in the unit disk and is the class of holomorphic functions f in such that log|f(z)| ≤ B _f p _f (z) + C _f , z ∈ , where B _f and C _f are constants and p _f ∈ . We obtain sufficient conditions for a given number sequence Λ = { λ_n} ⊂ to be a subsequence of zeros of some nonzero holomorphic function from , i.e., Λ is a nonuniqueness sequence for .__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 775–787.Original Russian Text Copyright ©2005 by L. Yu. Cherednikova.     

Purely infinite C*-Algebras: Ideal-preserving zero homotopies

We show that if A is a separable, nuclear, -absorbing (or strongly purely infinite) C*-algebra which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form where Γ is a finite connected graph (and is the algebra of continuous functions on Γ that vanish at a distinguished point ).We show further that if B is any separable, nuclear C*-algebra, then is isomorphic to a crossed product where D is an inductive limit of C*-algebras of the form (and D is -absorbing and homotopic to zero in an ideal-system preserving way).Received: December 2003 Revision: July 2004 Accepted: July 2004     

Isomorphic chain complexes of Hamiltonian dynamics on tori

In this work, for a given smooth, generic Hamiltonian ${H : \mathbb{S}^{1} \times \mathbb{T}^{2n} \rightarrow \mathbb{R}}$ on the torus ${\mathbb{T}^{2n} = \mathbb{R}^{2n}/\mathbb{Z}^{2n}}$ we construct a chain isomorphism ${\Phi_{*} : (C_{*}(H), \partial^{M}_{*}) \rightarrow (C_{*}(H), \partial^{F}_{*})}$ between the Morse complex of the Hamiltonian action AH on the free loop space of the torus ${\Lambda_{0}(\mathbb{T}^{2n})}$ and the Floer complex. Though both complexes are generated by the critical points of A _H , their boundary operators differ. Therefore, the construction of ${\Phi}$ is based on counting the moduli spaces of hybrid-type solutions which involves stating a new non-Lagrangian boundary value problem for Cauchy–Riemann type operators not yet studied in Floer theory. We finally want to note that the problem is completely symmetric. So we also could construct an isomorphism ${\Psi_{*} : (C_{*}(H), \partial^{F}_{*}) \rightarrow (C_{*}(H), \partial^{M}_{*})}$ .     

On the relative Wall-Witt groups

LetR* be a simplicial involutive ring. According to certain involutions onK(R*) and _ε L R _∗, there are 1/2-local splittings and . It is known [2] that _ε L _\ga ^α R _∗, the Wall-Witt group. SupposeI→R S is a split extension of discrete involutive rings withI ²=0, andI is a freeS-bimodule. Then we have and . The trace map Tr: Prim _n ∧*M(I ⊗ ℚ)→ ₀ ρ _n ;I ⊗ ℚ) is an isomorphism. We prove in Lemma 1 that the trace map Tr is ℤ/2-equivariant. In Theorem 2 we show that under a certain assumption the rational relative Wall-Witt group vanishes. Theorem 2 can be extended to a more general case (Theorem 3) by employing Goodwillie’s reduction technique [3]. This work was partially supported by KOSEF under Grant 923-0100-010-1.     

On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings f:D\{ b } ? \mathbbR³ f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR³ D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR³ b \in {\mathbb{R}^3} , we prove the following statement: Let a point y ₀ belong to [`(\mathbbR³)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRⁿ g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} such that A ⊂ U.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators

In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF _*, of classical distributions. In particular, we have the following inclusion $WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),$ where Ω is an open subset of ${\mathbb {R}^n}In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF _*, of classical distributions. In particular, we have the following inclusion

WF*(u) ì WF*(Pu) èS,     u ? D￠(W),WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),      10. Sums of Bisectorial Operators and Applications      Wolfgang Arendt  Shangquan Bu 《Integral Equations and Operator Theory》2005,52(3):299-321 We study sums of bisectorial operators on a Banach space X and show that interpolation spaces between X and D(A) (resp. D(B)) are maximal regularity spaces for the problem Ay + By = x in X. This is applied to the study of regularity properties of the evolution equation u′ + Au = f on for or and the evolution equation u′ + Au = f on [0, 2π] with periodic boundary condition u(0) = u(2π) in or       11. Sui fibrati con struttura quaternionale generalizzata      Stefano Marchiafava  Giuliano Romani 《Annali di Matematica Pura ed Applicata》1975,107(1):131-157 Summary Quaternion generalized fiber bundles are studied, both isomorphic to global tensorial product ordinary quaternion fiber bundles right and left respectively) and quite general ones. A cohomology class is considered which represents the obstruction in order the fiber bundle be a tensorial product. Several properties and a splitting principle are proved for bundles . On this ground and founding on a convenient bundle BE → X associated to jaz (that we call Bonan's bundle and for which ɛ( =ɛ(BE)) relations are stated among Stiefel-Whitney classes of , BE and the class ɛ. Entrata in Redazione il 14 agosto 1974. Lavoro eseguito con contributo del C.N.R., nell'ambito del Gruppo Nazionale per le Strutture Algebriche e Geometriche e loro Applicazioni.      12. Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$      Apoloniusz?TyszkaEmail author 《Aequationes Mathematicae》2004,67(3):225-235 Summary. Let We say that preserves the distance d 0 if for each implies Let A n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D n denote the set of all positive numbers d with the property: if and then there exists a finite set S xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and       13. Some prime factorization results for type II<Subscript>1</Subscript> factors      Narutaka?OzawaEmail author  Sorin?PopaEmail author 《Inventiones Mathematicae》2004,156(2):223-234 We prove several unique prime factorization results for tensor products of type II1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1. In particular, we show that if is isomorphic to a subfactor in , for some 2ri,sj, then mn. Mathematics Subject Classification (2000) Primary 46L10; Secondary 20F67      14. Relative MilnorK-theory      Marc Levine 《K-Theory》1992,6(2):113-175 LetR be a commutative, semi-local ring,I 1, ...,I s ideals. In this paper, we define therelative Milnor K-groups of (R;I 1, ...,I s ),K p M (R;I 1, ...,I s ), and show that these groups have many of the properties of the usual MilnorK-groups of a field. In particular, assuming a weak condition on the ideals, we show thatK p M (R;I 1, ...,I s ) is isomorphic to the weightp portion of the relative QuillenK-groupK p (R;I 1, ...,I s ), after inverting (p–1)!. We also define the relative group homology of GL n (R;I 1, ...,I s ), and show thatK p M (R;I 1, ...,I s ) is isomorphic toH p (GLp(R;I 1, ...,I s ))/Im(H p (GL p–1 (R;I 1, ...,I s ))). Finally, we consider a generalization to the relative setting of Kato's conjecture asserting that the Galois symbol gives an isomorphism fromK p M (F)/l v to , and show that this relative version of Kato's conjecture implies the Quillen-Lichtenbaum conjectures asserting the Chern class:       15. <Emphasis Type="Italic">K</Emphasis>-homology of <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-categories and symmetric spectra representing <Emphasis Type="Italic">K</Emphasis>-homology      Michael Joachim 《Mathematische Annalen》2003,327(4):641-670 We define K-homology groups K * () for small C * -categories in terms of Hilbert modules over the C * -category . We also define a functor A f from the category of small C * -categories into the category of C * -algebras and show that there is a natural isomorphism . In addition, we give an easy construction of a functor from the category of C * -algebras into the category of symmetric spectra which represents K-homology, i.e. we show that the functor comes with a natural isomorphism for C * -algebras A. It then follows that the composition A f provides a functor that can be used in the Davis-Lück approach for constructing the Baum-Connes map.      16. Dual partially harmonic tensors and Brauer-Schur-Weyl duality      J. Hu 《Transformation Groups》2010,15(2):333-370 Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let $ \mathfrak{B}_n^{(f)} Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let \mathfrakBn(f) \mathfrak{B}_n^{(f)} be the two-sided ideal of the Brauer algebra \mathfrakBn( - 2m ) {\mathfrak{B}_n}\left( { - 2m} \right) over K generated by e 1 e 3⋯ e 2f-1 where 0 ≤ f ≤ [n/2]. Let HTf ?n \mathcal{H}\mathcal{T}_f^{ \otimes n} be the subspace of partial-harmonic tensors of valence f in V ⊗n . In this paper we prove that dimHTf ?n \mathcal{H}\mathcal{T}_f^{ \otimes n} and dim \textEn\textdK\textSp(V)( V ?n \mathord / \vphantom V ?n V ?n V ?n\mathfrakBn(f) ) {\text{En}}{{\text{d}}_{K{\text{Sp}}(V)}}\left( {{{{V^{ \otimes n}}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}} {{V^{ \otimes n}}}}} \right.} {{V^{ \otimes n}}}}\mathfrak{B}_n^{(f)}} \right) are both independent of K, and the natural homomorphism from \mathfrakBn( - 2m ) \mathord / \vphantom ( - 2m ) \mathfrakBn(f) \mathfrakBn(f) {\mathfrak{B}_n}{{\left( { - 2m} \right)} \mathord{\left/{\vphantom {{\left( { - 2m} \right)} {\mathfrak{B}_n^{(f)}}}} \right.} {\mathfrak{B}_n^{(f)}}} to \textEn\textdK\textSp(V)( V ?n \mathord / \vphantom V ?n V ?n V ?n\mathfrakBn(f) ) {\text{En}}{{\text{d}}_{K{\text{Sp}}(V)}}\left( {{{{V^{ \otimes n}}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}} {{V^{ \otimes n}}}}} \right.} {{V^{ \otimes n}}}}\mathfrak{B}_n^{(f)}} \right) is always surjective. We show that HTf ?n \mathcal{H}\mathcal{T}_f^{ \otimes n} has a Weyl filtration and is isomorphic to the dual of V ?n\mathfrakBn(f) \mathord / \vphantom V ?n\mathfrakBn(f) V V ?n\mathfrakBn( f + 1 ) {{{{V^{ \otimes n}}\mathfrak{B}_n^{(f)}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}\mathfrak{B}_n^{(f)}} V}} \right.} V}^{ \otimes n}}\mathfrak{B}_n^{\left( {f + 1} \right)} as an \textSp(V) - ( \mathfrakBn( - 2m ) \mathord / \vphantom ( - 2m ) \mathfrakBn( f + 1 ) \mathfrakBn( f + 1 ) ) {\text{Sp}}(V) - \left( {{\mathfrak{B}_n}{{\left( { - 2m} \right)} \mathord{\left/{\vphantom {{\left( { - 2m} \right)} {\mathfrak{B}_n^{\left( {f + 1} \right)}}}} \right.} {\mathfrak{B}_n^{\left( {f + 1} \right)}}}} \right) -bimodule. We obtain an \textSp(V) - \mathfrakBn {\text{Sp}}(V) - {\mathfrak{B}_n} -bimodules filtration of V ⊗n such that each successive quotient is isomorphic to some ?( l) ?zg,l\mathfrakBn \nabla \left( \lambda \right) \otimes {z_{g,\lambda }}{\mathfrak{B}_n} with λ ⊢ n 2g, ℓ(λ)≤m and 0 ≤ g ≤ [n/2], where ∇(λ) is the co-Weyl module associated to λ and z g,λ is an explicitly constructed maximal vector of weight λ. As a byproduct, we show that each right \mathfrakBn {\mathfrak{B}_n} -module zg,l\mathfrakBn {z_{g,\lambda }}{\mathfrak{B}_n} is integrally defined and stable under base change.      17. Equivariant K-Homology and Restriction to Finite Cyclic Subgroups      Michel Matthey  Guido Mislin 《K-Theory》2004,32(2):167-179 For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K * G (E(G, in)), is isomorphic to K * G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.      18. Correction to: ‘The UCT, the Milnor Sequence, and a Canonical Decomposition of the Kasparov Groups’      Claude L. Schochet 《K-Theory》1998,14(2):197-199 In this note we correct a mistake in K-Theory 10 (1996), 49–72. In that paper we asserted that under bootstrap hypotheses the short exact sequence which arises in the computation ofKK(A,B) (is a split sequence. This is not always the case. ThusKK(A,B) (decomposes into the three components and However, this is a decomposition in the sense of composition series, not as three direct summands. The same correction applies to the Milnor sequence. If there is no primepfor which bothK(A) (andK(B) *haveptorsion then the decomposition is indeed as direct summands. The other results of the paper are unaffected.      19. Approximation of bandlimited functions by finite exponential sums      S. Norvidas 《Lithuanian Mathematical Journal》2009,49(2):185-189 For a compact set K in ℝ n , let B 2 K be the set of all functions f ∈ L 2(ℝ2) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B 2 K by finite exponential sumsin the space , as τ → ∞.      20. THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS      D. H. BOO  C. G. PARK 《数学年刊B辑(英文版)》2000,21(4):441-452 Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed&gt;1 and {0} if d&gt;1.Let Bcd be a cd-homogeneous C^*-algebra over S^2&#215;T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2&#215;Z^m-2) in Bcd &#215;C^*(Z^m-3) by a totally skew multiplier ρ on T^2&#215;Z^m-2。It is shown that Sρ^cd&#215;Mρ∞ is isomorphic to C(S^2)&#215;C^*(T^2&#215;Z^m-2,ρ）&#215; Mcd(C)&#215;Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.

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Sums of Bisectorial Operators and Applications

We study sums of bisectorial operators on a Banach space X and show that interpolation spaces between X and D(A) (resp. D(B)) are maximal regularity spaces for the problem Ay + By = x in X. This is applied to the study of regularity properties of the evolution equation u′ + Au = f on for or and the evolution equation u′ + Au = f on [0, 2π] with periodic boundary condition u(0) = u(2π) in or      

Sui fibrati con struttura quaternionale generalizzata

Summary Quaternion generalized fiber bundles are studied, both isomorphic to global tensorial product ordinary quaternion fiber bundles right and left respectively) and quite general ones. A cohomology class is considered which represents the obstruction in order the fiber bundle be a tensorial product. Several properties and a splitting principle are proved for bundles . On this ground and founding on a convenient bundle B_E → X associated to jaz (that we call Bonan's bundle and for which ɛ( =ɛ(B_E)) relations are stated among Stiefel-Whitney classes of , B_E and the class ɛ.

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Entrata in Redazione il 14 agosto 1974.

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Lavoro eseguito con contributo del C.N.R., nell'ambito del Gruppo Nazionale per le Strutture Algebriche e Geometriche e loro Applicazioni.     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

Some prime factorization results for type II<Subscript>1</Subscript> factors

We prove several unique prime factorization results for tensor products of type II₁ factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1. In particular, we show that if is isomorphic to a subfactor in , for some 2r_i,s_j, then mn. Mathematics Subject Classification (2000) Primary 46L10; Secondary 20F67     

Relative MilnorK-theory

LetR be a commutative, semi-local ring,I ₁, ...,I _s ideals. In this paper, we define therelative Milnor K-groups of (R;I ₁, ...,I _s ),K _p ^M (R;I ₁, ...,I _s ), and show that these groups have many of the properties of the usual MilnorK-groups of a field. In particular, assuming a weak condition on the ideals, we show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic to the weightp portion of the relative QuillenK-groupK _p (R;I ₁, ...,I _s ), after inverting (p–1)!. We also define the relative group homology of GL _n (R;I ₁, ...,I _s ), and show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic toH _p (GL_p(R;I ₁, ...,I _s ))/Im(H _p (GL _p–1 (R;I ₁, ...,I _s ))). Finally, we consider a generalization to the relative setting of Kato's conjecture asserting that the Galois symbol gives an isomorphism fromK _p ^M (F)/l ^v to , and show that this relative version of Kato's conjecture implies the Quillen-Lichtenbaum conjectures asserting the Chern class:

<Emphasis Type="Italic">K</Emphasis>-homology of <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-categories and symmetric spectra representing <Emphasis Type="Italic">K</Emphasis>-homology

We define K-homology groups K _* () for small C ^* -categories in terms of Hilbert modules over the C ^* -category . We also define a functor A ^f from the category of small C ^* -categories into the category of C ^* -algebras and show that there is a natural isomorphism . In addition, we give an easy construction of a functor from the category of C ^* -algebras into the category of symmetric spectra which represents K-homology, i.e. we show that the functor comes with a natural isomorphism for C ^* -algebras A. It then follows that the composition A ^f provides a functor that can be used in the Davis-Lück approach for constructing the Baum-Connes map.     

Dual partially harmonic tensors and Brauer-Schur-Weyl duality

Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let $ \mathfrak{B}_n^{(f)} Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let \mathfrakB_n^(f) \mathfrak{B}_n^{(f)} be the two-sided ideal of the Brauer algebra \mathfrakB_n( - 2m ) {\mathfrak{B}_n}\left( { - 2m} \right) over K generated by e ₁ e _3⋯ e _2f-1 where 0 ≤ f ≤ [n/2]. Let HT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} be the subspace of partial-harmonic tensors of valence f in V ^⊗n . In this paper we prove that dimHT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} and dim \textEn\textd_K\textSp(V)( V ^?n \mathord

Equivariant K-Homology and Restriction to Finite Cyclic Subgroups

For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K _* ^G (E(G, in)), is isomorphic to K _* ^G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.     

Correction to: ‘The UCT, the Milnor Sequence, and a Canonical Decomposition of the Kasparov Groups’

In this note we correct a mistake in K-Theory 10 (1996), 49–72. In that paper we asserted that under bootstrap hypotheses the short exact sequence

Approximation of bandlimited functions by finite exponential sums

For a compact set K in ℝ ⁿ , let B ² _K be the set of all functions f ∈ L ²(ℝ²) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B ² _K by finite exponential sums

THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS

Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed&gt;1 and {0} if d&gt;1.Let Bcd be a cd-homogeneous C^*-algebra over S^2&#215;T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2&#215;Z^m-2) in Bcd &#215;C^*(Z^m-3) by a totally skew multiplier ρ on T^2&#215;Z^m-2。It is shown that Sρ^cd&#215;Mρ∞ is isomorphic to C(S^2)&#215;C^*(T^2&#215;Z^m-2,ρ）&#215; Mcd(C)&#215;Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.