arXiv subjects
Publications and source records attributed to V. Manuilov.
We give an example of a non-trivial asymptotic representation of the reduced C*-algebra of a free group. This example allows to evaluate the asymptotic tensor C*-norm of some elements in tensor product C*-algebras and to show semi-invertibility of the non-invertible extension of $C^*_r(\mathbb F_2)$ considered by Haagerup and Thorbjornsen.
Let $A$ be a C*-algebra, $J \subset A$ a C*-subalgebra, and let $B$ be a stable C*-algebra. Under modest assumptions we organize invertible C*-extensions of $A$ by $B$ that are trivial when restricted onto $J$ to become a group $Ext_J^{-1}(A,B)$, which can be computed by a six-term exact sequence which generalizes the excision six-term exact sequence in the first variable of $KK$-theory. Subsequently we investigate the relative K-homology which arises from the group of relative extensions by specializing to abelian C*-algebras. It turns out that this relative K-homology carries substantial information also in the operator theoretic setting from which the BDF theory was developed and we conclude the paper by extracting some of this information on approximation of normal operators.
Using ideas of S. Wassermann on non-exact $C^*$-algebras and property T groups, we show that one of his examples of non-invertible C*-extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a C*-extension which is not even invertible up to homotopy.
The algebra $Ψ(M)$ of order zero pseudodifferential operators on a compact manifold $M$ defines a well-known $C^*$-extension of the algebra $C(S^*M)$ of continuous functions on the cospherical bundle $S^*M\subset T^*M$ by the algebra $\K$ of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphism $T$ from $C_0(T^*M)$ to $\K$, which plays the role of a deformation for the commutative algebra $C_0(T^*M)$. Similar constructions exist also for operators and symbols with coefficients in a $C^*$-algebra. We show that the image of the above extension under the Connes--Higson construction is $T$ and that this extension can be reconstructed out of $T$. This explains, why the classical approach to the index theory coincides with the one based on asymptotic homomorphisms.
We introduce a new asymptotic one-sided and symmetric tensor norm, the latter of which can be considered as the minimal tensor norm on the category of separable C*-algebras with homotopy classes of asymptotic homomorphisms as morphisms. We show that the one-sided asymptotic tensor norm differs in general from both the minimal and the maximal tensor norms and we discuss its relation to semi-invertibility of C*-extensions.
We consider the semigroup $Ext(A,B)$ of extensions of a separable C*-algebra $A$ by a stable C*-algebra $B$ modulo unitary equivalence and modulo asymptotically split extensions. This semigroup contains the group $Ext^{-1/2}(A,B)$ of invertible elements (i.e. of semi-invertible extensions). We show that the functor $Ext^{1/2}(A,B)$ is homotopy invariant and that it coincides with the functor of homotopy classes of asymptotic homomorphisms from $C(\mathbb T)\otimes A$ to $M(B)$ that map $SA\subseteq C(\mathbb T)\otimes A$ into $B$.
Let $A$, $B$ be C*-algebras; $A$ separable, $B$ $σ$-unital and stable. We introduce a notion of translation invariance for asymptotic homomorphisms from $SA=C_0(\mathbb R)\otimes A$ to $B$ and show that the Connes-Higson construction applied to any extension of $A$ by $B$ is homotopic to a translation invariant asymptotic homomorphism. In the other direction we give a construction which produces extensions of $A$ by $B$ out of such a translation invariant asymptotic homomorphism. This leads to our main result; that the homotopy classes of extensions coincide with the homotopy classes of translation invariant asymptotic homomorphisms.
Let $A$ be a separable $C^*$-algebra and $B$ a stable $C^*$-algebra containing a strictly positive element. We show that the group $\Ext(SA,B)$ of unitary equivalence classes of extensions of $SA$ by $B$, modulo the extensions which are asymptotically split, coincides with the group of homotopy classes of such extensions. This is done by proving that the Connes-Higson construction gives rise to an isomorphism between $\Ext(SA,B)$ and the $E$-theory group $E(A,B)$ of homotopy classes of asymptotic homomorphisms from $S^2A$ to $B$.
Let $A$ be a separable $C^*$-algebra and let $B$ be a stable $C^*$-algebra with a strictly positive element. We consider the (semi)group $\Ext^{as}(A,B)$ (resp. $\Ext(A,B)$) of homotopy classes of asymptotic (resp. of genuine) homomorphisms from $A$ to the corona algebra $M(B)/B$ and the natural map $i:\Ext(A,B)\ar\Ext^{as}(A,B)$. We show that if $A$ is a suspension then $\Ext^{as}(A,B)$ coincides with $E$-theory of Connes and Higson and the map $i$ is surjective. In particular any asymptotic homomorphism from $SA$ to $M(B)/B$ is homotopic to some genuine homomorphism.
We show that the E-theory of Connes and Higson can be formulated in terms of C*-extensions in a way quite similar to the way in which the KK-theory of Kasparov can. The essential difference is that the role played by split extensions should be taken by asymptotically split extensions. We call an extension of a C*-algebra $A$ by a stable C*-algebra $B$ asymptotically split if there exists an asymptotic homomorphism consisting of right inverses for the quotient map. An extension is called semi-invertible if it can be made asymptotically split by adding another extension to it. Our main result is that there exists a one-to-one correspondence between asymptotic homomorphisms from $SA$ to $B$ and homotopy classes of semi-invertible extensions of $S^2A$ by $B$.
We study the $C^*$-algebras related to Mishchenko's version of asymptotic homomorphisms. In particular we show that their different versions are weakly homotopy equivalent but not isomorphic to each other. We give also the continuous version for these algebras.
We define for discrete finitely presented groups a new property related to their asymptotic representations. Namely we say that a groups has the property AGA if every almost representation generates an asymptotic representation. We give examples of groups with and without this property. For our example of a group $G$ without AGA the group $K^0(BG)$ cannot be covered by asymptotic representations of $G$.