Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStable Elements and Property (S)
252
0
0.0
(
0
)
Authors
Joan Bosa
Ask ChatGPT about the research
No Arabic abstract
We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of its support, and study these concepts from different sides : hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions as Regularity, $omega$-comparison, Corona Factorization Property, property R, etc.. are equivalent under mild assumptions.
rate research
Read More
We consider a Banach algebra $A$ with the property that, roughly speaking, sufficiently many irreducible representations of $A$ on nontrivial Banach spaces do not vanish on all square zero elements. The class of Banach algebras with this property turns out to be quite large -- it includes $C^*$-algebras, group algebras on arbitrary locally compact groups, commutative algebras, $L(X)$ for any Banach space $X$, and various other examples. Our main result states that every derivation of $A$ that preserves the set of quasinilpotent elements has its range in the radical of $A$.
Let $A$ be a unital separable simple ${cal Z}$-stable C*-algebra which has rational tracial rank at most one and let $uin U_0(A),$ the connected component of the unitary group of $A.$ We show that, for any $epsilon>0,$ there exists a self-adjoint element $hin A$ such that $$ |u-exp(ih)|<epsilon. $$ The lower bound of $|h|$ could be as large as one wants. If $uin CU(A),$ the closure of the commutator subgroup of the unitary group, we prove that there exists a self-adjoint element $hin A$ such that $$ |u-exp(ih)| <epsilon and |h|le 2pi. $$ Examples are given that the bound $2pi$ for $|h|$ is the optimal in general. For the Jiang-Su algebra ${cal Z},$ we show that, if $uin U_0({cal Z})$ and $epsilon>0,$ there exists a real number $-pi<tle pi$ and a self-adjoint element $hin {cal Z}$ with $|h|le 2pi$ such that $$ |e^{it}u-exp(ih)|<epsilon. $$
We investigate the orthogonality preserving property for pairs of mappings on inner product $C^*$-modules extending existing results for a single orthogonality-preserving mapping. Guided by the point of view that the $C^*$-valued inner product structure of a Hilbert $C^*$-module is determined essentially by the module structure and by the orthogonality structure, pairs of linear and local orthogonality-preserving mappings are investigated, not a priori bounded. The intuition is that most often $C^*$-linearity and boundedness can be derived from the settings under consideration. In particular, we obtain that if $mathscr{A}$ is a $C^{*}$-algebra and $T, S:mathscr{E}longrightarrow mathscr{F}$ are two bounded ${mathscr A}$-linear mappings between full Hilbert $mathscr{A}$-modules, then $langle x, yrangle = 0$ implies $langle T(x), S(y)rangle = 0$ for all $x, yin mathscr{E}$ if and only if there exists an element $gamma$ of the center $Z(M({mathscr A}))$ of the multiplier algebra $M({mathscr A})$ of ${mathscr A}$ such that $langle T(x), S(y)rangle = gamma langle x, yrangle$ for all $x, yin mathscr{E}$. In particular, for adjointable operators $S$ we have $T=(S^*)^{-1}$, and any bounded invertible module operator $T$ may appear. Varying the conditions on the mappings $T$ and $S$ we obtain further affirmative results for local operators and for pairs of a bounded and of an unbounded module operator with bounded inverse, among others. Also, unbounded operators with disjoint ranges are considered. The proving techniques give new insights.
In this thesis, we investigate the proof of the Baum-Connes Conjecture with Coefficients for a-$T$-menable groups. We will mostly and essentially follow the argument employed by N. Higson and G. Kasparov in the paper [Nigel Higson and Gennadi Kasparov. $E$-theory and $KK$-theory for groups which act properly and isometrically on Hilbert space. Invent. Math., 144(1):23-74, 2001]. The crucial feature is as follows. One of the most important point of their proof is how to get the Dirac elements (the inverse of the Bott elements) in Equivariant $KK$-Theory. We prove that the group homomorphism used for the lifting of the Dirac elements is an isomorphism in the case of our interests. Hence, we get a clear and simple understanding of the lifting of the Dirac elements in the Higson-Kasparov Theorem. In the course of our investigation, on the other hand, we point out a problem and give a fixed precise definition for the non-commutative functional calculus which is defined in the paper In the final part, we mention that the $C^*$-algebra of (real) Hilbert space becomes a $G$-$C^*$-algebra naturally even when a group $G$ acts on the Hilbert space by an affine action whose linear part is of the form an isometry times a scalar and prove the infinite dimensional Bott-Periodicity in this case by using Fells absorption technique.
Working within the framework of free actions of countable amenable groups on compact metrizable spaces, we show that the small boundary property is equivalent to a density version of almost finiteness, which we call almost finiteness in measure, and that under this hypothesis the properties of almost finiteness, comparison, and $m$-comparison for some nonnegative integer $m$ are all equivalent. The proof combines an Ornstein-Weiss tiling argument with the use of zero-dimensional extensions which are measure-isomorphic over singleton fibres. These kinds of extensions are also employed to show that if every free action of a given group on a zero-dimensional space is almost finite then so are all free actions of the group on spaces with finite covering dimension. Combined with recent results of Downarowicz-Zhang and Conley-Jackson-Marks-Seward-Tucker-Drob on dynamical tilings and of Castillejos-Evington-Tikuisis-White-Winter on the Toms-Winter conjecture, this implies that crossed product C$^*$-algebras arising from free minimal actions of groups with local subexponential growth on finite-dimensional spaces are classifiable in the sense of Elliotts program. We show furthermore that, for free actions of countably infinite amenable groups, the small boundary property implies that the crossed product has uniform property $Gamma$, which under minimality confirms the Toms-Winter conjecture for such crossed products by the aforementioned work of Castillejos-Evington-Tikuisis-White-Winter.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
