Space Of Automorphisms Research Articles

On Uniformly Finitely Extensible Banach spaces

We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno and Plichko (2009) [39] and Castillo and Plichko (2010) [18]. We show that they have the Uniform Approximation Property of Pełczyński and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstrauss and Rosenthal – do there exist automorphic spaces other than c0(I) and ℓ2(I)? – showing that a space all whose subspaces are UFO must be automorphic when it is Hereditarily Indecomposable (HI), and a Hilbert space when it is either locally minimal or isomorphic to its square. We will finally show that most HI – among them, the super-reflexive HI space constructed by Ferenczi – and asymptotically ℓ2 spaces in the literature cannot be automorphic.

On the $KK$-theory of strongly self-absorbing $C^{*}$-algebras

Let $\mathcal D$ and $A$ be unital and separable $C^{*}$-algebras; let $\mathcal D$ be strongly self-absorbing. It is known that any two unital ${}^*$-homomorphisms from $\mathcal D$ to $A \otimes \mathcal D$ are approximately unitarily equivalent. We show that, if $\mathcal D$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\mathcal D$ is asymptotically inner. Moreover, the space of automorphisms of $\mathcal D$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,(\mathrm{Aut}(\mathcal D)]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\mathcal D$. As an application, we give a description of the Kasparov group $KK(\mathcal D, A\otimes \mathcal D)$ in terms of $^*$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\mathcal D, A\otimes \mathcal D)$ is isomorphic to $K_0(A\otimes \mathcal D)$.

On automorphic Banach spaces

A Banach space X will be called extensible if every operator E → X from a subspace E ⊂ X can be extended to an operator X → X. Denote by dens X. The smallest cardinal of a subset of X whose linear span is dense in X, the space X will be called automorphic when for every subspace E ⊂ X every into isomorphism T: E → X for which dens X/E = dens X/TE can be extended to an automorphism X → X. Lindenstrauss and Rosenthal proved that c0 is automorphic and conjectured that c0 and l2 are the only separable automorphic spaces. Moreover, they ask about the extensible or automorphic character of c0(Γ), for Γ uncountable. That c0(Γ) is extensible was proved by Johnson and Zippin, and we prove here that it is automorphic and that, moreover, every automorphic space is extensible while the converse fails. We then study the local structure of extensible spaces, showing in particular that an infinite dimensional extensible space cannot contain uniformly complemented copies of lnp, 1 ≤ p < ∞, p ≠ 2. We derive that infinite dimensional spaces such as Lp(μ), p ≠ 2, C(K) spaces not isomorphic to c0 for K metric compact, subspaces of c0 which are not isomorphic to c0, the Gurarij space, Tsirelson spaces or the Argyros-Deliyanni HI space cannot be automorphic.

MultiTowers, conjugacies and codes: Three theorems in Ergodic theory, one variation on Rokhlin’s Lemma

We show that three theorems about the measurable dynamics of a fixed aperiodic measure preserving transformation of a Lebesgue probability space are equivalent. One theorem asserts that the conjugates of are dense in the uniform topology on the space of automorphisms. The other two results assert the existence of a partition of the space with special properties. One partition result shows that given a mixing Markov chain, there is a code (i.e., a partition of the space) so that the itinerary process given by and the partition has the distribution of the given Markov Chain. The other partition result is a generalization of the Rokhlin Lemma, stating that the space can be partitioned into denumerably many columns and the measures of the columns can be prescribed in advance. Thus the first two results are equivalent to this strengthening of Rokhlin's Lemma.

Automorphisms ofp–compact groups and their root data

We construct a model for the space of automorphisms of a connected p-compact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a p-compact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper ``The classification of 2-compact groups'', where we prove the conjectured classification of 2-compact groups and determine their automorphism spaces.

Subdifferentiability of Real Functions

In this paper, we show that nowhere monotone functions are the key ingredients to construction of continuous functions, absolutely continuous functions, and Lipschitz functions with large subdifferentials on the real line. Let $\partial_{c}f, \partial_{a}f$ denote the Clarke subdifferential and approximate subdifferential respectively. We construct absolutely continuous functions on $\R$ such that $\partial_{a}f=\partial_{c}f\equiv \R$. In the Banach space of continuous functions defined on $[0,1]$, denoted by $C[0,1]$, with the uniform norm, we show that there exists a residual and prevalent set $D\subset C[0,1]$ such that $\partial_{a}f=\partial_{c}f\equiv \R$ on $[0,1]$ for every $f\in D$. In the space of automorphisms we prove that most functions $f$ satisfy $\partial_{a}f=\partial_{c}f\equiv [0,+\infty)$ on $[0,1]$. The subdifferentiability of the Weierstrass function and the Cantor function are completely analyzed. Similar results for Lipschitz functions are also given.

Shyness of Sets in the Space of Automorphisms on [0, 1

In a 1986 paper, Graf, Mauldin and Williams introduced the notion of a random homeomorphism by defining certain natural probability measures on the space of increasing homeomorphisms on the unit interval [0, 1]. We contrast their study by investigating the shyness and prevalence of some of the sets arising in their article.

Algebraic structure of games

Game theory and algebra become inextricably intertwined once one recognizes that the notion of a permutation of players gives rise to a representation of the symmetric group in the space of automorphisms of the vector space of games. In this paper the authors utilize this observation to attempt to turn the typical approach to game-theoretic problems on its head by analyzing the space of games from an algebraic viewpoint. In this way we find, quite surprisingly, that it is the notion of inessential, or additive, games, previously thought to be of little or no interest, that is of prime significance in describing the structure of the space of games.