arXiv math/06067942006-06-30math.OA

Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces

Uffe Haagerup, Agata Przybyszewska

英文摘要：

In this article it is proved, that every locally compact second countable group has a left invariant metric d, which generates the topology on G, and which is proper, ie. every closed d-bounded set in G is compact. Moreover, we obtain the following extension of a result due to N. Brown and E. Guentner: Every locally compact second countable $G$ admits a proper affine action on the reflexive and strictly convex Banach space $\bigoplus^{\infty}_{n=1} L^{2n}(G, dμ),$ where the direct sum is taken in the $l^2$-sense.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Hyperbolic groups admit proper affine isometric actions on $l^p$-spaces

In this paper, we show that hyperbolic groups admit proper affine isometric actions on $l^p$-spaces.

2004-11-10math.GRmath.OA

Evanescent affine isometric actions and weak identity excluding groups

We investiguate a property of affine isometric actions on Hilbert spaces called evanescence. Evanescent actions are the extreme opposite of irreducible actions. Every affine isometric action decomposes naturally into an evanescent part and an irreducible part. We study when this decomposition is unique. We also study when an action that has almost fixed points is automatically evanescent. We relate these questions to the identity excluding property for groups. We also relate them to the finiteness of the von Neumann algebras generated by the linear part of the action.

2021-05-21math.GRmath.OA

Automorphisms of Banach space projective tensor product of C*-algebras

For unital $C^*$-algebras $A$ and $B$, we completely characterize the isometric ($*$-) automorphisms of their Banach space projective tensor product $A\otimes^γB$. This leads to the characterization of inner and outer isometric $*$-automorphisms of $A\otimes^γB$, as well. As an application, we provide a partial affirmative answer to a question posed by Kaijser and Sinclair, viz., we prove that for unital $C^*$-algebras $A$ and $B$, the set of norm-one unitaries of $A\otimes^γB$ coincides with $U(A) \otimes U(B)$, where $U(A)$ is the unitary group of $A$. We also establish the fact that the rel…

2018-10-05math.OA

Classification of $L^p$ AF algebras

We define spatial $L^p$ AF algebras for $p \in [1, \infty) \setminus \{ 2 \}$, and prove the following analog of the Elliott AF algebra classification theorem. If $A$ and $B$ are spatial $L^p$ AF algebras, then the following are equivalent: 1) $A$ and $B$ have isomorphic scaled preordered $K_0$-groups. 2) $A \cong B$ as rings. 3) $A \cong B$ (not necessarily isometrically) as Banach algebras. 4) $A$ is isometrically isomorphic to $B$ as Banach algebras. 5) $A$ is completely isometrically isomorphic to $B$ as matrix normed Banach algebra. As background, we develop the theory of matrix normed $L…

2017-10-06math.OA

Irreducible affine isometric actions on Hilbert spaces

We undertake a systematic study of irreducible affine isometric actions of locally compact groups on Hilbert spaces. It turns out that, while that are a few parallels of this study to the by now classical theory of irreducible unitary representations, these two theories differ in several aspects (for instance, the direct sum of two irreducible affine actions can still be irreducible). One of the main tools we use is an affine version of Schur's lemma characterizing the irreducibility of an affine isometric group action. This enables us to describe for instance the irreducible affine isometric …

2014-11-09math.GRmath.OA

A Representation Theorem for Completely Contractive Dual Banach Algebras

In this paper, we prove that every completely contractive dual Banach algebra is completely isometric to a $w^\ast$-closed subalgebra the operator space of completely bounded linear operators on some reflexive operator space.

2007-02-07math.FAmath.OA

更多相关论文收起更多论文

Ergodic theory of affine isometric actions on Hilbert spaces

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine…

2020-10-22math.DSmath.GRmath.OA

Jordan weak amenability and orthogonal forms on JB*-algebras

We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized derivations from $\mathcal{J}$ into $\mathcal{J}^*$. We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all anti-symmetric orthogonal forms on $\mathcal{J}$, and of all Lie Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$.

2014-11-11math.FAmath.OA

A C(K) Banach space which does not have the Schroeder-Bernstein property

We construct a totally disconnected compact Hausdorff space N which has clopen subsets M included in L included in N such that N is homeomorphic to M and hence C(N) is isometric as a Banach space to C(M) but C(N) is not isomorphic to C(L). This gives two nonisomorphic Banach spaces of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces of the form C(K). N is obtained as a particular compactification of the pairwise disjoint union of a sequence of Ks for w…

2011-06-15math.FAmath.GNmath.OA

Harmonic cocycles, von Neumann algebras, and irreducible affine isometric actions

Let $G$ be a compactly generated locally compact group and $(π, \mathcal H)$ a unitary representation of $G.$ The $1$-cocycles with coefficients in $π$ which are harmonic (with respect to a suitable probability measure on $G$) represent classes in the first reduced cohomology $\bar{H}^1(G,π).$ We show that harmonic $1$-cocycles are characterized inside their reduced cohomology class by the fact that they span a minimal closed subspace of $\mathcal H.$ In particular, the affine isometric action given by a harmonic cocycle $b$ is irreducible (in the sense that $\mathcal H$ contains no non-empty,…

2016-12-28math.GRmath.OA

Uniform embeddings of bounded geometry spaces into reflexive Banach space

We show that every metric space with bounded geometry uniformly embeds into an explicit reflexive Banach space (a direct sum of l^p spaces). In the case of discrete groups we show the analogue of a-T-menability. That is, we construct a metrically proper affine isometric action on this Banach space.

2003-09-11math.FAmath.OA

Fourier spaces and completely isometric representations of Arens product algebras

Motivated by the definition of a semigroup compactification of a locally compact group and a large collection of examples, we introduce the notion of an (operator) "homogeneous left dual Banach algebra" (HLDBA) over a (completely contractive) Banach algebra $A$. We prove a Gelfand-type representation theorem showing that every HLDBA over $A$ has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^*$ with a compatible (matrix) norm and a type of left Arens product ${\scriptscriptstyle \square}$. Examples include all left Arens product alg…

2018-05-29math.FAmath.OA

Poincaré inequalities and rigidity for actions on Banach spaces

The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan's property (T) and is equivalent to the fact that $H^1(G,π)=0$ for every isometric representation $π$ of $G$ on $X$. The condition is expressed in terms of $p$-Poincaré constants and we provide examples of groups, which satisfy such conditions and for which $H^1(G,π)$ vani…

2014-03-30math.FAmath.GRmath.OA

Dynamical entropy in Banach spaces

We introduce a version of Voiculescu-Brown approximation entropy for isometric automorphisms of Banach spaces and develop within this framework the connection between dynamics and the local theory of Banach spaces discovered by Glasner and Weiss. Our fundamental result concerning this contractive approximation entropy, or CA entropy, characterizes the occurrence of positive values both geometrically and topologically. This leads to various applications; for example, we obtain a geometric description of the topological Pinsker factor and show that a C*-algebra is type I if and only if every mul…

2006-04-03math.DSmath.FAmath.OA

Quotients of Banach algebras acting on $L^p$-spaces

We show that the class of Banach algebras that can be isometrically represented on an $L^p$-space, for $p\neq 2$, is not closed under quotients. This answers a question asked by Le Merdy 20 years ago. Our methods are heavily reliant on our earlier study of Banach algebras generated by invertible isometries of $L^p$-spaces.

2016-03-29math.FAmath.OA

The Daugavet property of $C^$-algebras, $JB^$-triples, and of their isometric preduals

A Banach space $X$ is said to have the Daugavet property if every rank-one operator $T:X\longrightarrow X$ satisfies $\|Id + T\| = 1 + \|T\|$. We give geometric characterizations of this property in the settings of $C^*$-algebras, $JB^*$-triples and their isometric preduals. We also show that, in these settings, the Daugavet property passes to ultrapowers, and thus, it is equivalent to an stronger property called the uniform Daugavet property.

2004-12-13math.FAmath.OA

Regular normed bimodules

In this article, we will give a characterization of Banach bimodules over $C^*$-algebras of compact operators that arises from operator spaces as well as a characterization of (F)-Banach bundles amongst all (H)-Banach bundles over a hyper-Stonian space. These two characterizations are concerned with whether certain natural map from a Banach bimodule to its canonical bidual is isometric (we call such bimodule \emph{regular}).

2006-09-06math.FAmath.OA

Dilation theorem for p-approximate Schauder frames for separable Banach spaces

Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space $\mathcal{H}$ is image of a Riesz basis under an orthogonal projection from a separable Hilbert space $\mathcal{H}_1$ which contains $\mathcal{H}$ isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case.

2020-11-23math.FAmath.OA

Strong Banach property (T), after Vincent Lafforgue

This text takes, with more details and simplifying a proof in section 3, the parts of [Laf08] and [Laf09] treating p-adic groups. We prove that $SL_{3}$ over a non archimedian local field $F$ has strong Banach property (T). The applications are as follows: any connected almost $F$-simple algebraic group $G$ over $F$ whose Lie algebra contains $sl_3(F)$ has strong Banach property (T), the family of expanders constructed from a lattice of $G(F)$ do not embed uniformly in any Banach space of type $>1,$ and any isometric affine action of $G(F)$ on a Banach space of type $>1$ admits a fix point.

2012-12-19math.GRmath.OA

Strong Banach Property (T) for Simple Algebraic Groups of Higher Rank

In [Laf08], [Laf09], Vincent Lafforgue proved strong Banach property (T) for $SL_3$ over a non archimedean local field $F.$ In this paper, we extend his results to $Sp_4$ and therefore to any connected almost $F$-simple algebraic group with $F$-split rank $\geq 2.$ As applications, the family of expanders constructed from any lattice of such a group do not admit a uniform embedding into any Banach space of type $>1,$ and any isometric affine action of such a group, or its cocompact lattice, on a Banach space of type $>1$ has a fixed point.

2014-11-22math.GRmath.OA

On the axiomatization of convex subsets of Banach spaces

We prove that any convex-like structure in the sense of Nate Brown is affinely and isometrically isomorphic to a closed convex subset of a Banach space. This answers an open question of Brown. As an intermediate step, we identify Brown's algebraic axioms as equivalent to certain well-known axioms of abstract convexity. We conclude with a new characterization of convex subsets of Banach spaces.

2015-10-20math.FAmath.MGmath.OA