We establish a close link between the amenability of a unitary representation $π$ of a group $G$ (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system $(\s_π,G)$, where $\s_\H$ is the unit sphere the Hilbert space of representation. We prove that $π$ is amenable if and only if either $π$ contains a finite-dimensional subrepresentation or the maximal uniform compactification of $\s_π$ has a $G$-fixed point. Equivalently, the latter means that the $G$-space $(\s_π,G)$ has the concentration property: every finite cover of the sphere $\s_π$ contains a set $A$ such that for every $\e>0$ the $\e$-neighbourhoods of the translations of $A$ by finitely many elements of $G$ always intersect. As a corollary, amenability of $π$ is equivalent to the existence of a $G$-invariant mean on the uniformly continuous bounded functions on $\s_π$. As another corollary, a locally compact group $G$ is amenable if and only if for every strongly continuous unitary representation of $G$ in an infinite-dimensional Hilbert space $\mathcal H$ the system $(\s_\H,G)$ has the property of concentration.
A complete characterization of Hilbert space operators that generate weakly amenable algebras remains open, even in the case of compact operator. Farenick, Forrest and Marcoux proposed the question that if $T$ is a compact weakly amenable operator on a Hilbert space $\mathfrak{H}$, then is $T$ similar to a normal operator? In this paper we demonstrate an example of compact triangular operator on infinite-dimension Hilbert space which is a weakly amenable and character amenable operator but is not similar to a normal operator.
Suppose that S is a left amenable semitopological semigroup. We prove that if ${T_{t}: t \in S}$ is a uniformly k-Lipschitzian semigroup on a bounded closed and convex subset C of a Hilbert space and $k<\sqrt{2}$, then the set of fixed points of this semigroup is a Hölder continuous retract of C. This gives a qualitative complement to the Ishihara-Takahashi fixed point existence theorem.
We study actions of discrete groups on Hilbert $C^*$-modules induced from topological actions on compact Hausdorff spaces. We show non-amenability of actions of non-amenable and non-a-T-menable groups, provided there exists a quasi-invariant probability measure which is sufficiently close to being invariant.
It is proved that a discrete group $G$ is amenable if and only if for every unitary representation of $G$ in an infinite-dimensional Hilbert space $\cal H$ the maximal uniform compactification of the unit sphere $\s_{\cal H}$ has a $G$-fixed point, that is, the pair $(\s_{\cal H},G)$ has the concentration property in the sense of Milman. Consequently, the maximal $U({\cal H})$-equivariant compactification of the sphere in a Hilbert space $\cal H$ has no fixed points, which answers a 1987 question by Milman. This is a version as of November 19, 1998, incorporating some revisions.
We prove a new weak mean ergodic theorem (Theorem A) for 1-cocycles associated to weakly mixing representations of amenable groups. Let $G$ be a finitely generated, discrete, amenable group $G$ which admits a controlled Folner sequence. We use Theorem A to deduce that any affine action $G\ca^T \Cal H$ on Hilbert space with weakly mixing linear part admits a sequence of almost fixed points (Theorem B). Specializing to the case that $G$ is a finitely generated group of polynomial growth, we show that convex combinations of averages of the associated 1-cocycle over $n$-balls provide a sequence of…
We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenab…
Suppose that H is a complex Hilbert space and that B(H) denotes the bounded linear operators on H. We show that every abelian, amenable operator algebra is similar to a C*-algebra. We do this by showing that if A is an abelian subalgebra of B(H) with the property that given any bounded representation $\varrho: A \to B(H_\varrho)$ of A on a Hilbert space $H_\varrho$, every invariant subspace of $\varrho(A)$ is topologically complemented by another invariant subspace of $\varrho(A)$, then A is similar to an abelian $C^*$-algebra.
There has been a long-standing conjecture in Banach algebra that every amenable operator is similar to a normal operator. In this paper, we study the structure of amenable operators on Hilbert spaces. At first, we show that the conjecture is equivalent to every non-scalar amenable operator has a non-trivial hyperinvariant subspace and equivalent to every amenable operator is similar to a reducible operator and has a non-trivial invariant subspace; and then, we give two decompositions for amenable operators, which supporting the conjecture.
In this paper, we consider the CQ projection method and the shrinking projection method given by a finite family of nonexpansive mappings of the unit sphere of a real Hilbert space, and prove strong convergence theorems to common fixed points of the mappings.
Let G be an amenable group, let X be a Banach space and let π: G --> B(X) be a bounded representation. We show that if the set {π(t) : t \in G} is gamma-bounded then πextends to a bounded homomorphism w : C*(G) --> B(X) on the group C*-algebra of G. Moreover w is necessarily gamma-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G is equal to either the real numbers, the integers or the unit circle, and/or when X has prope…
We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $σ$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).
The paper deals with continuous homomorphisms $S \ni s \mapsto T_s \in L(E)$ of amenable semigroups $S$ into the algebra $L(E)$ of all bounded linear operators on a Banach space $E$. For a closed linear subspace $F$ of $E$, sufficient conditions are given under which there exists a projection $P \in L(E)$ onto $F$ that commutes with all $T_s$. And when $E$ is a Hilbert space, sufficient conditions are given for the existence of an invertible operator $R \in L(E)$ such that all $R T_s R^{-1}$ are isometries. Also certain results on extending intertwining operators, renorming as well as on opera…
In this paper, we introduce the concept of biamenability of Banach algebras and we show that despite the apparent similarities between amenability and biamenability of Banach algebras, they lead to very different, and somewhat opposed, theories. In this regard, we show that commutative Banach algebras such as R and C tend to lack biamenability, while they may be amenable and highly noncommutative Banach algebras such as B(H) for an infinite dimensional Hilbert space H tend to be biamenable, while they are not amenable. Also, we show that although the unconditional unitization of an amenable Ba…
We introduce a one-parameter family of transforms, $U^t_{(m)}$, $t>0$, from the Hilbert space of Clifford algebra valued square integrable functions on the $m$--dimensional sphere, $L^2(S^{m},dσ_{m})\otimes \mathbb{C}_{m+1}$, to the Hilbert spaces, ${\mathcal M}L^2(\mathbb{R}^{m+1} \setminus \{0\},dμ_t)$, of monogenic functions on $\mathbb{R}^{m+1}\setminus \{0\}$ which are square integrable with respect to appropriate measures, $dμ_t$. We prove that these transforms are unitary isomorphisms of the Hilbert spaces and are extensions of the Segal-Bargman coherent state transform, $U_{(1)} : L^2(…
We investigate how coarse embeddability of box spaces into Hilbert space behaves under group extensions. In particular, we prove a result which implies that a semidirect product of a finitely generated free group by a finitely generated residually finite amenable group has a box space which coarsely embeds into Hilbert space. This provides a new class of examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have property A, generalising the example of Arzhantseva, Guentner and Spakula.
If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here "almost" and "close" are in terms of the Hilbert-Schmidt norm. Using tracial 2-norm on $II_1$-factors we similarly define $II_1$-factor stability for groups. Our main result is that all 1-relator groups with non-trivial center are $II_{1}$-factor stable. Many of them are also matricially stable and RFD. For amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characte…
In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group $\Iso(\U)$ of isometries of Urysohn's universal complete separable metric space $\mathbb U$, equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if $\U$ is replaced with any generalized Urysohn metric space $U$ that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into…
The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(λ) = \exp(i(λ_1ψ_1 + ... + λ_nψ_n)) for λin R^n, gives rise to a manifold M of orthogonal projections for the subspaces u(λ)H^2(R) of L^2(R). For classes of admissible functions ψ_i the strong operator topology closures of M and M \cup M^\perp are determined explicitly as various n-balls and n-spheres. The arguments used are direct and rely on the analysis of oscillatory integrals and Hilbert space geometry. Some classes of these closed projection manifolds are classified up to unitary eq…
Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact sigma-compact groups (e.g. countable groups). Along the way, we introduce a "moderate" variant of the classical induction of representations and we generalize the Gaboriau--Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the "measure-theoretic solution" to the von Neumann problem for…
Let $K$ be a compact Hausdorff space and let $H$ be a real or complex Hilbert space with dim$(H_\mathbb{R})\geq 2$. We prove that the space $C(K,H)$ of all $H$-valued continuous functions on $K$, equipped with the supremum norm, satisfies the Mazur--Ulam property, that is, if $Y$ is any real Banach space, every surjective isometry $Δ$ from the unit sphere of $C(K,H)$ onto the unit sphere of $Y$ admits a unique extension to a surjective real linear isometry from $C(K,H)$ onto $Y$. Our strategy relies on the structure of $C(K)$-module of $C(K,H)$ and several results in JB$^*$-triple theory. For …
In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe--Ritter to the sphere $\mathbb{S}^n$. We determine the resulting Osterwalder--Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group $G^c = \mathrm{O}_{1,n}(\mathbb{R})^{\uparrow}$ and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key to…
Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere $S^m$ in the Euclidean space $\mathbb{R}^{m+1}$ are known to depend upon the Mercer's expansion of the compact and self-adjoint $L^2(S^m)$-operator associated to the kernel. The estimation of the Kolmogorov $n$-th width of the unit ball of the reproducing kernel Hilbert space in $L^2(S^m)$ and the identification of the so-called optimal subspace usually suffice. These Kolmogorov widths can be computed through the eigenvalues of the integral operator associated to the kernel. Thi…
We study Hilbert generated versions of nonseparable Banach spaces $\mathcal X$ considered by Shelah, Steprāns and Wark where the behavior of the norm on nonseparable subsets is so irregular that it does not allow any linear bounded operator on $\mathcal X$ other than a diagonal operator (or a scalar multiple of the identity) plus a separable range operator. We address the questions if these spaces admit uncountable equilateral sets and if their unit spheres admit uncountable $(1+)$-separated or $(1+\varepsilon)$-separated sets. We resolve some of the above questions for two types of these spac…
We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, which one-complemented subspaces inherit, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces, including that they are uniformly convex and uniformly smooth, and that they form a self-dual class. To this end, we relate the fact that the group of isometries acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobás property and use some known r…
We construct the first example of a coarsely non-amenable (= without Guoliang Yu's property A) metric space with bounded geometry which coarsely embeds into a Hilbert space.
We consider the sound ranging problem, which is to find the position of the source-point from the moments when the wave-sphere of linearly, with time, increasing radius reaches the sensor-points, in the infinite-dimensional separable Euclidean space H, and describe the solving methods, for entire space and for its unit sphere. In the former case, we give some sufficient conditions for uniqueness of the solution. We also provide two examples with the sets of sensors being a basis of H: 1st, when sound ranging problem and so-called dual problem both have single solutions, and 2nd, when sound ran…
Column and row operator spaces - which we denote by COL and ROW, respectively - over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group $G$ and $p,p' \in (1,\infty)$ with $\frac{1}{p} + \frac{1}{p'} = 1$, we use the operator space structure on $CB(COL(L^{p'}(G)))$ to equip the Figa-Talamanca-Herz algebra $A_p(G)$ with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for $p \leq q \leq 2$ or $2 \leq q \leq p$ and amenable $G$, the cano…
Upon improving and extending the concept of redundancy of frames, we introduce the notion of redundancy of fusion frames, which is concerned with the properties of lower and upper redundancies. These properties are achieved by considering the minimum and maximum values of the redundancy function which is defined from the unit sphere of the Hilbert space into the positive real numbers. In addition, we study the relationship between redundancy of frames (fusion frames) and dual frames (dual fusion frames). Moreover, we indicate some results about excess of fusion frames. We state the relationshi…
We prove that every surjective isometry from the unit sphere of the space $K(H),$ of all compact operators on an arbitrary complex Hilbert space $H$, onto the unit sphere of an arbitrary real Banach space $Y$ can be extended to a surjective real linear isometry from $K(H)$ onto $Y$. This is probably the first example of an infinite dimensional non-commutative C$^*$-algebra containing no unitaries and satisfying the Mazur--Ulam property. We also prove that all compact C$^*$-algebras and all weakly compact JB$^*$-triples satisfy the Mazur--Ulam property.
Given two complex Hilbert spaces $H$ and $K$, let $S(B(H))$ and $S(B(K))$ denote the unit spheres of the C$^*$-algebras $B(H)$ and $B(K)$ of all bounded linear operators on $H$ and $K$, respectively. We prove that every surjective isometry $f: S(B(K)) \to S(B(H))$ admits an extension to a surjective complex linear or conjugate linear isometry $T: B(K)\to B(H)$. This provides a positive answer to Tingley's problem in the setting of $B(H)$ spaces.
Let $V$ be a norm-closed subset of the unit sphere of a Hilbert space $H$ that is stable under multiplication by scalars of absolute value 1. A {\em maximal vector} (for $V$) is a unit vector $ξ\in H$ whose distance to $V$ is maximum $d(ξ,V)=\sup_{\|η\|=1}d(η,V)$, $d(ξ,V)$ denoting the distance from $ξ$ to the set $V$. Maximal vectors generalize the {\em maximally entangled} unit vectors of quantum theory. In general, under a mild regularity hypothesis on $V$, there is a {\em norm} on $H$ whose restriction to the unit sphere achieves its minimum precisely on $V$ and its maximum precisely on th…
For a given Hilbert space $\mathcal H$, consider the space of self-adjoint projections $\mathcal P(\mathcal H)$. In this paper we study the differentiable structure of a canonical sphere bundle over $\mathcal P(\mathcal H)$ given by $$ \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \|f\|=1\, \}. $$ We establish the smooth action on $\mathcal R$ of the group of unitary operators of $\mathcal H$, therefore $\mathcal R$ is an homogeneous space. Then we study the metric structure of $\mathcal R$ by endowing it first with the uniform quotient metric, which is a F…
Given a C$^*$-algebra $A$, let $S(A^+)$ denote the set of those positive elements in the unit sphere of $A$. Let $H_1$, $H_2,$ $H_3$ and $H_4$ be complex Hilbert spaces, where $H_3$ and $H_4$ are infinite-dimensional and separable. In this note we prove a variant of Tingley's problem by showing that every surjective isometry $Δ: S(B(H_1)^+)\to S(B(H_2)^+)$ or (respectively, $Δ: S(K(H_3)^+)\to S(K(H_4)^+)$) admits a unique extension to a surjective complex linear isometry from $B(H_1)$ onto $B(H_2))$ (respectively, from $K(H_3)$ onto $B(H_4)$). This provides a positive answer to a conjecture po…
In this note we address various algorithmic problems that arise in the computation of the operator norm in unitary representations of a group on Hilbert space. We show that the operator norm in the universal unitary representation is computable if the group is residually finite-dimensional or amenable with decidable word problem. Moreover, we relate the computability of the operator norm on the product of non-abelian free groups to Kirchberg's QWEP Conjecture, a fundamental open problem in the theory of operator algebras.
