On some applications of representable and continuous functionals of Banach quasi *-algebras
Maria Stella Adamo
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Contribution Proceedings of the International Workshop on Operator Theory and its Applications (IWOTA) 2019, Lisbon. arXiv admin note: text overlap with arXiv:1809.00470
This survey aims to highlight some of the consequences that representable (and continuous) functionals have in the framework of Banach quasi *-algebras. In particular, we look at the link between the notions of *-semisimplicity and full representability in which representable functionals are involved. Then, we emphasize their essential role in studying *-derivations and representability properties for the tensor product of Hilbert quasi *-algebras, a special class of Banach quasi *-algebras.
In the study of locally convex quasi *-algebras an important role is played by representable linear functionals; i.e., functionals which allow a GNS-construction. This paper is mainly devoted to the study of the continuity of representable functionals in Banach and Hilbert quasi *-algebras. Some other concepts related to representable functionals (full-representability, *-semisimplicity, etc) are revisited in these special cases. In particular, in the case of Hilbert quasi *-algebras, which are shown to be fully representable, the existence of a 1-1 correspondence between positive, bounded ele…
This note aims to highlight the link between representable functionals and derivations on a Banach quasi *-algebra, i.e. a mathematical structure that can be seen as the completion of a normed *-algebra in the case the multiplication is only separately continuous. Representable functionals and derivations have been investigated in previous papers for their importance concerning the study of the structure properties of a Banach quasi *-algebra and applications to quantum models.
In this work we are concerned with maximality of monotone operators representable by certain convex functions in non-reflexive Banach spaces. We also prove that these maximal monotone operators satisfy a Bronsted-Rockafellar type property. We show that if a function in XxX^* and its conjugate are above the duality product in their respective domains, then this function represents a maximal monotone operator.
In this paper, we unify the theory of SSD spaces and the theory of strongly representable sets, and we apply our results to the theory of the various classes of maximally monotone sets. In particular, we prove that type (ED), dense type, type (D), type (NI) and strongly representable are equivalent concepts and, consequently, that the known properties of strongly representable sets follow from known properties of sets of type (ED).
We study several properties and applications of the ultrapower $M_{\mathcal U}$ of a metric space $M$. We prove that the Lipschitz-free space $\mathcal F(M_{\mathcal U})$ is finitely representable in $\mathcal F(M)$. We also characterize the metric spaces that are finitely Lipschitz representable in a Banach space as those that biLipschitz embed into an ultrapower of the Banach space. Thanks to this link, we obtain that if $M$ is finitely Lipschitz representable in a Banach space $X$, then $\mathcal F(M)$ is finitely representable in $\mathcal F(X)$. We apply these results to the study of coty…
We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a Köthe function space $E$ is upper locally uniformly monotone if and only if its $p$-conv…
The injective tensor product of normal representable bimodules over von Neumann algebras is shown to be normal. The usual Banach module projective tensor product of central representable bimodules over an Abelian C$^*$-algebra is shown to be representable. A normal version of the projective tensor product is introduced for central normal bimodules.
We present a Lebesgue-type decomposition for a representable functional on a $^*$-algebra into absolutely continuous and singular parts with respect to an other. This generalizes the corresponding results of S. P. Gudder for unital Banach $^*$-algebras. We also prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.
In this paper, we unify the theory of SSD spaces, part of the theory of strongly representable multifunctions, and the theory of the equivalence of various classes of maximally monotone multifunctions.
A continuous quadratic form ("quadratic form", in short) on a Banach space $X$ is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator $T\colon X\to X^*$ factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if $T$ is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional $L_p(μ)$ space ($1\le p \le\infty$) is: (a) delta-semidefinite iff $p \ge 2$; (b) delta-convex iff $p>1$…
A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ such that $X$ is finitely representable (f.r.) in $Y$ and $Y$ is f.r. in $X$ then $Y$ is isometric to $X$. We provide a direct proof (without model theory) of the fact that every finitely determined space $X$ (isometrically) contains every (separable) space $Y$ which is finitely representable in $X$. We also point out how a similar argument proves the Krivine-Maurey theorem on stable Banach spaces, and give the model theoretic interpretations of some results.
We use the injective envelope to study quasimultipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasimultipliers. We obtain generalizations of the Banach-Stone theorem.
A Banach space X is superreflexive if each Banach space Y that is finitely representable in X is reflexive. Superreflexivity is known to be equivalent to J-convexity and to the non-existence of uniformly bounded factorizations of the summation operators S_n through X. We give a quantitative formulation of this equivalence. This can in particular be used to find a factorization of S_n through X, given a factorization of S_N through [L_2,X], where N is `large' compared to n.
It is known that, under certain conditions, *-representability and extensibility to the unitized *-algebra of a positive linear functional, defined on a *-algebra without unit, are equivalent. In this paper, a new condition for an analogous result is given for the case of a hermitian linear functional defined on a quasi *-algebra $(\A,\Ao)$ without unit.
Quasi *-algebras form an essential class of partial *-algebras, which are algebras of unbounded operators. In this work, we aim to construct tensor products of normed, respectively Banach quasi *-algebras, and study their capacity to preserve some important properties of their tensor factors, like for instance, *-semisimplicity and full representability.
We show that any faithful quasi-free actions of a finite group on the Cuntz algebra $\mathcal{O}_\infty$ are mutually conjugate, and that they are asymptotically representable.
This paper is devoted to the study of unbounded derivations on Banach quasi *-algebras with a particular emphasis to the case when they are infinitesimal generators of one parameter automorphisms groups. Both of them, derivations and automorphisms are considered in a weak sense; i.e., with the use of a certain families of bounded sesquilinear forms. Conditions for a weak *-derivation to be the generator of a *-automorphisms group are given.
Quasi-multipliers for a Hilbert C*-bimodule V were introduced by Brown, Mingo and Shen 1994 as a certain subset of the Banach bidual module V**. We give another (equivalent) definition of quasi-multipliers for Hilbert C*-bimodules using the centralizer approach and then show that quasi-multipliers are, in fact, universal (maximal) objects of a certain category. We also introduce quasi-multipliers for bimodules in Kasparov's sense and even for Banach bimodules over C*-algebras, provided these C*-algebras act non-degenerately. A topological picture of quasi-multipliers via the quasi-strict topol…
We present and study commutative Banach algebras generated by Toeplitz operators with generalized quasi-radial pseudo-homogeneous symbols acting on the Bergman space over the unit ball. We develop the Gelfand theory of these algebras and give some structural information about them. In particular, we provide a description of the radical of these algebras. This paper generalizes and completes the results from previous works related to Toeplitz operators with quasi-radial quasi-homogeneous symbols.
Let $n>2$ be a positive integer. In this paper, we obtain the general solution of the following functional equation n \sum_{1 \le i<j \le n} Q(x_i-x_j)=\sum_{i=1}^{n}Q(\sum_{j =1}^n x_j -n x_i) which is derived from the centroid of the $n$ distinct vectors $x_1, ..., x_n$ in an inner product space. Furthermore, we prove that a mapping $f$ between quasi-Banach modules over a $C^*$-algebra satisfying approximately the equation can be approximated by a quadratic mapping $Q$ satisfying exactly the equation such that $|f(x)-Q(x)|$ is bounded.
Let $A$ be a Banach Algebra, we say that $a\in A$ has generalized Hirano inverse if there exists some $b \in A$ such that $b=b.a.b, a.b=b.a$ and $a-a^2.b$ is quasi nil potent, if and only if there exists some $p=p^3\in A$ such that $a-p$ is quasi nil potent.
This note aims to investigate the tensor product of two given Hilbert quasi *-algebras and its properties. The construction proposed in this note turns out to be again a Hilbert quasi *-algebra, thus interesting representability properties studed in \cite{AT} are maintained. Furthermore, if two functionals are representable and continuous respectively on the two Hilbert quasi *-algebras, then so is their tensor product.
We continue with the study of greedy-type bases in quasi-Banach spaces started in [3]. In this paper, we study partially-greedy bases focusing our attention in two main results: -Characterization of partially-greedy bases in quasi-Banach spaces in terms of quasi-greediness and different conservative-like properties. -Given a $C$-partially-greedy basis in a quasi-Banach space, there exists a "renorming" such that the basis is $1$-partially-greedy.
We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.
We identify symmetric quasi-Banach range of the discrete Calderón operator and Hilbert transform acting on a symmetric quasi-Banach sequence space. As an application we present an example of optimal range in the case when the domain of those operators is the weak-$\ell_{1}$ space of sequences.
We investigate stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE's and several applications are presented.
Spaces of homogeneous polynomials on a Banach space are frequently equipped with quasinorms instead of norms. In this paper we develop a technique to replace the original quasi-norm by a norm in a dual preserving way, in the sense that the dual of the space with the new norm coincides with the dual of the space with the original quasi-norm. Applications to problems on the existence and approximation of solutions of convolution equations and on hypercyclic convolution operators on spaces of entire functions are provided.
We show that if $X$ is a quasi-Banach space with trivial dual then every continuous function $f:[0,1]\to X$ has a primitive, answering a question of M.M. Popov.
In this paper, we define for the first time the local Morrey-type space associated with ball quasi-Banach function spaces and show the related series of properties. In addition, Hardy-Littlewood maximal operator's boundedness is proved. We investigate nonsmooth decomposition of the local Morrey-type space associated with ball quasi-Banach function spaces via the Hardy local Morrey-type spaces associated with ball quasi-Banach function spaces. And we consider Hardy operator's boundedness.
The main result of this paper establishes that the known Arazy-Cwikel property holds for classes of uniformly K-monotone spaces in the quasi-Banach setting provided that the initial couple is mutually closed. As a consequence, we get that the class of all quasi-Banach K-spaces (i.e., interpolation spaces which are described by the real K-method) with respect to an arbitrary mutually closed Banach couple enjoys the Arazy-Cwikel property. Another consequence complements some previous results by Bykov and Ovchinnikov, showing that this property holds also for the class of all interpolation quasi-…
Let $X$ be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces $E$ of $X$-valued sequences, the sets $E-\bigcup _{q\inΓ}\ell_{q}(X)$, where $Γ$ is any subset of $(0,\infty]$, and $E-c_{0}(X)$ contain closed infinite-dimensional subspaces of $E$ (if non-empty, of course). This result is applied in several particular cases and it is also shown that the same technique can be used to improve a result on the existence of spaces formed by norm-attaining linear operators.
In this note, we extend to the setting of quasi-reflexive spaces a classical result of N. Kalton and L. Randrianarivony on the coarse Lipschitz structure of reflexive and asymptotically uniformly smooth Banach spaces. As an application, we show for instance, that for $1\le q<p$, a $q$-asymptotically uniformly convex Banach space does not coarse Lipschitz embed into a $p$-asymptotically uniformly smooth quasi-reflexive Banach space. This extends a recent result of B.M. Braga.
We generalize the classical coorbit space theory developed by Feichtinger and Gr"ochenig to quasi-Banach spaces. As a main result we provide atomic decompositions for coorbit spaces defined with respect to quasi-Banach spaces. These atomic decompositions are used to prove fast convergence rates of best $n$-term approximation schemes. We apply the abstract theory to time-frequency analysis of modulation spaces $M^{p,q}_m$, $0< p,q \leq \infty$.
We study how well a quasi-Banach space can be coarsely embedded into a Hilbert space. Given any quasi-Banach space X which coarsely embeds into a Hilbert space, we compute its Hilbert space compression exponent. We also show that the Hilbert space compression exponent of X is equal to the supremum of the amounts of snowflakings of X which admit a bi-Lipschitz embedding into a Hilbert space.
We generalize the theory of Wiener amalgam spaces on locally compact groups to quasi-Banach spaces. As a main result we provide convolution relations for such spaces. Also we weaken the technical assumption that the global component is invariant under right translations, which is even new for the classical Banach space case. To illustrate our theory we discuss in detail an example on the $ax+b$ group.
The main aim of this paper is to develop a general approach, which allows to extend the basics of Brudnyi-Kruglyak interpolation theory to the realm of quasi-Banach lattices. We prove that all $K$-monotone quasi-Banach lattices with respect to a $L$-convex quasi-Banach lattice couple have in fact a stronger property of the so-called $K(p,q)$-monotonicity for some $0<q\leq p\leq 1$, which allows us to get their description by the real $K$-method. Moreover, we obtain a refined version of the $K$-divisibility property for Banach lattice couples and then prove an appropriate version of this proper…
We deduce continuity, compactness and invariance properties for quasi-Banach Orlicz modulation spaces. We characterize such spaces in terms of Gabor expansions and by their images under the Bargmann transform.
We introduce new quasi-Banach modulation spaces on locally compact abelian (LCA) groups which coincide with the classical ones in the Banach setting and prove their main properties. Then we study Gabor frames on quasi-lattices, significantly extending the original theory introduced by Gröchenig and Strohmer. These issues are the key tools in showing boundedness results for Kohn-Nirenberg and localization operators on modulation spaces and studying their eigenfunctions' properties. In particular, the results in the Euclidean space are recaptured.
We prove the boundedness of a general class of multipliers and Fourier multipliers, in particular of the Hilbert transform, on quasi-Banach modulation spaces. We also deduce boundedness for multiplications and convolutions for elements in such spaces.
In three previous papers, we discussed quasidense multifunctions from a Banach space into its dual, or, equivalently, quasidense subsets of the product of a Banach space and its dual. In this paper, we survey (without proofs) some of the main results about quasidensity, and give some simple limiting examples in Hilbert spaces, reflexive Banach spaces, and nonreflexive Banach spaces.
There are two main aims of the paper. The first one is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second one is to extend the criterion for the precompactness of sets in the Lebesgue spaces $L_p(\mathbb R^n)$, $1 \leq p < \infty$, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces $B^s_{p,q}(\mathbb R^n)$, $0<s<1$, $0<p,q\leq \in…
The main focus of this paper is to define the notion of quasi-$(2,β)$-Banach space and show some properties in this new space, by help of it and under some natural assumptions, we prove that the fixed point theorem [16, Theorem 2.1] is still valid in the setting of quasi-$(2,β)$-Banach spaces, this is also an extension of the fixed point result of Brzdęk et al. [12, Theorem 1] in $2$-Banach spaces to quasi-$(2,β)$-Banach spaces. In the next part, we give a general solution of the radical-type functional equation (1.2). In addition, we study the hyperstability results for these functional equat…
In this paper, we achieve the general solution and the generalized Hyers-Ulam-Rassias stability of the following functional equation $$f(x+ky)+f(x-ky)=k^2f(x+y)+k^2f(x-y)+2(1-k^2)f(x)\eqno {2 cm}$$for fixed integers $k$ with $k\neq0,\pm1$ in the quasi-Banach spaces.
This paper is a continuation of our study of compact, power compact, Riesz, and quasicompact endomorphisms of commutative Banach algebras. Previously it has been shown that if $B$ is a unital commutative semisimple Banach algebra with connected character space, and $T$ is a unital endomorphism of $B$, then $T$ is quasicompact if and only if the operators $T^n$ converge in operator norm to a rank-one unital endomorphism of $B$. In this note the discussion is extended in two ways: we discuss endomorphisms of commutative Banach algebras which are semiprime and not necessarily semisimple; we also …
We investigate relations between symmetrizations of quasi-Banach function spaces and constructions such as Calderon-Lozanovskii spaces, pointwise product spaces and pointwise multipliers. We show that under reasonable assumptions the symmetrization commutes with these operations. We determine also the spaces of pointwise multipliers between Lorentz spaces and Cesaro spaces. Developed methods may be regarded as an arithmetic of quasi-Banach function spaces and proofs of Theorems 3, 4 and 6 give a kind of tutorial for these methods. Finally, the above results will be used in proofs of some facto…
In this paper, we establish the general solution of the functional equation $$f(nx+y)+f(nx-y)=n^2f(x+y)+n^2f(x-y)+2(f(nx)-n^2f(x))-2(n^2-1)f(y)\eqno {0 cm}$$for fixed integers $n$ with $n\neq0,\pm1$ and investigate the generalized Hyers-Ulam-Rassias stability of this equation in quasi-Banach spaces.
We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchenig and Rzeszotnik in [24] to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators [11], which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrö…
We consider linear operators defined on a subspace of a complex Banach space into its topological antidual acting positively in a natural sense. The goal of this paper is to investigate of this kind of operators. The main theorem is a constructive characterization of the bounded positive extendibility of these linear mappings. From this result we can characterize the compactness of the extended operators and that when the positive extensions have closed ranges. As a main application of our general extension theorem, we present some necessary and sufficient conditions that a positive functional…
We present a new approach to define a suitable integral for functions with values in quasi-Banach spaces. The integrals of Bochner and Riemann have deficiencies in the non-locally convex setting. The study of an integral for $p$-Banach spaces initiated by Vogt is neither totally satisfactory, since there are quasi-Banach spaces which are $p$-convex for all $0<p<1$, so it is not always possible to choose an optimal $p$ to develop the integration. Our method puts the emphasis on the galb of the space, which permits a precise definition of its convexity. The integration works for all spaces of ga…
We prove that for any two quasi-Banach spaces $X$ and $Y$ and any $α>0$ there exists a constant $γ_α>0$ such that $$ \sup_{1\le k\le n}k^αe_k(T)\le γ_α\sup_{1\le k\le n} k^αc_k(T) $$ holds for all linear and bounded operators $T:X\to Y$. Here $e_k(T)$ is the $k$-th entropy number of $T$ and $c_k(T)$ is the $k$-th Gelfand number of $T$. For Banach spaces $X$ and $Y$ this inequality is widely used and well-known as Carl's inequality. For general quasi-Banach spaces it is a new result.
