We generalize earlier results about connected components of idempotents in Banach algebras, due to B. Szőkefalvi Nagy, Y. Kato, S. Maeda, Z. V. Kovarik, J. Zemánek, J. Esterle. Let $A$ be a unital complex Banach algebra, and $p(λ) = \prod\limits_{i = 1}^n (λ- λ_i)$ a polynomial over $\Bbb C$, with all roots distinct. Let $E_p(A) := \{a \in A \mid p(a) = 0\}$. Then all connected components of $E_p(A)$ are pathwise connected (locally pathwise connected) via each of the following three types of paths: 1)~similarity via a finite product of exponential functions (via an exponential function); 2)~a polynomial path (a cubic polynomial path); 3)~a polygonal path (a polygonal path consisting of $n$ segments). If $A$ is a $C^*$-algebra, $λ_i \in \Bbb R$, let $S_p(A):= \{a\in A \mid a = a^*$, $p(a) = 0\}$. Then all connected components of $S_p(A)$ are pathwise connected (locally pathwise connected), via a path of the form $e^{-ic_mt}\dots e^{-ic_1t} ae^{ic_1t}\dots e^{ic_mt}$, where $c_i = c_i^*$, and $t \in [0, 1]$ (of the form $e^{-ict} ae^{ict}$, where $c = c^*$, and $t \in [0,1]$). For (self-adjoint) idempotents we have by these old papers that the distance of different connected components of them is at least~$1$. For $E_p(A)$, $S_p(A)$ we pose the problem if the distance of different connected components is at least $\min \bigl\{|λ_i - λ_j| \mid 1 \leq i,j \leq n, \ i \neq j\bigr\}$. For the case of $S_p(A)$, we give a positive lower bound for these distances, that depends on $λ_1, \dots, λ_n$. We show that several local and global lifting theorems for analytic families of idempotents, along analytic families of surjective Banach algebra homomorphisms, from our recent paper with B. Aupetit and M. Mbekhta, have analogues for elements of $E_p(A)$ and $S_p(A)$.
The Ando-Choi-Effros lifting theorem provides conditions under which a bounded linear mapping taking values in a quotient space can be lifted through the quotient map. We prove two versions of said theorem for regular maps between Banach lattices. Our conditions mirror the classical ones, but additionally taking into account the order structure.
Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Banach lattices and lattice homomorphisms. Namely, a Banach lattice $X$ satisfies the lattice-lifting property if every lattice homomorphism to $X$ having a bounded linear right-inverse must have a lattice homomorphism right-inverse. In terms of free Banach lattices, this can be rephrased into the following question: which Banach lattices embed into the free Banach lattice which they generate as a lattice-c…
Lifting properties for Banach spaces are studied. An alternate version of the lifting property due to Lindenstrass and Tzafriri is proposed and a characterization, up to isomorphism, is given. The quotient lifting property for pairs of Banach spaces $(X,J)$, with $J$ proximinal in $X$, is considered and several conditions for the property to hold are given.
Generalizing results of our earlier paper, we investigate the following question. Let $π(λ) : A \to B$ be an analytic family of surjective homomorphisms between two Banach algebras, and $q(λ)$ an analytic family of idempotents in $B$. We want to find an analytic family $p(λ)$ of idempotents in $A$, lifting $q(λ)$, i.e., such that $π(λ)p(λ) = q(λ)$, under hypotheses of the type that the elements of $\text{Ker}\, π(λ)$ have small spectra. For spectra which do not disconnect $\Bbb C$ we obtain a local lifting theorem. For real analytic families of surjective $^*$-homomorphisms (for continuous inv…
We prove a version of the Lebesgue Differentiation Theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon-Nikodým property.
Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a $C^*$-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebra, or of self-adjoint elements in a $C^*$-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we will prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the given equation. We also formulate open questions.
We generalize Wonham's theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to A*P+PA-PBB*P+C*C=0 when (A,B) is exponentially stabilizable and (C,A) is exponentially detectable. The proof is based on a new approach that treats the linear part of the equation as the generator of a positive semigroup on the space of symmetric operators from a Banach space to its dual, and the quadratic part as an order concave map. A direct analog of global Newton's iteration for concave functions is then used to approximate the solution, …
In this short note, we first consider some inequalities for comparison of some algebraic properties of two continuous algebra-multiplications on an arbitrary Banach space and then, as an application, we consider some very basic observations on the space of all continuous algebra-multiplications on a Banach space.
In this article, we apply the approach of relative algebraic geometry towards analytic geometry to the category of bornological and Ind-Banach spaces (non-Archimedean or not). We are able to recast the theory of Grosse-Klönne dagger affinoid domains with their weak G-topology in this new language. We prove an abstract recognition principle for the generators of their standard topology (the morphisms appearing in the covers). We end with a sketch of an emerging theory of dagger affinoid spaces over the integers, or any Banach ring, where we can see the Archimedean and non-Archimedean worlds com…
In this paper, we introduce $p$-amenability, bounded $s$-symmetric approximate and $s$-symmetric virtual diagonals for a Banach algebra $\mathfrak{A}$ where $s$ is a non-zero element of algebraic center of $\mathfrak{A}$ that is denoted by $Z(\mathfrak{A})$. We show that if a Banach algebra $\mathfrak{A}$ is $p$-amenable then it has bounded $s$-symmetric approximate and $s$-symmetric virtual diagonals and by this fact we prove that if the Banach algebra $\mathfrak{A}$ is unital then $p$-amenability and symmetric amenability are equivalent.
Let $X$ and $E$ be $f$-algebras and $p:X \to E_+$ be a monotone vector norm. Then the triple $(X,p,E)$ is called a lattice-normed $f$-algebraic space. In this paper, we show a generalization of the extension of the Hahn-Banach theorem for operators on the lattice-normed $f$-algebras, in which the extension of one step of that is not similar to the other Hahn-Banach theorems. Also, we give some applications and results.
Let $E$ be the Banach space constructed by Read (J. London Math. Soc. 1989) such that the Banach algebra $\mathscr{B}(E)$ of bounded operators on $E$ admits a discontinuous derivation. We show that $\mathscr{B}(E)$ has a singular, admissible extension which splits algebraically, but does not split strongly. This answers a natural question going back to the work of Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), and complements recent results of Laustsen and Skillicorn (C. R. Math. Acad. Sci. Paris, to appear).
We prove that for a Banach algebra $A$ having a bounded $\mathcal{Z}(A)$-approximate identity and for every $\bf[IN]$ group $G$ with weight $w$ which is either constant on conjugacy classes or $w \geq 1$, $\mathcal{Z}\big(L^1_w(G) \otimes^γA\big) \cong \mathcal{Z}(L^1_w(G)) \otimes^γ\mathcal{Z}(A)$. As an application, we discuss the conditions under which $\mathcal{Z}\big(L^1_w(G,A)\big)$ enjoys certain Banach algebraic properties, for example, weak amenability, semisimplicity etc.
We introduce a "dual-space approach" to mixed Nevanlinna-Pick/Carathéodory-Schur interpolation in Banach spaces X of holomorphic functions on the disk. Our approach can be viewed as complementary to the well-known commutant lifting approach of D. Sarason and B. Nagy-C.Foiaş. We compute the norm of the minimal interpolant in X by a version of the Hahn-Banach theorem, which we use to extend functionals defined on a subspace of kernels without increasing their norm. This Functional extensions lemma plays a similar role as Sarason's Commutant lifting theorem but it only involves the predual of X a…
Let $A$ and $X$ be Banach algebras and let $X$ be an algebraic Banach $A-$module. Then the $\ell^1-$direct sum $A\times X$ equipped with the multiplication $$(a,x)(b,y)=(ab, ay+xb+xy)\quad (a,b\in A, x,y\in X)$$ is a Banach algebra, denoted by $A\bowtie X$, which will be called "\textit{a generalized module extension Banach algebra}". Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. We characterize the structure of $n-$dual valued ($n\in\mathbb N)$) derivations on $A\bowtie X$ from which we investigate the $n-$we…
Let $A$ and $B$ be Banach algebras and let $B$ be an algebraic Banach $A-$bimodule. Then the $\ell^1-$direct sum $A\times B$ equipped with the multiplication $$(a_1,b_1)(a_2,b_2)=(a_1a_2,a_1\cdot b_2+b_1\cdot a_2+b_1b_2),~~ (a_1, a_2\in A, b_1, b_2\in B)$$ is a Banach algebra denoted by $A\bowtie B$. Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. We obtain characterizations of bounded approximate identities, spectrum, and topological center of this product. This provides a unified approach for obtaining some kn…
We show that there exists a Banach space $E$ with the following properties: the Banach algebra $\mathscr{B}(E)$ of bounded, linear operators on $E$ has a singular extension which splits algebraically, but it does not split strongly, and the homological bidimension of $\mathscr{B}(E)$ is at least two. The first of these conclusions solves a natural problem left open by Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), while the second answers a question of Helemskii. The Banach space $E$ that we use was originally introduced by Read (J. London Math. Soc. 1989).
In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a contribution towards the found…
Let $G$ be a locally compact group. We prove that the augmentation ideal in $L^1(G)$ is (algebraically) finitely-generated as a left ideal if and only if $G$ is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use t…
Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule. We introduce and study the notions of $n$-multipliers and approximate local $n$-multipliers by generalizing the classical concept of multipliers from $A$ into $X$. As an algebraic result, we construct a Banach algebra consisting of $n$-multipliers on $A$ and under some mild conditions, we give a nice relation of this algebra with $n$-homomorphisms from $A$ into $\mathbb{C}$.
If $X$ is a compact Hausdorff space and $σ$ is a homeomorphism of $X$, then a Banach algebra $\ell^1(Σ)$ of crossed product type is naturally associated with this topological dynamical system $Σ=(X,σ)$. If $X$ consists of one point, then $\ell^1(Σ)$ is the group algebra of the integers. We study the algebraically irreducible representations of $\ell^1(Σ)$ on complex vector spaces, its primitive ideals and its structure space. The finite dimensional algebraically irreducible representations are determined up to algebraic equivalence, and a sufficiently rich family of infinite dimensional algebr…
我们定义了一个单子M一类可测量的代数集，我们展示了如何产生这个单子的理论向量值的整合，是有关的概念佩蒂斯积分。我们证明了这个单子的代数X是一个代数学上的局部凸向量空间，它被赋予了将X中的向量\int f dm与传入映射f：T --> X和T上的测度m相关联的运算。证明了只要Banach空间中的每一个有界弱可测函数都有Pettis积分，则Banach空间是M-代数。因此，所有可分的巴拿赫空间和所有自反的巴拿赫空间都是M-代数。
Blecher and Read have recently introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. In the present paper we generalize some part of this, and some other facts, to larger classes of Banach algebras.
The goal of this paper is to study band-dominated operators on Banach spaces with Schauder basis with respect to uniformly locally finite metric spaces as well as the Banach algebras generated by them: the so called uniform Roe algebras. We investigate several kinds of isomorphisms between those Banach spaces (e.g., isomorphisms preserving norm, order, algebraic structure, etc) and prove several rigidity results on when a certain kind of isomorphism between the uniform Roe algebras implies that the base metric spaces are (bijectively) coarsely equivalent.
The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally-unbounded infinitesimal generators. In conclusion the concept of module over a Banach algebra is proposed as the generalization of Banach algebra. As an application to mathematical physics, the rigorous formulation of rotation group, which consists of unbounded operators being written by differen…
One important class of tools in the study of the connections between algebraic and topological structures are the "Banach-Stone type theorems", which describe algebraic isomorphisms of algebras (or groups, lattices, etc.) of functions in terms of homeomorphisms between the underlying topological spaces. Several such theorems have been proven throughout the last century, however not all of them are comparable, and in particular no single one is the strongest. In this article, we describe a general framework which encompasses several of these results, and which allows for new applications relate…
With Charles Read we have introduced and studied a new notion of (real) positivity in operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. As motivation note that the `completely' real positive maps on C*-algebras or operator systems are precisely the completely positive maps in the usual sense; however with real positivity one may develop a useful order theory for more general spaces and algebras. This is intimately connected to new relationships between an operator algebra and the C*-algebra it generates. We have continued this work …
