In this paper we extend a result of Semrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms of matrix algebras of which statement we present a short proof. The need to get such a proof was formulated in Semrl's paper.
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.
In the present paper we study the structure of a Banach algebra B(A, T_g) generated by a certain Banach algebra $A$ of operators acting in a Banach space D and a group {T_g}_{g \in G} of isometries of D such that T_g A T^{-1}_g = A. We investigate the interrelations between the existence of the expectation of B(A, T_g) onto A, topological freedom of the automorphisms of A induced by T_g and the dual action of the group G on B(A, T_g). The results obtained are applied to the description of the structure of Banach algebras generated by 'weighted composition operators' acting in various spaces.
In the present paper we continue the study of the structure of a Banach algebra B(A, T_g) generated by a certain Banach algebra $A$ of operators acting in a Banach space $D$ and a group {T_g}_{g \in G} of isometries of D such that T_g A T^{-1}_g = A. We investigate the interrelations between the existence of the expectation of B(A, T_g) onto $A$, metrical freedom of the automorphisms of A induced by T_g and the dual action of the group G on B(A, T_g). The results obtained are applied to the description of the structure of Banach algebras generated by 'weighted composition operators' acting in …
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…
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…
We investigate Banach space automorphisms $T:\ell_\infty/c_0\rightarrow\ell_\infty/c_0 $ focusing on the possibility of representing their fragments of the form $$T_{B,A}:\ell_\infty(A)/c_0(A)\rightarrow \ell_\infty(B)/c_0(B)$$ for $A, B\subseteq N$ infinite by means of linear operators from $\ell_\infty(A)$ into $\ell_\infty(B)$, infinite $A\times B$-matrices, continuous maps from $B^*=βB\setminus B$ into $A^*$, or bijections from $B$ to $A$. This leads to the analysis of general linear operators on $\ell_\infty/c_0$. We present many examples, introduce and investigate several classes of oper…
In the present paper we investigate Banach--Kantorovich algebras over faithful solid subalgebras of algebras measurable functions. We prove that any Banach--Kantorovich algebra over faithful solid subalgebras of algebra measurable functions represented as a measurable bundle of Banach algebras with vector-valued lifting. We apply such representation to the spectrum of elements Banach--Kantorovich algebras.
A Banach algebra A is self-induced if the multiplication is an isomorphism from the A-balanced projective tensor-square of A to A. The class of self-induced Banach algebras is a natural generalization of unital Banach algebras, providing a fertile framework for developing homological aspects of unital Banach algebras. Elementary results with applictions to computations of the bounded Hochschild cohomology groups H^1(A,A^*) with emphasis on A=A(X), the approximable operators on a Banach space X, are given.
We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C$^*$- and W$^*$-algebras; we show that interpolation space techniques can be used in the place of GNS type arguments. We define a notion of injectivity for dual Banach algebras, and show that this i…
In the sequel we establish the Banach Principle for semifinite JW-algebras without direct summand of type I sub 2, which extends the recent results of Chilin and Litvinov on the Banach Principle for semifinite von Neumann algebras to the case of JW-algebras.
In this paper, among other things, we study those conditions under which the weighted semigroup algebra $\ell^1(S,ω)$ is a dual Banach algebra with respect to predual $c_0(S)$. Some useful examples, illustrating the results, are also included.
We answer, by counterexample, several open questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the `nonselfadjoint analogue' of a W*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors is sharp.
We define a Banach algebra A to be dual if $A = (A_\ast)^\ast$ for a closed submodule $A_\ast$ of $A^\ast$. The class of dual Banach algebras includes all $W^\ast$-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus finite-dimensional. We…
We introduce a class of Banach algebras of generalized matrices and study the existence of approximate units, ideal structure, and derivations of them.
We work on $\ell_p$ uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of I. Farah and the authors to mutual embeddings of uniform Roe algebras of operators on $\ell_p$ spaces. Simultaneously, we obtain rigidity results for the classic uniform Roe $\mathrm{C}^*$-algebras which depend only on their Banach algebra structure.
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.
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.
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.
In this paper, we define the first topological $(σ,τ)$-cohomology group and examine vanishing of the first $(σ,τ)$-cohomology groups of certain triangular Banach algebras. We apply our results to study the $(σ,τ)$-weak amenability and $(σ,τ)$-amenability of triangular Banach algebras.
We give a formula for the stable rank of inclusions of unital Banach algebras in the sense of finite Watatani index. As an application we show that the stable rank of $\ell^1$-algebras of Disk algebras by any action of finite groups is 2.
We present two new forms in which the Frechet differential of a power series in a Banach algebra can be expressed in terms of absolutely convergent series involving the commutant $C(T):A\mapsto [A,T]$.Then we apply the results to the analytic functional calculus in a complex Banach space.
We extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra $\mathcal A$, we charaterise the maximal and the minimal $\mathcal A$-bimodules having a given essential support function or support function pair. These characterisations are complete except for the minimal $\mathcal A$-bimodule corresponding to a support function pair, in which case we make some headway. We also show that the wea…
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 paper is devoted to 2-local automorphisms on $AW^\ast$-algebras. Using the technique of matrix algebras over a unital Banach algebra we prove that any 2-local automorphism on an arbitrary $AW^\ast$-algebra without finite type~I direct summands is a global automorphism.
The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra $\mathcal{A}$ with the inner derivation property we prove that any 2-local derivation on the algebra $M_{2^n}(\mathcal{A}),$ $n\geq 2,$ is a derivation. We apply this result to $AW^\ast$-algebras and show that any 2-local derivation on an arbitrary $AW^\ast$-algebra is a derivation.
We investigate Banach algebras of convolution operators on the $L^p$ spaces of a locally compact group, and their K-theory. We show that for a discrete group, the corresponding K-theory groups depend continuously on $p$ in an inductive sense. Via a Banach version of property RD, we show that for a large class of groups, the K-theory groups of the Banach algebras are independent of $p$.
We show that several known facts concerning roots of matrices generalize to operator algebras and Banach algebras. We show for example that the so-called Newton, binomial, Visser, and Halley iterative methods converge to the root in Banach and operator algebras under various mild hypotheses. We also show that the `sign' and `geometric mean' of matrices generalize to Banach and operator algebras, and we investigate their properties. We also establish some other facts about roots in this setting.
We give a Banach algebra approach to the additiveness property of Voiculescu's amalgamated R-transform. We also define an amalgamated S-transform and prove that it is multiplicative on products of amalgamated free random variables when the algebra of amalgamation is commutative.
Let $A$ be a Banach algebra, not necessarily unital, and let $B$ be a closed subalgebra of $A$. We establish a connection between the Banach cyclic cohomology group $ {\cal{HC}}^n(A)$ of $A$ and the Banach $B$-relative cyclic cohomology group $ {\cal{HC}}^n_B(A) $ of $A$. We prove that, for a Banach algebra $A$ with a bounded approximate identity and an amenable closed subalgebra $B$ of $A$, up to topological isomorphism, ${\cal{HC}}^n(A) = {\cal{HC}}^n_B(A) $ for all $n \ge 0$. We also establish a connection between the Banach simplicial or cyclic cohomology groups of $A$ and those of the quo…
We show that the Gurarij space $\mathbb{G}$ has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. We also compute the universal minimal flow of the automorphism group of the Poulsen simplex $\mathbb{P}$ and we prove that it consists of the canonical action on $\mathbb{P}$ itself. This answers a question of Conley and Törnquist. We show that the pointwise stabilizer of any closed proper face of $\mathbb{P}$ is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simp…
Is there a mod Hilbert-Schmidt analogue of the BDF-theorem, with the Pincus g-function playing the role of the index ? We show that part of the question is about the K-theory of certain Banach algebras. These Banach algebras, related to Lipschitz functions and dirichlet algebras have nice Banach space duality properties. Moreover their corona algebras are C*-algebras.
The Green-Julg theorem states that K_0^G(B) is isomorphic to K_0(L^1(G,B)) for every compact group G and every G-C*-algebra B. We formulate a generalisation of this result to proper groupoids and Banach algebras and deduce that the Bost assembly map is surjective for proper Banach algebras. On the way, we show that the spectral radius of an element in a C_0(X)-Banach algebra can be calculated from the spectral radius in the fibres.
Given two equivalent locally compact Hausdorff groupoids, the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C*-coefficients. To show these results, the functoriality of Lafforgue's KK-theory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L^1-algebras (with Banach algebra coefficients).
We propose a new construction of Banach-Lie groups and algebras relying on nonstandard analysis. A major standard application is the Local Theorem which to certain extent reduces the problem of associating a Lie group to a given banach-Lie algebra to a similar problem for finitely generated Lie subalgebras. We discuss possible applications, e.g., to gauge theories.
Classical Banach principle is an essential tool for the investigation of the ergodic properties of Cesaro subsequences. The aim of this work is to extend Banach principle to the case of the stochastic convergence in the operator algebras. We start by establishing a sufficient condition for the stochastic convergence (stochastic Banach principle). Then we formulate stochastic convergence for the bounded Besicovitch sequences, and, as consequence for uniform subsequences.
We prove that every $2$-local automorphism on a finite-dimensional semi-simple Lie algebra $\mathcal{L}$ over an algebraically closed field of characteristic zero is an automorphism. We also show that each finite-dimensional nilpotent Lie algebra $\mathcal{L}$ with $\dim \mathcal{L}\geq 2$ admits a $2$-local automorphism which is not an automorphism.
