Algebra AP Research Articles

A Characterization of Multipliers of the Herz Algebra

For the characterization of multipliers of Lp(Rd) or more generally, of Lp(G) for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra Ap(G) plays an important role. Following Larsen’s book, we describe multipliers as bounded linear operators that commute with translations. The main result of this paper is the characterization of the multipliers of Ap(G). In fact, we demonstrate that it coincides with the space of multipliers of Lp(G),∥·∥p. Given a multiplier T of (Ap(G),∥·∥Ap(G)) and using the embedding (Ap(G),∥·∥Ap(G))↪C0(G),∥·∥∞, the linear functional f↦[T(f)(0)] is bounded, and T can be written as a moving average for some element in the dual PMp(G) of (Ap(G),∥·∥Ap(G)). A key step for this identification is another elementary fact: showing that the multipliers from Lp(G),∥·∥p to C0(G),∥·∥∞ are exactly the convolution operators with kernels in Lq(G),∥·∥q for 1<p<∞ and 1/p+1/q=1. The proofs make use of the space of mild distributions, which is the dual of the Segal algebra S0(G),∥·∥S0, and the fact that multipliers T from S0(G) to S0′(G) are convolution operators of the form T:f↦σ∗f for some uniquely determined σ∈S0′. This setting also allows us to switch from the description of these multipliers as convolution operators (by suitable pseudomeasures) to their description as Fourier multipliers, using the extended Fourier transform in the setting of S0′(G),∥·∥S0′. The approach presented here extends to other function spaces, but a more detailed discussion is left to future publications.

On 2nd-stage quantization of quantum cluster algebras

Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the 2nd-stage quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its 2nd-stage quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses a mutually alternating quantum cluster algebra such that their 2nd-stage quantization can be essentially the same.As an example, we give the 2nd-stage quantized cluster algebra Ap,q(SL(2)) of FunC(SLq(2)) in §7.1 and show that it is a non-trivial 2nd-stage quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. As another example, we present a class of quantum cluster algebras with coefficients which possess a non-trivial 2nd-stage quantization. In particular we obtain a class of quantum cluster algebras from surfaces with coefficients which possess non-trivial 2nd-stage quantization.Finally, we prove that the compatible Poisson structure of a quantum cluster algebra without coefficients is always a locally standard Poisson structure. Following this, it is shown that the 2nd-stage quantization of a quantum cluster algebra without coefficients is in fact trivial.

Weak⁎-continuity of invariant means on spaces of matrix coefficients

With every locally compact group G, one can associate several interesting bi-invariant subspaces X(G) of the weakly almost periodic functions WAP(G) on G, each of which captures parts of the representation theory of G. Under certain natural assumptions, such a space X(G) carries a unique invariant mean and has a natural predual, and we view the weak⁎-continuity of this mean as a rigidity property of G. Important examples of such spaces X(G), which we study explicitly, are the algebra McbAp(G) of p-completely bounded multipliers of the Figà-Talamanca–Herz algebra Ap(G) and the p-Fourier–Stieltjes algebra Bp(G). In the setting of connected Lie groups G, we relate the weak⁎-continuity of the mean on these spaces to structural properties of G. Our results generalise results of Bekka, Kaniuth, Lau and Schlichting.

Bases in the mod p Steenrod algebra, β included

In [3] well known results of Wall and Arnon on the monomial bases in the mod 2 Steenrod algebra (see [9,1]) were generalized to the subalgebra A‾p of the mod p Steenrod algebra, p>2, generated by the reduced powers. In the present paper we considered the case of the full Steenrod algebra Ap. We constructed βX-, βZ-, βC-, ZA-, and XC-bases. We proved extremal properties of the βX-, βZ-, ZA-, and XC-bases. Also we constructed a new polynomial generators of the ring H⁎(K(Z/p,n),Z/p) in terms of the βC-basis.

Dynamical systems on Hilbert modules over locally C*-algebras

Let A be a locally C*-algebra and S(A) be the family of continuous C*-seminorms and let E be a Hilbert A-module. We prove that every dynamical system of unitary operators on E defines a dynamical system of automorphisms on the compact operators on E and show that under certain conditions, the converse is true. We define a generalized derivation on E and prove that if E is a full Hilbert A-module and δ : E → E is a bounded generalized derivation, then δp : Ep → Ep is a generalized derivation on the Hilbert module Ep over the C*-algebra Ap for each p _ S(A).

Relations between kernels and images of reduced powers for some right Ap-modules

We investigate the right action of the mod p Steenrod algebra Ap on the homology H*(L^s,Zp) where L=BZp is the lens space. Following ideas of Ault and Singer we investigate the relation between intersection of kernels of the reduced powers Ppi and Bockstein element ? and the intersection of images of Ppi+1?1 and of ?. Namely one can check that ?ki=0 imPpi+1?1 ? ?ki=0 ker Ppi and ?ki=0 imPpi+1?1 ? im? ? ?k i=0 ker Ppi ? ker ?. We generalize Ault?s homotopy systems to p > 2 and examine when the reverse inclusions are true.

Finite dimensional invariant subspaces for algebras of linear operators and amenable Banach algebras

We study a finite dimensional invariant subspace property similar to Fan's Theorem on semigroups for arbitrary Banach algebras A in terms of amenability of X(A,ϕ), the closed subalgebra of A generated by the set of all maximal elements in A with respect to a character ϕ. As a consequence, we offer some applications to the measure algebra M(G) and the generalized Fourier algebra Ap(G) of a locally compact group G.

Bass and Topological Stable Ranks of Complex and Real Algebras of Measures, Functions and Sequences

We compute the Bass stable rank and the topological stable rank of several convolution Banach algebras of complex measures on (-∞,∞) or on [0,∞) consisting of a discrete measure and/or of an absolutely continuous measure. We also compute the stable ranks of the convolution algebras , , l1(S) and , where S is an arbitrary subgroup of , of the almost periodic algebra AP and of , etc. We answer affirmatively the question posed by Mortini (Studia Mathematica 103(3):275–281, 1992). For the above algebras, the polydisc algebra , the algebra of continuous functions, and others, we also study their subsets (real Banach algebras) of real-valued measures, real-valued sequences or real-symmetric functions, and of corresponding exponentially stable algebras (for example, the Callier–Desoer algebra of causal exponentially decaying measures and L1 functions), and we compute their stable ranks. Finally, we show that in some of these real algebras a variant of the parity interlacing property is equivalent to reducibility of a unimodular (or coprime) pair. Also corona theorems are presented and the existence of coprime fractions is studied; in particular, we list which of these algebras are Bezout domains.

On character amenable Banach algebras

In this thesis, we survey the properties of character amenable Banach algebras. Character amenability is a cohomological property weaker than the classical amenability introduced by B.E. Johnson. We give characterization of character amenability in terms of bounded approximate identities and certain topologically invariant elements of the second dual. In addition, we obtain equivalent characterizations of character amenability of Banach algebras in terms of variances of the approximate diagonal and the virtual diagonal. We show that character amenability for either the group algebra L(G) or the Herz–Figa-Talamanca algebra Ap(G) is equivalent to the amenability of the underlying group G. We also discuss hereditary properties of character amenability. In the case of uniform algebras we obtain complete characterization of character amenability in term of the Choquet boundary of the underlying space. In addition, we discuss character amenable version of the reduction of dimension formula and splitting properties of modules over character amenable Banach algebras.

Higher homotopy commutativity of H-spaces and homotopy localizations

In this paper, we prove that the homotopy localization of an ACn-space is an ACn-space so that the universal map is an ACn-map. This result is used to study the higher homotopy commutativity of H-spaces with finitely generated cohomology over the Steenrod algebra Ap*. Our result shows that for any prime p, if X is a connected ACp-space whose mod p cohomology H*(X; Z / p) is finitely generated as an algebra over Ap*, then X has the mod p homotopy type of a Postnikov H-space.

Compactifications of Discrete Quantum Groups

Given a discrete quantum group A we construct a certain Hopf *-algebra AP which is a unital *-subalgebra of the multiplier algebra of A. The structure maps for AP are inherited from M(A) and thus the construction yields a compactification of A which is analogous to the Bohr compactification of a locally compact group. This algebra has the expected universal property with respect to homomorphisms from multiplier Hopf algebras of compact type (and is therefore unique). This provides an easy proof of the fact that for a discrete quantum group with an infinite dimensional algebra the multiplier algebra is never a Hopf algebra.

Representations of locally compact groups on QSLp-spaces and a p-analog of the Fourier–Stieltjes algebra

For a locally compact group G and p ∈ (1, oo), we define Bp(G) to be the space of all coefficient functions of isometric representations of G on quotients of subspaces of L p spaces. For p = 2, this is the usual Fourier-Stieltjes algebra. We show that Bp(G) is a commutative Banach algebra that contractively (isometrically, if G is amenable) contains the Figa-Talamanca-Herz algebra Ap(G). If 2 ≤ q ≤ p or p ≤ q ≤ 2, we have a contractive inclusion B q (G) ⊂ Bp(G). We also show that Bp(G) embeds contractively into the multiplier algebra of Ap(G) and is a dual space. For amenable G, this multiplier algebra and Bp(G) are isometrically isomorphic.

On Classification of Subalgebras of the Poincaré Algebra

The substantiation of the algorithm for classifying subalgebras of the Poincaré algebra AP (1, n) up to P (1, n)-conjugacy is completed

Conditional Symmetry and Exact Solutions of the Multidimensional Nonlinear d'Alembert Equation

The symmetry reduction of the equation (1) was considered in [1–5]. Maximal subalgebras of rank n of the algebra AP (1, n) were described in [3, 4]. These results allow to construct all symmetry ansatzes reducing the equation (1) to ordinary differential equations. Using the solutions of the reduced equations, some classes of multiparametric exact solutions of the equation (1) were found. Classification of the maximal subalgebras of rank n − 1 of the algebra AP (1, n) was given in [5]. Using these subalgebras ansatzes reducing equation (1) to equations of two variables were constructed. The present paper continues the research which was carried out in [3–5]. Here new operators of conditional symmetry of the d’Alembert equation are found. Using these operators conditionally-invariant ansatzes reducing equation (1) to ordinary differential equations are built. New classes of exact solutions of this equation are constructed. Consider the subalgebra L = where Ga = J0a − Jam, a = 1, 2, . . . ,m− 1. Let solutions of the equation (1) satisfy the conditions G1u = 0, . . . , Gm−1u = 0, Pm+1u = 0, . . . , Pnu = 0. (2)

On reducibility of ultrametric almost periodic linear representations

AbstractLet G be a group and K be a complete ultrametric valued field. Let AP(G, K) be the algebra of the generalized almost periodic functions of G in K. We have shown in a previous paper that when AP(G, K) has an invariant mean, then any almost periodic linear representation is quasi-reducible. Here, we show that with the same hypothesis, any topologically irreducible almost periodic linear representation is finite dimensional; also, any almost periodic linear representation is the topological sum of irreducible representations. Furthermore, we obtain a Peter-Weyl theorem for the algebra AP(G, K).We use the technical tools of Hopf algebra theory.

Homotopical uniqueness of classifying spaces

If G is a connected compact Lie group, then for almost all prime numbers p the mod p cohomology ring of the classifying space BG is a finitely generated polynomial algebra. In 1961, N. Steenrod [24] asked in general for a determination of all spaces X such that H∗(X,Fp) is a finitely generated polynomial algebra (i.e., such that X has a polynomial cohomology ring); at that time, the only examples known were spaces of the form X = BG. There has been a lot of subsequent progress on this problem. On one hand, the topological constructions of Sullivan [25, p. 4.28], as exploited by Clark-Ewing [8] and Wilkerson [27], have led to the discovery of exotic spaces X with polynomial cohomology rings. On the other hand, the algebraic arguments of Wilkerson [26] and Adams-Wilkerson [1] have shown in some generality that if X is any space with a polynomial cohomology ring then H∗(X,Fp) must be one of the polynomial algebras listed in [8]. However, there is still a gap here between topology and algebra; in this paper we act to narrow the gap and in some cases to close it. Suppose that p is an odd prime. Let K be the category of unstable algebras [18] over the mod p Steenrod algebra Ap and Kpoly the full subcategory of K consisting of objects which as rings are finitely generated polynomial algebras. If X is a p-complete space with H(X,Fp) ∈ Kpoly we extend the ideas of [1] by associating to X a finite p-adic linear group WX generated by pseudoreflections (see 1.1); given any such finite group W such that p does not divide the order of W, we then show (1.2) that there is up to homotopy exactly one p-complete space X with H∗(X,Fp) ∈ Kpoly and WX = W. In a variety of particular situations (1.3, 1.4) this gives a bijective correspondence between finite group data and homotopy types of p-complete spaces with polynomial cohomology rings.

Reduction of a multidimensional poincare-invariant nonlinear equation to two-dimensional equations

We study the structure of the invariants of the extended isochronous Galilean algebra $$A\tilde G (0, n - 1)$$ which is a subalgebra of the Poincare algebra AP(1, n). Using these results we classify maximal subalgebras of rank n−2 and n−1 of AP(1, n). With respect to subalgebras of rank n−1 of AP(1, n) ansatzes are constructed reducing the equation Φ(□u, (▽u)2, u)=0 to differential equations in two invariant variables.

Maximal subalgebras of rank n?1 of the algebra AP(1, n) and the reduction of nonlinear wave equations. II

Maximal subalgebras of rank n?1 of the algebra AP(1, n) and the reduction of nonlinear wave equations. II

Maximal subalgebras of rank n ?1 of the algebra AP(1, n) and reduction of nonlinear wave equations. I

The concept of canonical decomposition of an arbitrary subalgebra of the algebraAO(1,n) is introduced. With the help of this decomposition all maximal subalgebras L of rankn−1 of the algebraAP(1,n), satisfying the conditionL ∩V= , whereV= is the space of translation are described.

Exact solutions of the d'Alembert and Liouville equations in the pseudo-Euclidean space R2, 2. II

Invariants of the maximal rank-3 and rank-4 subalgebras of the Poincare algebras AP(2,2), which is the maximal invariance algebra of the d'Alembert equation □u+λuk=0,k=1, and the Liouville equation □u+λ expu=0 are constructed. The reduction of the given equations with respect to the maximal rank-3 subalgebra is carried out and some exact solutions of these equations are found.