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Ebrahim Samei

Publications and source records attributed to Ebrahim Samei.

36 records · Page 2Linked to original sources
arXiv2014

Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra

If $D:A \to X$ is a derivation from a Banach algebra to a contractive, Banach $A$-bimodule, then one can equip $X^{**}$ with an $A^{**}$-bimodule structure, such that the second transpose $D^{**}: A^{**} \to X^{**}$ is again a derivation. We prove an analogous extension result, where $A^{**}$ is replaced by $\F(A)$, the \emph{enveloping dual Banach algebra} of $A$, and $X^{**}$ by an appropriate kind of universal, enveloping, normal dual bimodule of $X$. Using this, we obtain some new characterizations of Connes-amenability of $\F(A)$. In particular we show that $\F(A)$ is Connes-amenable if and only if $A$ admits a so-called WAP-virtual diagonal. We show that when $A=L^1(G)$, existence of a WAP-virtual diagonal is equivalent to the existence of a virtual diagonal in the usual sense. Our approach does not involve invariant means for $G$.

math.FA↗
arXiv2014

Some Beurling-Fourier algebras on compact groups are operator algebras

Let $G$ be a compact connected Lie group. The question of when a weighted Fourier algebra on $G$ is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on $G$ with the order of growth strictly bigger than the half of the dimension of the group. The case of SU(n) will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.

math.FA↗
arXiv2013

ZL-amenability and characters for the restricted direct products of finite groups

Let $G$ be a restricted direct product of finite groups $\{G_i \}_{i\in I}$, and let $\Zl^1(G)$ denote the centre of its group algebra. We show that $\Zl^1(G)$ is amenable if and only if $G_i$ is abelian for all but finitely many $i$, and characterize the maximal ideals of $\Zl^1(G)$ which have bounded approximate identities. We also study when an algebra character of $\Zl^1(G)$ belongs to $c_0$ or $\ell^p$ and provide a variety of examples.

math.FA↗
arXiv2013

ZL-amenability constants of finite groups with two character degrees

We calculate the exact amenability constant of the centre of $\ell^1(G)$ when $G$ is one of the following classes of finite group: dihedral; extraspecial; or Frobenius with abelian complement and kernel. This is done using a formula which applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk (J. Funct. Anal. 2009).

math.FA↗
arXiv2013

Some weighted group algebras are operator algebras

Let $G$ be a finitely generated group with polynomial growth, and let $\om$ be a weight, i.e. a sub-multiplicative function on $G$ with positive values. We study when the weighted group algebra $\ell^1(G,\om)$ is isomorphic to an operator algebra. We show that $\ell^1(G,\om)$ is isomorphic to an operator algebra if $\om$ is a polynomial weight with large enough degree or an exponential weight of order $0<α<1$. We will demonstrate the order of growth of $G$ plays an important role in this question. Moreover, the algebraic centre of $\ell^1(G,\om)$ is isomorphic to a $Q$-algebra and hence satisfies a multi-variable von Neumann inequality. We also present a more detailed study of our results when $G$ is the $d$-dimensional integers $\Z^d$ and 3-dimensional discrete Heisenberg group $\mathbb{H}_3(\Z)$. The case of the free group with two generators will be considered as a counter example of groups with exponential growth.

math.FA↗
arXiv2013

Quotients of Fourier algebras, and representations which are not completely bounded

We observe that for a large class of non-amenable groups $G$, one can find bounded representations of $A(G)$ on Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $A(G)$, equipped with the natural operator space structure, and ask whether such algebras can be completely isomorphic to operator algebras; partial results are obtained, using a modified notion of Helson set which takes account of operator space structure. In particular, we show that if $G$ is virtually abelian, then the restriction algebra $A_G(E)$ is completely isomorphic to an operator algebra if and only if $E$ is finite.

math.FA↗
arXiv2012

Completely bounded representations of convolution algebras of locally compact quantum groups

Given a locally compact quantum group $\mathbb G$, we study the structure of completely bounded homomorphisms $π:L^1(\mathbb G)\rightarrow\mathcal B(H)$, and the question of when they are similar to $\ast$-homomorphisms. By analogy with the cocommutative case (representations of the Fourier algebra $A(G)$), we are led to consider the associated map $π^*:L^1_\sharp(\mathbb G) \rightarrow \mathcal B(H)$ given by $π^*(ω) = π(ω^\sharp)^*$. We show that the corepresentation $V_π$ of $L^\infty(\mathbb G)$ associated to $π$ is invertible if and only if both $π$ and $π^*$ are completely bounded. Moreover, we show that the co-efficient operators of such representations give rise to completely bounded multipliers of the dual convolution algebra $L^1(\hat \mathbb G)$. An application of these results is that any (co)isometric corepresentation is automatically unitary. An averaging argument then shows that when $\mathbb G$ is amenable, $π$ is similar to a *-homomorphism if and only if $π^*$ is completely bounded. For compact Kac algebras, and for certain cases of $A(G)$, we show that any completely bounded homomorphism $π$ is similar to a *-homomorphism, without further assumption on $π^*$. Using free product techniques, we construct new examples of compact quantum groups $\mathbb G$ such that $L^1(\mathbb G)$ admits bounded, but not completely bounded, representations.

math.OA↗
arXiv2012

The Moment Problem for Continuous Positive Semidefinite Linear functionals

Let $τ$ be a locally convex topology on the countable dimensional polynomial $\reals$-algebra $\rx:=\reals[X_1,...,X_n]$. Let $K$ be a closed subset of $\reals^n$, and let $M:=M_{\{g_1, ... g_s\}}$ be a finitely generated quadratic module in $\rx$. We investigate the following question: When is the cone $\Pos(K)$ (of polynomials nonnegative on $K$) included in the closure of $M$? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of $M=\sos$ with respect to weighted norm-$p$ topologies. We show that this closure coincides with the cone $\Pos(K)$ where $K$ is a certain convex compact polyhedron.

math.AG↗
arXiv2010

Projectivity of modules over Fourier algebras

In this paper we will study the homological properties of various natural modules associated to the Fourier algebra of a locally compact group. In particular, we will focus on the question of identifying when such modules will be projective in the category of operator spaces. We will show that projectivity often implies that the underlying group is discrete and give evidence to show that amenability also plays an important role.

math.FA↗
arXiv2010

Beurling-Fourier algebras, operator amenability and Arens regularity

We introduce the class of Beurling-Fourier algebras on locally compact groups and show that they are non-commutative analogs of classical Beurling algebras. We obtain various results with regard to the operator amenability, operator weak amenability and Arens regularity of Beurling-Fourier algebras on compact groups and show that they behave very similarly to the classical Beurling algebras of discrete groups. We then apply our results to study explicitly the Beurling-Fourier algebras on SU(2), the 2 $\times$ 2 unitary group. We demonstrate that how Beurling-Fourier algebras are closely connected to the amenability of the Fourier algebra of SU(2). Another major consequence of our results is that our investigation allows us to construct families of unital infinite-dimensional closed Arens regular subalgebras of the Fourier algebra of certain products of SU(2).

math.FA↗
arXiv2009

Projectivity of modules over Segal algebras

In this paper we will study the projetivity of various natural modules associated to operator Segal algebras of the Fourier algebra of a locally compact group. In particular, we will focus on the question of identifying when such modules will be projective in the category of operator spaces. Projectivity often implies that the underlying group is discrete or even finite. We will also look at the projectivity for modules of $A_{cb}(G)$, the closure of $A(G)$ in the space of its completely bounded mutipliers. Here we give an evidence to show that weak amenability of $G$ plays an important role.

math.FA↗
arXiv2008

Smooth and weak synthesis of the anti-diagonal in Fourier algebras of Lie groups

Let $G$ be a Lie group of dimension $n$, and let $A(G)$ be the Fourier algebra of $G$. We show that the anti-diagonal $\checkΔ_G=\{(g,g^{-1})\in G\times G \mid g\in G\}$ is both a set of local smooth synthesis and a set of local weak synthesis of degree at most $[\frac{n}{2}]+1$ for $A(G\times G)$. We achieve this by using the concept of the cone property in \cite{ludwig-turowska}. For compact $G$, we give an alternative approach to demonstrate the preceding results by applying the ideas developed in \cite{forrest-samei-spronk}. We also present similar results for sets of the form $HK$, where both $H$ and $K$ are subgroups of $G\times G\times G\times G$ of diagonal forms. Our results very much depend on both the geometric and the algebraic structure of these sets.

math.FA↗
arXiv2008

Weak amenability of Fourier algebras on compact groups

We give for a compact group G, a full characterisation of when its Fourier algebra A(G) is weakly amenable: when the connected component of the identity G_e is abelian. This condition is also equivalent to the hyper-Tauberian property for A(G), and to having the anti-diagonal D^v={(s,s^{-1}):s is in G} being a set of spectral synthesis for A(GXG). We show the relationship between amenability and weak amenability of A(G), and (operator) amenability and (operator) weak amenability of A_D(G), an algebra defined by the authors in arXiv:0705.4277. We close by extending our results to some classes of non-compact, locally compact groups, including small invariant neighbourhood groups and maximally weakly almost periodic groups.

math.FA↗
arXiv2008

Convolutions on compact groups and Fourier algebras of coset spaces

In this note we study two related questions. (1) For a compact group G, what are the ranges of the convolution maps on A(GXG) given for u,v in A(G) by $u X v |-> u*v' ($v'(s)=v(s^{-1})$) and $u X v |-> u*v$? (2) For a locally compact group G and a compact subgroup K, what are the amenability properties of the Fourier algebra of the coset space A(G/K)? The algebra A(G/K) was defined and studied by the first named author. In answering the first question, we obtain for compact groups which do not admit an abelian subgroup of finite index, some new subalgebras of A(G). Using those algebras we can find many instances in which A(G/K) fails the most rudimentary amenability property: operator weak amenability. However, using different techniques, we show that if the connected component of the identity of G is abelian, then A(G/K) always satisfies the stronger property that it is hyper-Tauberian, which is a concept developed by the second named author. We also establish a criterion which characterises operator amenability of A(G/K) for a class of groups which includes the maximally almost periodic groups.

math.FA↗
arXiv2008

Amenability properties of the centres of group algebras

Let G be a locally compact group, and ZL1(G) be the centre of its group algebra. We show that when $G$ is compact ZL1(G) is not amenable when G is either nonabelian and connected, or is a product of infinitely many finite nonabelian groups. We also, study, for some non-compact groups G, some conditions which imply amenability and hyper-Tauberian property, for ZL1(G).

math.FA↗
arXiv2008

Weak amenability and 2-weak amenability of Beurling algebras

Let $L^1_\om(G)$ be a Beurling algebra on a locally compact abelian group $G$. We look for general conditions on the weight which allows the vanishing of continuous derivations of $L^1_\om(G)$. This leads us to introducing vector-valued Beurling algebras and considering the translation of operators on them. This is then used to connect the augmentation ideal to the behavior of derivation space. We apply these results to give examples of various classes of Beurling algebras which are weakly amenable, 2-weakly amenable or fail to be even 2-weakly amenable.

math.FA↗
arXiv2007

On local properties of Hochschild cohomology of a C$^*$- algebra

Let $A$ be a C$^*$-algebra, and let $X$ be a Banach $A$-bimodule. B. E. Johnson showed that local derivations from $A$ into $X$ are derivations. We extend this concept of locality to the higher cohomology of a $C^*$-algebra %for $n$-cocycles from $A^{(n)}$ into $X$ and show that, for every $n\in \N$, bounded local $n$-cocycles from $A^{(n)}$ into $X$ are $n$-cocycles.

math.OA↗