arXiv subjects
Publications and source records attributed to A. Pourabbas.
Let $\mathcal G$ be an ultragraph and let $K$ be a field. We describe prime and primitive ideals in the ultragraph Leavitt path algebra $L_K(\mathcal G)$. We identify the graded prime ideals in terms of downward directed sets and then we characterize the non-graded prime ideals. We show that the non-graded prime ideals of $L_K(\mathcal G)$ are always primitive.
We introduce the Leavitt path algebras of ultragraphs and we characterize their ideal structures. We then use this notion to introduce and study the algebraic analogous of Exel-Laca algebras.
In this paper, we study the notion of $ϕ$-biflatness for some Banach algebras, where $ϕ$ is a non-zero multiplicative linear functional. We show that the Segal algebra $S(G)$ is left $ϕ$-biflat if and only if $G$ is amenable. Also, we characterize left $ϕ$-biflatness of semigroup algebra $\ell^{1}(S)$ in the term of biflatness, where $S$ is a Clifford semigroup.
In this paper, we introduce a new notion of biprojectivity, called $WAP$-biprojectivity for $F(\mathcal{A})$, the enveloping dual Banach algebra associated to a Banach algebra $\mathcal{A}$. We find some relations between Connes biprojectivity, Connes amenability and this new notion. We show that, for a given dual Banach algebra $\mathcal{A}$, if $F(\mathcal{A})$ is Connes amenable, then $\mathcal{A}$ is Connes amenable. For an infinite commutative compact group $G$, we show that the convolution Banach algebra $F(L^2(G))$ is not $WAP$-biprojective. Finally, we provide some examples of the enveloping dual Banach algebras and we study their $WAP$-biprojectivity and Connes amenability.
In this paper, we introduce a notion of approximate Connes-biprojectivity for dual Banach algebras. We study the relation between approximate Connes-biprojectivity, Johnson pseudo-Connes amenability and $φ$-Connes amenability. We propose a criterion to show that some certain dual triangular Banach algebras are not approximately Connes-biprojective. Next we show that for a locally compact group $G$, the Banach algebra $M(G)$ is approximately Connes-biprojective if and only if $G$ is amenable. Finally for an infinite commutative compact group $G$ we show that the Banach algebra $L^2(G)$ with convolution product is approximately Connes-biprojective, but it is not Connes-biprojective.
In this paper, we study the notion of Johnson pseudo-contractibility for certain Banach algebras. For a bicyclic semigroup $S$, we show that $\ell^{1}(S)$ is not Johnson pseudo-contractible. Also for a Johnson pseudo-contractible Banach algebra $A$, we show that $A$ has no non-zero complemented closed nilpotent ideal.
In this paper, we introduce the new notion of strong pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion to the various notions of Connes amenability. Also we show that for every non-empty set $I$, $M_I(\mathbb{C})$ is strong pseudo-Connes amenable if and only if $I$ is finite. We provide some examples of certain dual Banach algebras and we study its strong pseudo-Connes amenability. In the last section, we investigate the property ultra central approximate identity for a Banach algebra $\mathcal{A}$ and its second dual $\mathcal{A}^{**}$. Also we show that for a left cancellative regular semigroup $S$, ${\ell^{1}(S)}^{**}$ has an ultra central approximat identity if and only if $S$ is a group. Finally we study this property for $φ$-Lau product Banach algebras and the module extension Banach algebras.
In this paper, we study the notion of Connes amenability for a class of $I\times{I}$-upper triangular matrix algebra $UP(I,\mathcal{A})$, where $\mathcal{A}$ is a dual Banach algebra with a non-zero $wk^\ast$-continuous character and $I$ is a totally ordered set. For this purpose, we characterize the $ϕ$-Connes amenability of a dual Banach algebra $\mathcal{A}$ through the existence of a specified net in $\mathcal{A}\hat{\otimes}\mathcal{A}$, where $ϕ$ is a non-zero $wk^\ast$-continuous character. Using this, we show that $UP(I,\mathcal{A})$ is Connes amenable if and only if $I$ is singleton and $\mathcal{A}$ is Connes amenable. In addition, some examples of $ϕ$-Connes amenable dual Banach algebras, which is not Connes amenable are given.
We introduce the notion of Johnson pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion with the various notions of Connes amenability like Connes amenability, approximate Connes amenability and pseudo Connes amenability. We also investigate some hereditary properties of this new notion. We prove that for a locally compact group $G$, $M(G)$ is Johnson pseudo-Connes amenable if and only if $G$ is amenable. Also we show that for every non-empty set $I$, $\mathbb{M}_I(\mathbb{C})$ under this new notion is forced to have a finite index. Finally, we provide some examples of certain dual Banach algebras and we study their Johnson pseudo-Connes amenability.
Given Banach algebras $ A $ and $ B $ with $ θ\inΔ(B) $. We shall study the Johnson pseudo-contractibility and pseudo-amenability of $ θ$-Lau product $ A\times_θ B $. We show that if $ A\times_θ B $ is Johnson pseudo-contractible, then $ A $ is Johnson pseudo-contractible and has a bounded approximate identity and $ B $ is Johnson pseudo-contractible. In some particular cases complete characterization of Johnson pseudo-contractibility of $ A\times_θ B $ are given. Also, we show that pseudo-amenability of $ A\times_θ B $ implies approximate amenability of $ A $ and pseudo-amenability of $ B $.
Let $A$ and $B$ be Banach algebras and let $T$ be an algebra homomorphism from $B$ into $A$. The Cartesian product space $A\times B$ by $T$- Lau product and $\ell^{1}$- norm becomes a Banach algebra $A\times_{T}B$. We investigate the notions such as injectivity, projectivity and flatness for the Banach algebra $A\times_{T}B$. We also characterize Hochschild cohomology for the Banach algebra $A\times_{T}B$.
We introduce the notions of approximate cohomology and approximate homotopy in Banach algebras and we study the relation between them. We show that the approximate homotopically equivalent cochain complexes give the same approximate cohomologies. As an special case, approximate Hochschild cohomology is introduced and studied.
In this paper we are going to investigate the approximate biprojectivity and the $ϕ$-biflatness of some Banach algebras related to the locally compact groups. We show that a Segal algebra $S(G)$ is approximate biprojective if and only if $G$ is compact. Also for a continuous weight $w\geq 1$, we show that $L^{1}(G,w)$ is a approximate biprojective if and only if $G$ is compact. We study $ϕ$-biflatness of some Banach algebras, where $ϕ:A\rightarrow \mathbb{C}$ is a multiplicative linear functional. We show that if $S(G)$ is $ϕ$-biflat, then $G$ is amenable group. Also we show that the $ϕ$-biflatness of $L^{1}(G)^{**}$ implies the amenability of $G$.
For a Banach algebra $ \mathcal{A} $, we introduce various approximate virtual diagonals such as approximate WAP-virtual diagonal and approximate virtual diagonal. For the enveloping dual Banach algebra $ F(\mathcal{A}) $ of $ \mathcal{A} $, we show that $ F(\mathcal{A}) $ is approximately Connes-amenable if and only if $ \mathcal{A} $ has an approximate WAP-virtual diagonal. Further, for a discrete group $ G $, we show that if the group algebra $ \ell^1(G) $ has an approximate WAP-virtual diagonal, then it has an approximate virtual diagonal.
In this paper, we introduce a new notion of biprojectivity, called Connes-biprojective, for dual Banach algebras. We study the relation between this new notion to Connes-amenability and we show that, for a given dual Banach algebra $ \mathcal{A} $, it is Connes-amenable if and only if $ \mathcal{A} $ is Connes-biprojective and has a bounded approximate identity. Also, for an Arens regular Banach algebra $ \mathcal{A} $, we show that if $ \mathcal{A} $ is biprojective, then the dual Banach algebra $ \mathcal{A} ^{**} $ is Connes-biprojective.
We study the relation between module and Hochschild cohomology groups of Banach algebras with a compatible module structure. More precisely, we show that for every commutative Banach $ \mathcal{A} $-$ \mathfrak{A}$-bimodule $ X $ and every $ k \in \mathbb{N}$, the seminormed spaces $ \mathcal{H}^{k}_{\mathfrak{A}} (\mathcal{A},X^*)$ and $ \mathcal{H}^k (\frac{\mathcal{A}}{J}, X^*) $ are isomorphic, where $ J $ is the closed ideal of $ \mathcal{A} $ generated by the elements of the form $ a (α\cdot b)-(a\cdot α)b$ with $ a,b \in \mathcal{A} $ and $α\in \mathfrak{A}. $ As an example, we calculate the module cohomologies of inverse semigroup algebras with coefficients in some related function algebras. In particular, we show that for an inverse semigroup $ S $ with the set of idempotents $ E $, when $\ell^1(E) $ acts on $\ell^1(S) $ by multiplication from right and trivially from left, the first module cohomology $\mathcal{H}^1_{\ell^1(E)} (\ell^1(S), \ell^1(G_S)^{(2n+1)})$ is trivial for each $ n \in \mathbb{N} $. As a consequence we conclude that the second module cohomology $\mathcal{H}^2_{\ell^1(E)} (\ell^1(S),\ell^1(G_S)^{(2n+1)})$ is a Banach space, where $ G_S $ is the maximal group homomorphic image of $ S $.
Let T be a homomorphism from a Banach algebra B to a Banach algebra A.The Cartesian product space A * B with T-Lau multiplication and l^1-norm becomes a new Banach algebra A *_T B. We investigate the notions such as approximate amenability, pseudo amenability, phi-pseudo amenability,phi-biflatness and phi-biprojectivity for Banach algebra A *_T B. We also present an example to show that approximate amenability of A and B is not stable for A *_T B. Finally we characterize the double centralizer algebra of A *_T B and present an application of this characterization.
In this paper, we investigate the notion of approximate biprojectivity for semigroup algebras and for some Banach algebras related to semigroup algebras. We show that $\ell^{1}(S)$ is approximately biprojective if and only if $\ell^{1}(S)$ is biprojective, provided that $S$ is a uniformly locally finite inverse semigroup. Also for a Clifford semigroup $S$, we show that approximate biprojectivity $\ell^{1}(S)^{**}$ gives pseudo amenability of $\ell^{1}(S)$. We give a class of Banach algebras related to semigroup algebras which is not approximately biprojective.