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Massoud Amini

Publications and source records attributed to Massoud Amini.

At least 55 records · Page 3Linked to original sources
arXiv2009

Module Biprojective and Module Biflat Banach Algebras

In this paper we define module biprojctivity and module biflatness for a Banach algebra which is a Banach module over another Banach algebra with compatible actions, and find their relation to classical biprojectivity and biflatness. As a typical example, We show that for an inverse semigroup $S$ with an upward directed set of idempotents $E$, the semigroup algebra $ \ell ^{1}(S)$, as an $\ell ^{1}(E)$-module, is module biprojective if and only if an appropriate group homomorphic image of $S$ is finite. Also we show that $\ell ^{1}(S)$ is module biflat if and only if $S$ is amenable.

math.FA↗
arXiv2006

Entangled Graphs

In this paper we prove a separability criterion for mixed states in $\mathbb C^p\otimes\mathbb C^q$. We also show that the density matrix of a graph with only one entangled edge is entangled.

cs.IT↗
arXiv2006

Hidden Subhypergroup Problem

The Hidden Subgroup Problem is used in many quantum algorithms such as Simon's algorithm and Shor's factoring and discrete log algorithms. A polynomial time solution is known in case of abelian groups, and normal subgroups of arbitrary finite groups. The general case is still open. An efficient solution of the problem for symmetric group $S_n$ would give rise to an efficient quantum algorithm for Graph Isomorphism Problem. We formulate a hidden sub-hypergroup problem for finite hypergroups and solve it for finite commutative hypergroups. The given algorithm is efficient if the corresponding QFT could be calculated efficiently.

quant-ph↗
arXiv2003

Restricted algebras on inverse semigroups I, representation theory

The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. This is the first in a series of papers in which we have investigated a similar relation on inverse semigroups. We use a new concept of "restricted" representations and study the restricted semigroup algebras and corresponding $C^*$-algebras.

math.OA↗
arXiv2003

Restricted algebras on inverse semigroups II, positive definite functions

The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. This is the second in a series of papers in which we have investigated the concept of "restricted" positive definite functions and their relation with representations.

math.OA↗
arXiv2003

Restricted algebras on inverse semigroups III, Fourier algebra

The Fourier and Fourier-Stieltjes algebras $A(G)$ and $B(G)$ of a locally compact group $G$ are introduced and studied in 60's by Piere Eymard in his PhD thesis. If $G$ is a locally compact abelian group, then $A(G)\simeq L^1(\hat{G})$, and $B(G)\simeq M(\hat{G})$, via the Fourier and Fourier-Stieltjes transforms, where $\hat{G}$ is the Pontryagin dual of $G$. Recently these algebras are defined on a (topological or measured) groupoid and have shown to share many common features with the group case. This is the last in a series of papers in which we have investigated a "restricted" form of these algebras on a unital inverse semigroup $S$.

math.OA↗
arXiv2003

Tannaka-Krein duality for compact groupoids I, representation theory

In a series of papers, we have shown that from the representatio theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study continuous representations of compact groupoids. We show that irreducible representations have finite dimensional fibres. We prove the Schur's lemma, Gelfand-Raikov theorem and Peter-Weyl theorem for compact groupoids.

math.OA↗
arXiv2003

Tannaka-Krein duality for compact groupoids II, Fourier transform

In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study the Fourier and Fourier-Plancherel transforms and prove the Plancherel theorem for compact groupoids. We also study the central functions in the algebra of square integrable functions on the isotropy groups.

math.OA↗
arXiv2003

Tannaka-Krein duality for compact groupoids III, duality theory

In a series of papers, we have shown that from the \repn theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we introduce the Tannaka groupoid of a compact groupoid and show how to recover the original groupoid from its Tannaka groupoid.

math.OA↗
arXiv2003

Control-target inversion property on Abelian groups

We show that the quantum Fourier transform on finite fields used to solve query problems is a special case of the usual quantum Fourier transform on finite abelian groups. We show that the control/target inversion property holds in general. We apply this to get a sharp query complexity separation between classical and quantum algorithms for a hidden homomorphism problem on finite Abelian groups.

quant-ph↗
arXiv2002

Locally Compact Pro-$C^{*}$-Algebras

Let $X$ be a locally compact non compact Hausdorff topological space. Consider the algebras $C(X)$, $C_b(X)$, $C_0(X)$, and $C_{00}(X)$ of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on $X$. From these, the second and third are $C^{*}$-algebras, the forth is a normed algebra, where as the first is only a topological algebra. The interesting fact about these algebras is that if one of them is given, the rest can be obtained using functional analysis tools. For instance, given the $C^{*}$-algebra $C_0(X)$, one can get the other three algebras by $C_{00}(X)=K(C_0(X))$, $C_b(X)=M(C_0(X))$, $C(X)=Γ(K(C_0(X)))$, that is by forming the Pedersen's ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen's ideal, respectively.In this article we consider the possibility of these transitions for general $C^{*}$-algebra . The difficult part is to start with a pro_$C^{*}$-algebra $A$ and to construct a $C^{*}$-algebra $A_0$ such that $A=Γ(K(A_0))$. The pro-$C^{*}$-algebras for which this is possible are called {\it locally compact} and we have characterized them using a concept similar to approximate identities.

math.OA↗
arXiv2002

On generalized Stone's Theorem

It is known that the generator of a strictly continuous one parameter unitary group in the multiplier algebra of a $C^{*}$-algebra is affiliated to that $C^{*}$-algebra . We show that under natural non degeneracy conditions, this self adjoint unbounded operator lies indeed in the (unbounded) multiplier algebra of the Pedersen's ideal of the $C^{*}$-algebra.

math.OA↗
arXiv2002

Fourier Algebras On Topological Foundation *-Semigroups

We introduce the notion of the Fourier and Fouier-Stieltjes algebra of a topological *-semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras.

math.OA↗
arXiv2002

Amenability of the Algebras R(S), F(S) of a Topological Semigroup S

For a locally compact Hausdorff semigroup S, the $L^\infty$ representation algebra R(S) was extensively studied by Dunkl and Ramirez. The Fourier-Stieltjes algebra F(S) of a topological semigroup was studied by Lau. The aim of this paper is to investigate these two algebras and study the amenability of them with respect to the structure of S.

math.OA↗
arXiv2002

Module Amenability for Semigroup Algebras

We extend the concept of amenability of a Banach algebra $A$ to the case that there is an extra $\mathfrak A$-module structure on $A$, and show that when $S$ is an inverse semigroup with subsemigroup $E$ of idempotents, then $A=\ell^1(S)$ as a Banach module over $\mathfrak A=\ell^1(E)$ is module amenable iff $S$ is amenable. When $S$ is a discrete group, $\ell^1(E)=\mathbb C$ and this is just the celebrated Johnson's theorem.

math.FA↗
arXiv2002

Non Commutative Topology and Local Structure of Operator Algebras

Starting with a $W^{*}$-algebra $M$ we use the inverse system obtained by cutting down $M$ by its (central) projections to define an inverse limit of $W^{*}$-algebras, and show that how this pro-$W^{*}$-algebra encodes the local structure of $M$. For the $C^{*}$-algebras we do the same thing using their atomic enveloping $W^{*}$-algebra . We investigate the relation of these ideas to the Akemann-Giles-Kummer non commutative topology. Finally we use these ideas to look at the local structure of Kac algebras.

math.OA↗