Searcharxiv⌕ Search

arXiv subjects

Massoud Amini

Publications and source records attributed to Massoud Amini.

At least 37 records · Page 2Linked to original sources
arXiv2020

The category of ordered Bratteli diagrams

A category structure for ordered Bratteli diagrams is proposed in which isomorphism coincides with the notion of equivalence of Herman, Putnam, and Skau. It is shown that the natural one-to-one correspondence between the category of Cantor minimal systems and the category of simple properly ordered Bratteli diagrams is in fact an equivalence of categories. This gives a Bratteli-Vershik model for factor maps between Cantor minimal systems. We give a construction of factor maps between Cantor minimal systems in terms of suitable maps (called premorphisms) between the corresponding ordered Bratteli diagrams, and we show that every factor map between two Cantor minimal systems is obtained in this way. Moreover, solving a natural question, we are able to characterize Glasner and Weiss's notion of weak orbit equivalence of Cantor minimal systems in terms of the corresponding C*-algebra crossed products.

math.OA↗
arXiv2019

$C^*$-Operator systems and crossed products

The purpose of this paper is to introduce a consistent notion of universal and reduced crossed products by actions and coactions of groups on operator systems and operator spaces. In particular we shall put emphasis to reveal the full power of the universal properties of the the universal crossed products. It turns out that to make things consistent, it seems useful to perform our constructions on some bigger categories which allow the right framework for studying the universal properties and which are stable under the construction of crossed products even for non-discrete groups. In the case of operator systems, this larger category is what we call a $C^*$-operator system, i.e., a selfadjoint subspace $X$ of some $\mathcal B(H)$ which contains a $C^*$-algebra $A$ such that $AX=X=XA$. In the case of operator spaces, the larger category is given by what we call $C^*$-operator bimodules. After we introduced the respective crossed products we show that the classical Imai-Takai and Katayama duality theorems for crossed products by group (co-)actions on $C^*$-algebras extend one-to-one to our notion of crossed products by $C^*$-operator systems and $C^*$-operator bimodules.

math.OA↗
arXiv2019

Equivariant homologies for operator algebras

This is a survey of a variety of equivariant (co)homology theories for operator algebras. We briefly discuss a background on equivariant theories, such as equivariant $K$-theory and equivariant cyclic homology. As the main focus, we discuss a notion of equivariant $ L^2 $-cohomology and equivariant $ L^2 $-Betti numbers for subalgebras of a von Neumann algebra. For graded $C^*$-algebras (with grading over a group) we elaborate on a notion of graded $ L^2 $-cohomology and its relation to equivariant $L^2$-cohomology.

math.OA↗
arXiv2018

Smooth cohomology of $ C^* $-algebras

We define a notion of smooth cohomology for $ C^* $-algebras which admit a faithful trace. We show that if $ \A\subseteq B(\h) $ is a $ C^* $-algebra with a faithful normal trace $ τ$ on the ultra-weak closure $ \bar{\A} $ of $ \mathcal{A} $, and $ X $ is a normal dual operatorial $ \bar{\A}$-bimodule, then the first smooth cohomology $ \mathcal{H}^1_{s}(\mathcal{A},X) $ of $ \mathcal{A} $ is equal to $ \mathcal{H}^1(\mathcal{A},X_τ)$, where $ X_τ $ is a closed submodule of $ X $ consisting of smooth elements.

math.OA↗
arXiv2017

Hilbert C*-bimodules of finite index and approximation properties of C*-algebras

Let $A$ and $B$ be arbitrary $C^*$-algebras, we prove that the existence of a Hilbert $A$-$B$-bimodule of finite index ensures that the WEP, QWEP, and LLP along with other finite-dimensional approximation properties such as CBAP and (S)OAP are shared by $A$ and $B$. For this, we first study the stability of the WEP, QWEP and LLP under Morita equivalence of $C^*$-algebras. We present examples of Hilbert $A$-$B$-bimodules which are not of finite index, while such properties are shared between $A$ and $B$. To this end, we study twisted crossed products by amenable discrete groups.

math.OA↗
arXiv2016

Extensions and Dilations of module maps

We study completely positive module maps on $C^{*}$-algebras which are $C^*$-module over another $C^*$-algebra with compatible actions. We extend several well known dilation and extension results to this setup, including the Stinespring dilation theorem and Wittstock, Arveson, and Voiculescu extension theorems.

math.OA↗
arXiv2016

$C^*$-simplicity and Ozawa conjecture for groupoid $C^*$-algebras, part I: injective envelopes

This paper studies injective envelopes of groupoid dynamical systems and the corresponding boundaries. Analogue to the group case, we associate a bundle of compact Hausdorff spaces to any (discrete) groupoid (the Hamana boundary of the groupoid). We study bundles of compact topological spaces equipped with an action of a groupoid. We show that any groupoid has a minimal boundary (the Furstenberg boundary of the groupoid). We prove that the Hamana and Furstenberg boundaries are the same, for (discrete) groupoids. We find the relation between the reduced crossed product of the $\G$-injective envelop of a groupoid dynamical system and the injective envelope of the reduced crossed product of the original system.

math.OA↗
arXiv2016

Dual space and hyperdimension of compact hypergroups

We characterize dual spaces and compute hyperdimensions of irreducible representations for two classes of compact hypergroups namely conjugacy classes of compact groups and compact hypergroups constructed by joining compact and finite hypergroups. Also studying the representation theory of finite hypergroups, we highlight some interesting differences and similarities between the representation theories of finite hypergroups and finite groups. Finally, we compute the Heisenberg inequality for compact hypergroups.

math.RT↗
arXiv2016

Crossed products of operator spaces

Let $V$ be an operator space and $\iso(V)$ be the group of all completely isometric bijective linear mappings on $V$. Let $G$ act on $V$ via a strongly continuous group homomorphism $α:G \to \iso (V)$. We define the full (and reduced) operator space crossed product $V\rtimes^{\op}_{α,(r)}G$ and show that for a $C^*$-algebra with its canonical operator space structure, it coincides with the corresponding $C^*$-algebra crossed product. Unfortunately, the proof of Theorem 4.3 of the paper contains a serious gap, which leads to the withdrawal of the paper.

math.OA↗
arXiv2014

Discretization of topological spaces

There are several compactification procedures in topology, but there is only one standard discretization, namely, replacing the original topology with the discrete topology. We give a notion of discretization which is dual (in categorical sense) to compactification and give examples of discretizations. Especially, a discretization functor from the category of $α$-scattered Stonean spaces to the category of discrete spaces is constructed which is the converse of the Stone-Čech compactification functor. The interpretations of discretization in the level of algebras of functions are given.

math.GN↗
arXiv2014

The C*-algebra generated by irreducible Toeplitz and composition operators

We describe the C*-algebra generated by an irreducible Toeplitz operator $T_ψ$, with continuous symbol $ψ$ on the unit circle $\mathbb{T}$, and finitely many composition operators on the Hardy space $H^2$ induced by certain linear-fractional self-maps of the unit disc, modulo the ideal of compact operators $K(H^2)$. For composition operators with automorphism symbols, we show that this algebra is not isomorphic to the one generated by the shift and composition operators.

math.OA↗
arXiv2014

The category of Bratteli diagrams

A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli's notion of equivalence, we obtain in particular a functorial formulation of Bratteli's classification of AF algebras (and at the same time, of Glimm's classification of UHF algebras). It is shown that the three approaches to classification of AF algebras, namely, through Bratteli diagrams, K-theory, and abstract classifying categories, are essentially the same from a categorical point of view.

math.OA↗
arXiv2014

Harmonic Operators of Ergodic Quantum Group Actions

In this paper we study the harmonic elements of (convolution) Markov maps associated to (ergodic) actions of locally compact quantum groups on ($σ$-finite) von Neumann algebras. We give several equivalent conditions under which the harmonic elements are trivial.

math.OA↗
arXiv2013

Homology for one-dimensional solenoids

Smale space is a particular class of hyperbolic topological dynamical systems, defined by David Ruelle. The third author constructed a homology theory for Smale spaces which is based on Krieger's dimension group invariant for shifts of finite type. In this paper, we compute this homology for the one-dimensional generalized solenoids of R.F. Williams.

math.DS↗
arXiv2013

Which multiplier algebras are $W^*$-algebras?

We consider the question of when the multiplier algebra $M(\mathcal{A})$ of a $C^*$-algebra $\mathcal{A}$ is a $ W^*$-algebra, and show that it holds for a stable $C^*$-algebra exactly when it is a $C^*$-algebra of compact operators. This implies that if for every Hilbert $C^*$-module $E$ over a $C^*$-algebra $\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $ W^*$-algebra, then $\mathcal{A}$ is a $C^*$-algebra of compact operators. Also we show that a unital $C^*$-algebra $\mathcal{A}$ which is Morita equivalent to a $ W^*$-algebra must be a $ W^*$-algebra.

math.OA↗
arXiv2010

Module amenability of the second dual and module topological center of semigroup algebras

Let $S$ be an inverse semigroup with an upward directed set of idempotents $E$. In this paper we define the module topological center of second dual of a Banach algebra which is a Banach module over another Banach algebra with compatible actions, and find it for $ \ell ^{1}(S)^{**}$ (as an $\ell^{1}(E)$-module). We also prove that $ \ell ^{1}(S)^{**}$ is $\ell^{1}(E)$-module amenable if and only if an appropriate group homomorphic image of $S$ is finite.

math.FA↗
arXiv2009

Module super-amenability for semigroup algebras

Let $S$ be an inverse semigroup with the set of idempotents $E$. In this paper we define the module super-amenability of a Banach algebra which is a Banach module over another Banach algebra with compatible actions, and show that when $E$ is upward directed and acts on $S$ trivially from left and by multiplication from right, the semigroup algebra $ \ell ^{1}(S)$ is $\ell^{1}(E)$-module super-amenable if and only if an appropriate group homomorphic image of $S$ is finite.

math.FA↗