arXiv subjects
Publications and source records attributed to Saeed Ghasemi.
Tensoring with type I algebras preserves elementary equivalence in the category of tracial von Neumann algebras. The proof involves a novel and general Feferman--Vaught-type theorem for direct integrals of metric structures.
We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal{P}(\mathbb{N})/\text{Fin}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of $\mathbb{N}$, while under the Continuum Hypothesis this rigidity fails and $\mathcal{P}(\mathbb{N})/\text{Fin}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, \v{C}ech-Stone remainders, and $\mathrm{C}^\ast$-algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible.
We show that the Fraïssé limit of a category of unital separable $C^*$-algebras which is sufficiently closed under tensor products of its objects and morphisms is strongly self-absorbing, given that it has approximate inner half-flip. We use this connection between Fraïssé limits and strongly self-absorbing $C^*$-algebras to give a self-contained and rather elementary proof for the well known fact that the Jiang-Su algebra is strongly self-absorbing.
We study the approximately finite-dimensional (AF) $C^*$-algebras that appear as inductive limits of sequences of finite-dimensional $C^*$-algebras and left-invertible embeddings. We show that there is such a separable AF-algebra $\mathcal A_\mathfrak{F}$ with the property that any separable AF-algebra is isomorphic to a quotient of $\mathcal A_\mathfrak{F}$. Equivalently, by Elliott's classification of separable AF-algebras, there are surjectively universal countable scaled (or with order-unit) dimension groups. This universality is a consequence of our result stating that $\mathcal A_\mathfrak{F}$ is the Fra\"ıssé limit of the category of all finite-dimensional $C^*$-algebras and left-invertible embeddings. With the help of Fra\"ıssé theory we describe the Bratteli diagram of $\mathcal A_\mathfrak{F}$ and provide conditions characterizing it up to isomorphisms. $\mathcal A_\mathfrak{F}$ belongs to a class of separable AF-algebras which are all Fra\"ıssé limits of suitable categories of finite-dimensional $C^*$-algebras, and resemble $C(2^\mathbb N)$ in many senses. For instance, they have no minimal projections, tensorially absorb $C(2^\mathbb N)$ (i.e. they are $C(2^\mathbb N)$-stable) and satisfy similar homogeneity and universality properties as the Cantor set.
We will show that separable unital AF-algebras whose Bratteli diagrams do not allow converging two nodes into one node, can be classified up to the tensor product with the universal UHF-algebra $\Q$ only by their trace spaces. That is, if $\A$ and $\B$ are such AF-algebras, then $T(A)=T(B)$ if and only if $\A\otimes \Q \cong \B\otimes \Q$.
We construct an essential extension of $\mathcal K(\ell_2({\mathfrak{c}}))$ by $\mathcal K(\ell_2)$, where ${\mathfrak{c}}$ denotes the cardinality of continuum, i.e., a $C^*$-algebra $\mathcal A\subseteq \mathcal B(\ell_2)$ satisfying the short exact sequence $$0\rightarrow \mathcal K(\ell_2)\xrightarrowι \mathcal A \rightarrow\mathcal K(\ell_2({\mathfrak{c}}))\rightarrow 0,$$ where $ι[\mathcal K(\ell_2)]$ is an essential ideal of $\mathcal A$ such that the algebra of multipliers $\mathcal M(\mathcal A)$ of $\mathcal A$ is equal to the unitization of $\mathcal A$. In particular $\mathcal A$ is not stable which sheds light on permanence properties of the stability in the nonseparable setting. Namely, an extension of a nonseparable algebra of compact operators, even by $\mathcal K(\ell_2)$, does not have to be stable. This construction can be considered as a noncommutative version of Mrówka's $Ψ$-space; a space whose one point compactification equals to its Cech-Stone compactification and is induced by a special uncountable family of almost disjoint subsets of ${\mathbb{N}}$. The role of the almost disjoint family is played by an almost orthogonal family of projections in $\mathcal B(\ell_2)$, but the almost matrix units corresponding to the matrix units in $\mathcal K(\ell_2({\mathfrak{c}}))$ must be constructed with extra care. This example may also contribute in the future to our understanding of the semigroups $Ext(\mathcal K(\ell_2(κ)))$ for $ω_1\leq κ\leq\mathfrak{c}$ which are unexplored at the moment.
We analyze the sequence obtained by consecutive applications of the Cantor-Bendixson derivative for a noncommutative scattered $C^*$-algebra $\mathcal A$, using the ideal $\mathcal I^{At}(\mathcal A)$ generated by the minimal projections of $\mathcal A$. With its help, we present some fundamental results concerning scattered $C^*$-algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the "cardinal sequences" programme in the classical topology, in the noncommutative context. This leads to some new constructions of noncommutative scattered $C^*$-algebras and new open problems. In particular, we construct a type $I$ $C^*$-algebra which is the inductive limit of stable ideals $\mathcal A_α$, along an uncountable limit ordinal $λ$, such that $\mathcal A_{α+1}/\mathcal A_α$ is $*$-isomorphic to the algebra of all compact operators on a separable Hilbert space and $\mathcal A_{α+1}$ is $σ$-unital and stable for each $α<λ$, but $\mathcal A$ is not stable and where all ideals of $\mathcal A$ are of the form $\mathcal A_α$. In particular, $\mathcal A$ is a nonseparable $C^*$-algebra with no ideal which is maximal among the stable ideals. This answers a question of M. R\ordam in the nonseparable case. All the above $C^*$-algebras $A_α$s and $A$ satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers.
A C*-algebra $A$ is said to be stable if it is isomorphic to $A \otimes K(\ell_2)$. Hjelmborg and Rørdam have shown that countable inductive limits of separable stable C*-algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra $A$ of $B(\ell_2)$, which is the inductive limit of length $ω_1$ of its separable stable ideals $I_α$ ($α<ω_1$) satisfying $I_{α+1}/I_α\cong K(\ell_2)$ for each $α<ω_1$, while $A$ is not stable. The sequence $(I_α)_{α\leqω_1}$ is the GCR composition series of $A$ which in this case coincides with the Cantor-Bendixson composition series as a scattered C*-algebra. $A$ has the property that all of its proper two-sided ideals are listed as $I_α$s for some $α<ω_1$ and therefore the family of stable ideals of $A$ has no maximal element. By taking $A'=A\otimes K(\ell_2)$ we obtain a stable C*-algebra with analogous composition series $(J_α)_{α<ω_1}$ whose ideals $J_α$s are isomorphic to $I_α$s for each $α<ω_1$. In particular, there are nonisomorphic scattered C*-algebras whose GCR composition series $(I_α)_{α\leqω_1}$ satisfy $I_{α+1}/I_α\cong K(\ell_2)$ for all $α<ω_1$, for which the composition series differ first at $α=ω_1$.
We extend the classical Feferman-Vaught theorem to logic for metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent, and therefore they are isomorphic under the Continuum Hypothesis. We also prove the existence of two separable C*-algebras of the form $\bigoplus_i M_{k(i)}(\mathbb{C})$ such that the assertion that their coronas are isomorphic is independent from ZFC, which gives the first example of genuinely non-commutative coronas of separable C*-algebras with this property.
We consider isomorphisms between quotient algebras of $\prod_{n=0}^{\infty} \mathbb{M}_{k(n)}(\mathbb{C})$ associated with Borel ideals on $\mathbb{N}$ and prove that it is relatively consistent with \textbf{ZFC} that all of these isomorphisms are trivial, in the sense that they lift to a *-homomorphism from $\prod_{n=0}^{\infty} \mathbb{M}_{k(n)}(\mathbb{C})$ into itself. This generalizes a result of Farah-Shelah who proved this result for centers of these algebras (in its dual form). We also use a simpler forcing notion and completely remove the large cardinal assumption used by Farah-Shelah.
In this paper we solve a question of Simon Wassermann, whether the Calkin algebra can be written as a C*-tensor product of two infinite dimensional C*-algebras. More generally we show that there is no surjective *-homomorphism from a SAW*-algebra onto C*-tensor product of two infinite dimensional C*-algebras.