Related papers: Trivial automorphisms

We prove that it is consistent with $\mathfrak c>\aleph_2$ that all automorphisms of $\mathcal P(\omega)/\mbox{fin}$ are trivial.

We study conditions on automorphisms of Boolean algebras of the form $P(\lambda)/I_\kappa$ (where $\lambda$ is an uncountable cardinal and $I_\kappa$ is the ideal of sets of cardinality less than $\kappa$) which allow one to conclude that a…

A trivial automorphism of the Boolean algebra $\mathcal P(\mathbb N) / \mathrm{Fin}$ is an automorphism induced by the action of some function $\mathbb N \rightarrow \mathbb N$. In models of forcing axioms all automorphisms are trivial, and…

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…

We investigate the statement ``all automorphisms of $\mathcal P(\lambda)/[\lambda]^{<\lambda}$ are trivial''. We show that MA implies the statement for regular uncountable $\lambda<2^{\aleph_0}$; that the statement is false for measurable…

The Proper Forcing Axiom implies all automorphisms of every Calkin algebra associated with an infinite-dimensional complex Hilbert space and the ideal of compact operators are inner. As a means of the proof we introduce the notion of Polish…

For suitable groups G we will show that one can add a Boolean algebra B by forcing in such a way that Aut(B) is almost isomorphic to G. In particular, we will give a positive answer to the following question due to J.Roitman: Is…

In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the…

We prove in ZFC that for mu >= aleph_2 there is a sigma --ideal I on mu and a Boolean sigma --subalgebra B of the family of subsets of mu which includes I such that the natural homomorphism from B onto B/I cannot be lifted.

Let $\mathcal{A}$ be the class of unital separable simple amenable $C$*-algebras $A$ which satisfy the Universal Coefficient Theorem for which $A\otimes M_{\texttt{P}}$ has tracial rank zero for some supernatural number $\texttt{p}$ of…

We prove that the so-called uniadic graph and its adic automorphism are Borel universal, i.e., every aperiodic Borel automorphism is isomorphic to the restriction of this automorphism to a subset invariant under the adic transformation, the…

The goal of this short note is to prove that when $A$ is a closed *-subalgebra of a C*-algebra $B$ satisfying the ideal intersection property plus a mild axiom (INV), then the map $J\mapsto J\cap A$ establishes an isomorphism from the…

Complete Boolean algebras proved to be an important tool in topology and set theory. Two of the most prominent examples are B(kappa), the algebra of Borel sets modulo measure zero ideal in the generalized Cantor space {0,1}^kappa equipped…

We prove that for every countable discrete group $G$, there is a $G$-flow on $\omega^*$ that has every $G$-flow of weight $\leq\! \aleph_1$ as a quotient. It follows that, under the Continuum Hypothesis, there is a universal $G$-flow of…

Let $\pi: Y\rightarrow X$ be a continuous surjection between compact Hausdorff spaces $Y$ and $X$ which is irreducible in the sense that if $F\subsetneq Y$ is closed, then $\pi(F)\neq X$. We exhibit isomorphisms between various Boolean…

We introduce a general method for showing under weak forcing axioms that reduced products of countable models of a theory $T$ have as few automorphisms as possible. We show that such forcing axioms imply that reduced products of countably…

If $\lambda$ is (strongly) inaccessible and $2^\lambda = \lambda^+$, then there is a nowhere trivial automorphism of the Boolean algebra $\mathcal P(\lambda)/[\lambda]^{<\lambda}$.

Assuming the continuum hypothesis, we show that the Calkin algebra has 2^{aleph_1} outer automorphisms.

We prove the consistency of the following statement: for some kappa<2^{aleph_0}, there is a kappa-complete ideal on kappa such that the Boolean algebra P(kappa)/I is sigma-centered and there are Q-sets of reals.

We show that for any countable homogeneous ordered graph $G$, the conjugacy problem for automorphisms of $G$ is Borel complete. In fact we establish that each such $G$ satisfies a strong extension property called ABAP, which implies that…