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Roger R. Smith

Publications and source records attributed to Roger R. Smith.

At least 19 recordsLinked to original sources
arXiv2023

Norming in Discrete Crossed Products

Let $G \curvearrowright A$ be an action of a discrete group on a unital $C^*$-algebra by $*$-automorphisms. In this note, we give two sufficient dynamical conditions for the $C^*$-inclusion $A \subseteq A \rtimes_r G$ to be norming in the sense of Pop, Sinclair, and Smith. As a consequence of our results, when $A$ is separable or simple, the inclusion $A \subseteq A \rtimes_r G$ is norming provided it has a unique pseudo-expectation in the sense of Pitts.

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arXiv2019

A Galois correspondence for reduced crossed products of unital simple C$^*$-algebras by discrete groups

Let a discrete group $G$ act on a unital simple C$^*$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes_{α,r}H$ between subgroups of $G$ and C$^*$-algebras $B$ satisfying $A\subseteq B \subseteq A\rtimes_{α,r}G$, where $A\rtimes_{α,r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,α,σ)$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes^σ_{α,r}G$.

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arXiv2016

Intermediate subalgebras and bimodules for crossed products of general von Neumann algebras

Let $G$ be a discrete group acting on a von Neumann algebra $M$ by properly outer $*$-automorphisms. In this paper we study the containment $M \subseteq M\rtimes_αG$ of $M$ inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the $M$-bimodules that are closed in the Bures topology and which coincide with the $w^*$-closed ones under a mild hypothesis on $G$. We use these results to obtain a general version of Mercer's theorem concerning the extension of certain isometric $w^*$-continuous maps on $M$-bimodules to $*$-automorphisms of the containing von Neumann algebras.

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arXiv2016

Structural properties of close II$_1$ factors

We show that a number of key structural properties transfer between sufficiently close II$_1$ factors, including solidity, strong solidity, uniqueness of Cartan masas and property $Γ$. We also examine II$_1$ factors close to tensor product factors, showing that such factors also factorise as a tensor product in a fashion close to the original.

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arXiv2014

Kadison-Kastler stable factors

A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For $n\geq 3$ and a free ergodic probability measure preserving action of $SL_n(\mathbb Z)$ on a standard nonatomic probability space $(X,μ)$, write $M=((L^\infty(X,μ)\rtimes SL_n(\mathbb Z))\,\overline{\otimes}\, R$, where $R$ is the hyperfinite II$_1$ factor. We show that whenever $M$ is represented as a von Neumann algebra on some Hilbert space $\mathcal H$ and $N\subseteq\mathcal B(\mathcal H)$ is sufficiently close to $M$, then there is a unitary $u$ on $\mathcal H$ close to the identity operator with $uMu^*=N$. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler's conjecture. We also obtain stability results for crossed products $L^\infty(X,μ)\rtimesΓ$ whenever the comparison map from the bounded to usual group cohomology vanishes in degree 2 for the module $L^2(X,μ)$. In this case, any von Neumann algebra sufficiently close to such a crossed product is necessarily isomorphic to it. In particular, this result applies when $Γ$ is a free group.

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arXiv2014

Bimodules in crossed products and regular inclusions of finite factors

In this paper, we study bimodules over a von Neumann algebra $M$ in two related contexts. The first is an inclusion $M \subseteq M \rtimes_αG$, where $G$ is a discrete group acting on a factor $M$ by outer automorphisms. The second is a regular inclusion $M \subseteq N$ of finite factors. In the case of crossed products, we characterize the $M$-bimodules $X$ that lie between $M$ and $M \rtimes_αG$ and are closed in the Bures topology, in terms of the subsets of $G$. We show that this characterization also holds for $w^*$-closed bimodules when $G$ has the approximation property ($AP$), a class of groups that includes all amenable and weakly amenable ones. As an application, we prove a version of Mercer's extension theorem for certain $w^*$-continuous isometric maps on $X$. We establish a similar theorem for bimodules arising from regular inclusions of finite factors, which generalizes the crossed product situation when $G$ acts on a finite factor. In the final section we apply these ideas to provide new examples of singly generated finite factors.

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arXiv2013

Strong singularity for subalgebras of finite factors

In this paper we develop the theory of strongly singular subalgebras of von Neumann algebras, begun in earlier work. We mainly examine the situation of type $\tto$ factors arising from countable discrete groups. We give simple criteria for strong singularity, and use them to construct strongly singular subalgebras. We particularly focus on groups which act on geometric objects, where the underlying geometry leads to strong singularity.

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arXiv2012

Type II_1 factors satisfying the spatial isomorphism conjecture

This paper addresses a conjecture of Kadison and Kastler that a von Neumann algebra M on a Hilbert space H should be unitarily equivalent to each sufficiently close von Neumann algebra N and, moreover, the implementing unitary can be chosen to be close to the identity operator. This is known to be true for amenable von Neumann algebras and in this paper we describe new classes of non-amenable factors for which the conjecture is valid. These are based on tensor products of the hyperfinite II_1 factor with crossed products of abelian algebras by suitably chosen discrete groups.

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arXiv2010

The Relative Weak Asymptotic Homomorphism Property for Inclusions of Finite von Neumann Algebras

A triple of finite von Neumann algebras $B\subseteq N\subseteq M$ is said to have the relative weak asymptotic homomorphism property if there exists a net of unitary operators $\{u_λ\}_{λ\in Λ}$ in $B$ such that $$\lim_λ|\mathbb{E}}_B(xu_λy)-{\mathbb{E}}_B({\mathbb{E}}_N(x)u_λ{\mathbb{E}}_N(y))\|_2=0$$ for all $x,y\in M$. We prove that a triple of finite von Neumann algebras $B\subseteq N\subseteq M$ has the relative weak asymptotic homomorphism property if and only if $N$ contains the set of all $x\in M$ such that $Bx\subseteq \sum_{i=1}^n x_iB$ for a finite number of elements $x_1,...,x_n$ in $M$. Such an $x$ is called a one sided quasi-normalizer of $B$, and the von Neumann algebra generated by all one sided quasi-normalizers of $B$ is called the one sided quasi-normalizer algebra of $B$. We characterize one sided quasi-normalizer algebras for inclusions of group von Neumann algebras and use this to show that one sided quasi-normalizer algebras and quasi-normalizer algebras are not equal in general. We also give some applications to inclusions $L(H)\subseteq L(G)$ arising from containments of groups. For example, when $L(H)$ is a masa we determine the unitary normalizer algebra as the von Neumann algebra generated by the normalizers of $H$ in $G$.

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arXiv2010

Groupoid normalisers of tensor products: infinite von Neumann algebras

The groupoid normalisers of a unital inclusion $B\subseteq M$ of von Neumann algebras consist of the set $\mathcal{GN}_M(B)$ of partial isometries $v\in M$ with $vBv^*\subseteq B$ and $v^*Bv\subseteq B$. Given two unital inclusions $B_i\subseteq M_i$ of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion $B_1\ \overline{\otimes}\ B_2\subseteq M_1\ \overline{\otimes}\ M_2$ establishing the formula $$ \mathcal{GN}_{M_1\,\overline{\otimes}\,M_2}(B_1\ \overline{\otimes}\ B_2)''=\mathcal{GN}_{M_1}(B_1)''\ \overline{\otimes}\ \mathcal{GN}_{M_2}(B_2)'' $$ when one inclusion has a discrete relative commutant $B_1'\cap M_1$ equal to the centre of $B_1$ (no assumption is made on the second inclusion). This result also holds when one inclusion is a generator masa in a free group factor. We also examine when a unitary $u\in M_1\ \overline{\otimes}\ M_2$ normalising a tensor product $B_1\ \overline{\otimes}\ B_2$ of irreducible subfactors factorises as $w(v_1\otimes v_2)$ (for some unitary $w\in B_1\ \overline{\otimes}\ B_2$ and normalisers $v_i\in\mathcal{N}_{M_i}(B_i)$). We obtain a positive result when one of the $M_i$ is finite or both of the $B_i$ are infinite. For the remaining case, we characterise the II$_1$ factors $B_1$ for which such factorisations always occur (for all $M_1, B_2$ and $M_2$) as those with a trivial fundamental group.

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arXiv2009

Perturbations of C*-algebraic invariants

Kadison and Kastler introduced a metric on the set of all C$^*$-algebras on a fixed Hilbert space. In this paper structural properties of C$^*$-algebras which are close in this metric are examined. Our main result is that the property of having a positive answer to Kadison's similarity problem transfers to close C$^*$-algebras. In establishing this result we answer questions about closeness of commutants and tensor products when one algebra satisfies the similarity property. We also examine $K$-theory and traces of close C$^*$-algebras, showing that sufficiently close algebras have isomorphic Elliott invariants when one algebra has the similarity property.

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arXiv2008

Groupoid normalizers of tensor products

We consider an inclusion $B\subseteq M$ of finite von Neumann algebras satisfying $B'\cap M\subseteq B$. A partial isometry $v\in M$ is called a groupoid normalizer if $vBv^*, v^*Bv\subseteq B$. Given two such inclusions $B_i\subseteq M_i$, $i=1,2$, we find approximations to the groupoid normalizers of $B_1 \vnotimes B_2$ in $M_1\vnotimes M_2$, from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis $B_i'\cap M_i\subseteq B_i$, $i=1,2$. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries $v\in M$ satisfying $vBv^*\subseteq B$ and $v^*v, vv^*\in B$.

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arXiv2007

Normalizers of Irreducible Subfactors

We consider normalizers of an irreducible inclusion $N\subseteq M$ of $\mathrm{II}_1$ factors. In the infinite index setting an inclusion $uNu^*\subseteq N$ can be strict, forcing us to also investigate the semigroup of one-sided normalizers. We relate these normalizers of $N$ in $M$ to projections in the basic construction and show that every trace one projection in the relative commutant $N'\cap < M,e_N>$ is of the form $u^*e_Nu$ for some unitary $u\in M$ with $uNu^*\subseteq N$. This enables us to identify the normalizers and the algebras they generate in several situations. In particular each normalizer of a tensor product of irreducible subfactors is a tensor product of normalizers modulo a unitary. We also examine normalizers of irreducible subfactors arising from subgroup--group inclusions $H\subseteq G$. Here the normalizers are the normalizing group elements modulo a unitary from $L(H)$. We are also able to identify the finite trace $L(H)$-bimodules in $\ell^2(G)$ as double cosets which are also finite unions of left cosets.

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arXiv2004

Representations of Group Algebras in Spaces of Completely Bounded Maps

Let G be a locally compact group, M(G) denote its measure algebra and L^1(G) denote its group algebra. Also, let pi:G->U(H) be a strongly continuous unitary representation, and let CB^{sigma}(B(H)) be the space of normal completely bounded maps on B(H). We study the range of the map Gamma_pi:M(G)->CB^sigma(B(H)), Gamma_pi(mu)= int_G pi(s)\otimes pi(s)^*dmu(s) where we identify CB^sigma(B(H)) with the extended Haagerup tensor product B(H)\otimes^{eh}B(H)$. We use the fact that the C*-algebra generated by integrating pi to L^1(G) is unital exactly when pi is norm continuous to show that Gamma_pi(L^1(G))\subset B(H)\otimes^{eh}B(H) exactly when pi is norm continuous. For the case that G is abelian, we study Gamma_pi(M(G)) as a subset of the Varopoulos algebra. We also characterise positive definite elements of the Varopoulos algebra in terms of completely positive operators.

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arXiv2003

The completely bounded approximation property for extended Cuntz-Pimsner algebras

The extended Cuntz-Pimsner algebra E(H), introduced by Pimsner, is constructed from a Hilbert B,B-bimodule H over a C*-algebra B. In this paper we investigate the Haagerup invariant Λ(.) for these algebras, the main result being that Λ(E(H))=Λ(B) when H is full over B. In particular, E(H) has the completely bounded approximation property if and only if the same is true for B.

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arXiv2003

Crossed products and entropy of automorphisms

Let A be an exact C^*-algebra, let G be a locally compact group, and let (A,G,α) be a C*-dynamical system. Each automorphism α_g induces a spatial automorphism Ad_{\lamba_g} on the reduced crossed product A\times_αG. In this paper we examine the question, first raised by E. Stormer, of when the topological entropies of α_g and Ad_{α_g} coincide. This had been answered by N. Brown for the particular case of discrete abelian groups. Using different methods, we extend his result to a wider class of groups called locally [FIA]^-. This class includes all abelian groups, both discrete and continuous, as well as all compact groups.

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