arXiv subjects

Vrej Zarikian

Publications and source records attributed to Vrej Zarikian.

13 recordsLinked to original sources

arXiv2026

Unique Pseudo-Expectations for Dynamical $C^*$-Inclusions

Let $(\mathcal{A},G,α)$ be a $C^*$-dynamical system. Unique extension properties for the $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_{α,r} G$ have been studied extensively. In this paper, we investigate unique extension properties for some other natural ``dynamical'' $C^*$-inclusions, namely: $\bullet$ $C_r^*(G) \subseteq \mathcal{A} \rtimes_{α,r} G$; $\bullet$ $\mathcal{A}^G \subseteq \mathcal{A}$, where $\mathcal{A}^G$ is the fixed-point subalgebra; $\bullet$ $Z(\mathcal{A})^c \subseteq \mathcal{A} \rtimes_{α,r} G$, where $Z(\mathcal{A})^c = Z(\mathcal{A})' \cap (\mathcal{A} \rtimes_{α,r} G)$ is the relative commutant of the center. Our hope is that a larger selection of examples will enable progress on some lingering open problems, in particular (1) the relationship between aperiodicity and the unique pseudo-expectation property and (2) the relationship between the faithful unique pseudo-expectation property and norming.

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arXiv2025

Unique Pseudo-Expectations for Hereditarily Essential $C^*$-Inclusions

The $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{B}$ is said to be hereditarily essential if for every intermediate $C^*$-algebra $\mathcal{A} \subseteq \mathcal{C} \subseteq \mathcal{B}$ and every non-zero ideal $\{0\} \neq \mathcal{J} \unlhd \mathcal{C}$, we have that $\mathcal{J} \cap \mathcal{A} \neq \{0\}$. That is, $\mathcal{A}$ detects ideals in every intermediate $C^*$-algebra $\mathcal{A} \subseteq \mathcal{C} \subseteq \mathcal{B}$. By a result of Pitts and Zarikian, a unital $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{B}$ is hereditarily essential if and only if every pseudo-expectation $θ:\mathcal{B} \to I(\mathcal{A})$ for $\mathcal{A} \subseteq \mathcal{B}$ is faithful. A decade-old open question asks whether hereditarily essential $C^*$-inclusions must have unique pseudo-expectations? In this note, we answer the question affirmatively for some important classes of $C^*$-inclusions, in particular those of the form $\mathcal{A} \subseteq \mathcal{A} \rtimes_{α,r}^σG$, for a twisted $C^*$-dynamical system $(\mathcal{A},G,α,σ)$. On the other hand, we settle the general question negatively by exhibiting $C^*$-irreducible inclusions of the form $C_r^*(G) \subseteq C(X) \rtimes_{α,r} G$ with multiple conditional expectations. Our results leave open the possibility that the question might have a positive answer for regular hereditarily essential $C^*$-inclusions.

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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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arXiv2022

Exotic Ideals in Free Transformation Group $C^*$-Algebras

Let $Γ$ be a discrete group acting freely via homeomorphisms on the compact Hausdorff space $X$ and let $C(X) \rtimes_ηΓ$ be the completion of the convolution algebra $C_c(Γ,C(X))$ with respect to a $C^*$-norm $η$. A non-zero ideal $J \unlhd C(X) \rtimes_ηΓ$ is exotic if $J \cap C(X) = \{0\}$. We show that exotic ideals are present whenever $Γ$ is non-amenable and there is an invariant probability measure on $X$. This fact, along with the recent theory of exotic crossed product functors, allows us to provide answers to two questions of K. Thomsen. Using the Koopman representation and a recent theorem of Elek, we show that when $Γ$ is a countably-infinite group having property (T) and $X$ is the Cantor set, there exists a free and minimal action of $Γ$ on $X$ and a $C^*$-norm $η$ on $C_c(Γ, C(X))$ such that $C(X)\rtimes_ηΓ$ contains the compact operators as an exotic ideal. We use this example to provide a positive answer to a question of A. Katavolos and V. Paulsen. The opaque and grey ideals in $C(X)\rtimes_ηΓ$ have trivial intersection with $C(X)$, and a result from arXiv:1901.09683 shows they coincide when the action of $Γ$ is free, however the problem of whether these ideals can be non-zero was left unresolved. We present an example of a free action of $Γ$ on a compact Hausdorff space $X$ along with a $C^*$-norm $η$ for which these ideals are non-trivial, in particular, they are exotic ideals.

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arXiv2020

A Generalization of Whyburn's Theorem, and Aperiodicity for Abelian C*-Inclusions

Let $j:Y \to X$ be a continuous surjection of compact metric spaces. Whyburn proved that $j$ is irreducible, meaning that $j(F) \subsetneq X$ for any proper closed subset $F \subsetneq Y$, if and only if $j$ is almost one-to-one, in the sense that \[ \overline{\{y \in Y: j^{-1}(j(y)) = y\}} = Y. \] In this note we prove the following generalization: There exists a unique minimal closed set $K \subseteq Y$ such that $j(K) = X$ if and only if \[ \overline{\{x \in X: card(j^{-1}(x)) = 1\}} = X. \] Translated to the language of operator algebras, this says that if $A \subseteq B$ is a unital inclusion of separable abelian $C^*$-algebras, then there exists a unique pseudo-expectation (in the sense of Pitts) if and only if the almost extension property of Nagy-Reznikoff holds. More generally, we prove that a unital inclusion of (not necessarily separable) abelian $C^*$-algebras has a unique pseudo-expectation if and only if it is aperiodic (in the sense of Kwaśniewski-Meyer).

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arXiv2017

The pure extension property for discrete crossed products

Let $G$ be a discrete group acting on a unital $C^*$-algebra $\mathcal{A}$ by $*$-automorphisms. In this note, we show that the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_r G$ has the pure extension property (so that every pure state on $\mathcal{A}$ extends uniquely to a pure state on $\mathcal{A} \rtimes_r G$) if and only if $G$ acts freely on $\mathcal{\widehat{A}}$, the spectrum of $\mathcal{A}$. The same characterization holds for the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes G$. This generalizes what was already known for $\mathcal{A}$ abelian.

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arXiv2017

Unique expectations for discrete crossed products

Let $G$ be a discrete group acting on a unital $C^*$-algebra $\mathcal{A}$ by $*$-automorphisms. We characterize (in terms of the dynamics) when the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_r G$ has a unique conditional expectation, and when it has a unique pseudo-expectation (in the sense of Pitts). Likewise for the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes G$. As an application, we (slightly) strengthen results of Kishimoto and Archbold-Spielberg concerning $C^*$-simplicity of $\mathcal{A} \rtimes_r G$.

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arXiv2016

Unique conditional expectations for abelian $C^*$-inclusions

Let $D \subseteq A$ be an inclusion of unital abelian $C^*$-algebras. In this note we characterize (in topological terms) when there is a unique conditional expectation $E:A \to D$, at least when $A$ is separable. As an application, we provide the first example of an inclusion with a unique conditional expectation, but multiple pseudo-expectations (in the sense of Pitts).

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arXiv2015

Unique Pseudo-Expectations for $C^*$-Inclusions

Given an inclusion D $\subseteq$ C of unital C*-algebras, a unital completely positive linear map $Φ$ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. The set PsExp(C,D) of all pseudo-expectations is a convex set, and for abelian D, we prove a Krein-Milman type theorem showing that PsExp(C,D) can be recovered from its extreme points. When C is abelian, the extreme pseudo-expectations coincide with the homomorphisms of C into I(D) which extend the inclusion of D into I(D), and these are in bijective correspondence with the ideals of C which are maximal with respect to having trivial intersection with D. Natural classes of inclusions have a unique pseudo-expectation (e.g., when D is a regular MASA in C). Uniqueness of the pseudo-expectation implies interesting structural properties for the inclusion. For example, when D $\subseteq$ C $\subseteq$ B(H) are W*-algebras, uniqueness of the pseudo-expectation implies that D' $\cap$ C is the center of D; moreover, when H is separable and D is abelian, we characterize which W*-inclusions have the unique pseudo-expectation property. For general inclusions of C*-algebras with D abelian, we characterize the unique pseudo-expectation property in terms of order structure; and when C is abelian, we are able to give a topological description of the unique pseudo-expectation property. Applications include: a) if an inclusion D $\subseteq$ C has a unique pseudo-expectation $Φ$ which is also faithful, then the C*-envelope of any operator space X with D $\subseteq$ X $\subseteq$ C is the C*-subalgebra of C generated by X; b) for many interesting classes of C*-inclusions, having a faithful unique pseudo-expectation implies that D norms C. We give examples to illustrate the theory, and conclude with several unresolved questions.

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arXiv2013

Bimodules over Cartan MASAs in von Neumann Algebras, Norming Algebras, and Mercer's Theorem

In a 1991 paper, R. Mercer asserted that a Cartan bimodule isomorphism between Cartan bimodule algebras A_1 and A_2 extends uniquely to a normal *-isomorphism of the von Neumann algebras generated by A_1 and A_2 [13, Corollary 4.3]. Mercer's argument relied upon the Spectral Theorem for Bimodules of Muhly, Saito and Solel [15, Theorem 2.5]. Unfortunately, the arguments in the literature supporting [15, Theorem 2.5] contain gaps, and hence Mercer's proof is incomplete. In this paper, we use the outline in [16, Remark 2.17] to give a proof of Mercer's Theorem under the additional hypothesis that the given Cartan bimodule isomorphism is weak-* continuous. Unlike the arguments contained in [13, 15], we avoid the use of the Feldman-Moore machinery from [8]; as a consequence, our proof does not require the von Neumann algebras generated by the algebras A_i to have separable preduals. This point of view also yields some insights on the von Neumann subalgebras of a Cartan pair (M,D), for instance, a strengthening of a result of Aoi [1]. We also examine the relationship between various topologies on a von Neumann algebra M with a Cartan MASA D. This provides the necessary tools to parametrize the family of Bures-closed bimodules over a Cartan MASA in terms of projections in a certain abelian von Neumann algebra; this result may be viewed as a weaker form of the Spectral Theorem for Bimodules, and is a key ingredient in the proof of our version of Mercer's theorem. Our results lead to a notion of spectral synthesis for weak-* closed bimodules appropriate to our context, and we show that any von Neumann subalgebra of M which contains D is synthetic. We observe that a result of Sinclair and Smith shows that any Cartan MASA in a von Neumann algebra is norming in the sense of Pop, Sinclair and Smith.

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arXiv2003

The Calculus of One-Sided $M$-Ideals and Multipliers in Operator Spaces

The theory of one-sided $M$-ideals and multipliers of operator spaces is simultaneously a generalization of classical $M$-ideals, ideals in operator algebras, and aspects of the theory of Hilbert $C^*$-modules and their maps. Here we give a systematic exposition of this theory; a reference tool for `noncommutative functional analysts' who may encounter a one-sided $M$-ideal or multiplier in their work.

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arXiv2002

One-sided M-Ideals and Multipliers in Operator Spaces, I

The theory of M-ideals and multiplier mappings of Banach spaces naturally generalizes to left (or right) M-ideals and multiplier mappings of operator spaces. These subspaces and mappings are intrinsically characterized in terms of the matrix norms. In turn this is used to prove that the algebra of left adjointable mappings of a dual operator space X is a von Neumann algebra. If in addition X is an operator A--B-bimodule for $C^{*}$-algebras A and B, then the module operations on X are automatically weak$^{*}$ continuous. One sided L-projections are introduced, and analogues of various results from the classical theory are proved. An assortment of examples is considered.

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arXiv2002

One-Sided Projections on C*-algebras

In [BEZ] the notion of a complete one-sided M-ideal for an operator space X was introduced as a generalization of Alfsen and Effros' notion of an M-ideal for a Banach space [AE72]. In particular, various equivalent formulations of complete one-sided M-projections were given. In this paper, some sharper equivalent formulations are given in the special situation that $X = \mathcal{A}$, a $C^*$-algebra (in which case the complete left M-projections are simply left multiplication on $\mathcal{A}$ by a fixed orthogonal projection in $\mathcal{A}$ or its multiplier algebra). The proof of the first equivalence makes use of a technique which is of interest in its own right--a way of ``solving'' multi-linear equations in von Neumann algebras. This technique is also applied to show that preduals of von Neumann algebras have no nontrivial complete one-sided M-ideals. In addition, we show that in a $C^*$-algebra, the intersection of finitely many complete one-sided M-summands need not be a complete one-sided M-summand, unlike the classical situation.

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