arXiv subjects

Chaitanya J. Kulkarni

Publications and source records attributed to Chaitanya J. Kulkarni.

11 recordsLinked to original sources

arXiv2026

Functional Models of Abelian Locally von Neumann Algebras and Direct Integrals of Locally Hilbert Spaces

We obtain a functional model for an arbitrary Abelian locally von Neumann algebra acting on a representing locally Hilbert space under the assumption that the index directed set is countable, in terms of locally essentially bounded functions on strictly inductive systems of measure spaces, which can be viewed as the reduction theory of this kind of operator algebras. Then, we single out the concept of a direct integral of locally Hilbert spaces and the concepts of locally decomposable and locally diagonlisable operators and we show that these form locally von Neumann algebras that are commutant one to each other. Finally, we show that any Abelian locally von Neumann algebra, which acts on separable representing locally Hilbert spaces and such that the index set is a sequentially finite directed set, is spatially isomorphic with the Abelian locally von Neumann algebra of all locally diagonlisable operators on a certain direct integral of locally Hilbert spaces with respect to a certain strictly inductive system of locally finite measure spaces on standard Borel spaces.

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arXiv2026

On the Haagerup property for partial crossed products

Let $(A,G,α)$ be a partial dynamical system and let $A\rtimes_{α,r} G$ denote the associated reduced partial crossed product. In this article, we introduce the Haagerup property for partial actions of discrete groups on $C^*$-algebras. We prove that the partial crossed product $A\rtimes_{α,r} G$ has the Haagerup property if and only if both $A$ and the partial action $α$ have the Haagerup property. As a consequence, we obtain an equivalence between the Haagerup property of the partial crossed product and that of the underlying $C^*$-algebra and the acting group. We also show that the Haagerup property is preserved under inductive limits and apply this result to study the Haagerup property of inductive limits of partial crossed products.

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arXiv2026

$C^\ast$-extreme points of unital completely positive maps invariant under group action

In this work, we study a sub-collection of unital completely positive maps from a unital $C^\ast$-algebra $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, the algebra of bounded linear operators on a Hilbert space $\mathcal{H}$ in the setting of $C^\ast$-convexity. Let $τ$ be an action of a group $G$ on the $C^\ast$-algebra $\mathcal{A}$ through $C^\ast$-automorphisms. We focus our attention to the set of all unital completely positive maps from $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, which remain invariant under $τ$. We denote this collection by the notation $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. This collection forms a $C^\ast$-convex set. We characterize the set of $C^\ast$-extreme points of $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. Further, we conclude the article by proving the Krein--Milman type theorem in the setting of $C^\ast$-convexity for the set $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$.

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arXiv2025

Direct integral of locally Hilbert spaces

In this work, we introduce the concept of the direct integral of locally Hilbert spaces. This notion is formulated such that the direct integral of locally Hilbert spaces forms a locally Hilbert space. We then define two classes of locally bounded operators on the direct integral of locally Hilbert spaces namely the class of decomposable locally bounded operators and the class of diagonalizable locally bounded operators. We prove that the set of all decomposable locally bounded operators forms a locally von Neumann algebra, while the set of diagonalizable locally bounded operators forms an abelian von Neumann algebra, and we show that they are commutant of each other.

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arXiv2025

Direct integral of locally Hilbert spaces over a locally measure space

In this work, we introduce the concept of the direct integral of locally Hilbert spaces by generalizing the classical notion of a measure space to that of a locally measure space. We establish that the direct integral of a family of locally Hilbert spaces over a locally measure space forms a locally Hilbert space. We then define two important subclasses of locally bounded operators on such direct integrals, namely decomposable locally bounded operators and diagonalizable locally bounded operators. We show that each of these subclasses forms a locally von Neumann algebra, and in particular, that the locally von Neumann algebra of diagonalizable operators is abelian. Finally, we prove that the locally von Neumann algebra of diagonalizable operators coincides with the commutant of the locally von Neumann algebra of decomposable operators.

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arXiv2025

Extreme points of unital completely positive maps invariant under partial action

The classical Choquet theorem establishes a barycentric decomposition for elements in a compact convex subset of a locally convex topological vector space. This decomposition is achieved through a probability measure that is supported on the set of extreme points of the subset. In this work, we consider a partial action $τ$ of a group $G$ on a $C^\ast$-algebra $\mathcal{A}$. For a fixed Hilbert space $\mathcal{H}$, we consider the set of all unital completely positive maps from $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$ that are invariant under the partial action $τ$. This set forms a compact convex subset of a locally convex topological vector space. To complete the picture of the barycentric decomposition provided by the classical Choquet theorem, we characterize the set of extreme points of this set.

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arXiv2024

Ergodic decomposition in the space of unital completely positive maps

The classical decomposition theory for states on a C*-algebra that are invariant under a group action has been studied by using the theory of orthogonal measures on the state space \cite{BR1}. In \cite{BK3}, we introduced the notion of \textit{generalized orthogonal measures} on the space of unital completely positive (UCP) maps from a C*-algebra $A$ into $B(H)$. In this article, we consider a group $G$ that acts on a C*-algebra $A$, and the collection of $G$-invariant UCP maps from $A$ into $B(H)$. This article examines a $G$-invariant decomposition of UCP maps by using the theory of generalized orthogonal measures on the space of UCP maps, developed in \cite{BK3}. Further, the set of all $G$-invariant UCP maps is a compact and convex subset of a topological vector space. Hence, by characterizing the extreme points of this set, we complete the picture of barycentric decomposition in the space of $G$-invariant UCP maps. We establish this theory in Stinespring and Paschke dilations of completely positive maps. We end this note by mentioning some examples of UCP maps admitting a decomposition into $G$-invariant UCP maps.

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arXiv2024

Direct Integral and Decompoisitions of Locally Hilbert spaces

In this work, we introduce the concept of direct integral of locally Hilbert spaces by using the notion of locally standard measure space (analogous to standard measure space defined in the classical setup), which we obtain by considering a strictly inductive system of measurable spaces along with a projective system of finite measures. Next, we define a locally Hilbert space given by the direct integral of a family of locally Hilbert spaces. Following that we introduce decomposable locally bounded and diagonalizable locally bounded operators. Further, we show that the class of diagonalizable locally bounded operators is an abelian locally von Neumann algebra, and this can be seen as the commutant of decomposable locally bounded operators. Finally, we discuss the following converse question: For a locally Hilbert space $\mathcal{D}$ and an abelian locally von Neumann algebra $\mathcal{M}$, does there exist a locally standard measure space and a family of locally Hilbert spaces such that (1) the locally Hilbert space $\mathcal{D}$ is identified with the direct integral of family of locally Hilbert spaces; (2) the abelian locally von Neumann algebra $\mathcal{M}$ is identified with the abelian locally von Neumann algebra of all diagonalizable locally bounded operators ? We answer this question affirmatively for a certain class of abelian locally von Neumann algebras.

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arXiv2023

Generalized Orthogonal Measures on the Space of Unital Completely Positive Maps

A classical result by Effros connects the barycentric decomposition of a state on a C*-algebra to the disintegration of the GNS representation of the state with respect to an orthogonal measure on the state space of the C*-algebra. In this note, we take this approach to the space of unital completely positive maps on a C*-algebra with values in B(H), connecting the barycentric decomposition of the unital completely positive map and the dis-integration of the minimal Stinespring dilation of the same. This generalizes Effros' work in the non-commutative setting. We do this by introducing a special class of barycentric measures which we call generalized orthogonal measures. We end this note by mentioning some examples of generalized orthogonal measures.

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arXiv2023

Decomposable Weak Expectations

In this article we define a special class of weak expectations for a representation of a separable unital C*-algebra, called decomposable weak expectation. We give necessary and sufficient conditions for such kind of weak expectations to exist for a given representation. Then we define decomposable measures on the state space of a C*-algebra and show that the GNS representation of a state admits a decomposable weak expectation if and only if there is a decomposable measure on the state space. Further we give an example of a decomposable weak expectation.

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arXiv2023

Barycentric decompositions in the space of weak expectations

The space of weak expectations for a given representation of a (unital) separable C*-algebra is a compact convex set of (unital) completely positive maps in the BW topology, when it is non-empty. An application of the classical Choquet theory gives a barycentric decomposition of a weak expectation in that set. However, to complete the barycentric picture, one needs to know the extreme points of the compact convex set in question. In this article, we explicitly identify the set of extreme points of the space of weak expectations for a given representation, using operator theoretic techniques.

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