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2024 | OriginalPaper | Chapter

On de Finetti-Type Theorems

Author : Paola Zurlo

Published in: Operator and Matrix Theory, Function Spaces, and Applications

Publisher: Springer Nature Switzerland

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Abstract

The investigation of distributional symmetries was initiated by de Finetti’s celebrated theorem, which shows that any finite joint-distribution of sequences of two-point valued exchangeable random variables is obtained by randomization of the binomial distribution. This result has since found several generalizations both in classical and noncommutative settings. In this paper, we discuss a series of recent results that extend de Finetti’s theorem to various non-commutative models, including Bolean, monotone, CAR algebra as well as \(\mathbb {Z}_2\)-graded \(C^*\)-algebras.

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previous chapter Kippenhahn’s Construction Revisited

See [13].

See [17].

We recall that, given an inclusion \(\mathfrak {B}\subset \mathfrak {A}\) of \(C^*\)-algebras, a conditional expectation \(E: \mathfrak {A}\rightarrow \mathfrak {B}\) is a positive linear map such that \(E(b)=b\), for any \(b\in \mathfrak {B}\), and \(E(b_1ab_2)=b_1E(a)b_2\), for any \(b_1, b_2\in \mathfrak {B}\), \(a\in \mathfrak {A}\).

\(\mathfrak {A}_+:=\mathfrak {A}_1\) and \(\mathfrak {A}_{-}:=\mathfrak {A}_{-1}\) are also referred to as the even part and the odd part of \(\mathfrak {A}\) respectively.

The involution is still denoted by \(*\) with a minor abuse of notation.

\(a_1\hat {\otimes } a_2\) is nothing but \(a_1\otimes a_2\) thought of as an element of the \(\mathbb {Z}_2\)-graded \(*\)-algebra \(\mathfrak {A}_1\hat {\otimes } \mathfrak {A}_2\), since the underlying vector spaces of \(\mathfrak {A}_1\hat {\otimes } \mathfrak {A}_2\) and \(\mathfrak {A}_1\otimes \mathfrak {A}_2\,\) are identical. When no confusion can occur one can use the two notations \(a_1\otimes a_2\) and \(a_1\hat {\otimes } a_2\) interchangeably.

(cf. [11] again for further details).

When \(\mathfrak {A}\) is a fixed unital \(C^*\)-algebra, the minimal Fermi tensor product of \(\mathfrak {A}\) with itself n times is denoted by \(\mathfrak {A}^{(n)}\) and the corresponding infinite graded tensor product by \(\hat {\otimes }_{\mathrm {min}}^{\mathbb {N}}\mathfrak {A}\).

If \(\omega _i=\omega ,\, \forall i \in \mathbb {N}\), we then simply denote by \(\times ^{\mathbb {N}}\omega \) the product state \(\underset {i\in \mathbb {N}}{\times }\omega _i\) on \(\hat {\otimes }^{\mathbb {N}}_{\mathrm {min}}\mathfrak {A}\).

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Title: On de Finetti-Type Theorems
Author: Paola Zurlo
Publisher: Springer Nature Switzerland
Book: Operator and Matrix Theory, Function Spaces, and Applications
Print ISBN: 978-3-031-50612-3

Electronic ISBN: 978-3-031-50613-0

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-50613-0_19

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