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Free Complementation of Certain MASAs in $L(\mathbb{F}_d)$ via Conditional Transport of Measure Jekel, David
Description
The free group factor $L(\mathbb{F}_d)$ can realized as the von Neumann algebra generated by $d$ freely independent semicircular varables $S_1, \ldots, S_d$, the free probabilistic analogue of an independent family of Gaussians. We consider non-commutative random variables $X_1, \ldots, X_m$ whose non-commutative distribution satisfy the integration-by-parts relation $\tau(D_{X_j} V(X) p(X)) = \tau \otimes \tau(\partial_{X_j} p(X))$, where $V$ is a suitably regular convex function. By studying conditional transport of measure" for the associated $N \times N$ random matrix models in the large $N$ limit, we show that there is an isomorphism $W^*(X_1, \ldots, X_d) \to W^*(S_1, \ldots, S_d)$ such that the $W^*$-algebra of the first $k$ generators is mapped to the $W^*$-algebra of the first $k$ generators for every $k$. In particular, $W^*(X_1)$ is sent to the generator MASA in $L(F_d)$, so it is freely complemented. As an application, we deduce that for every non-commutative polynomial $p$, the algebra $W^*(S_1 + \epsilon p(S))$ is a freely complemented MASA in $L(F_d)$ for sufficiently small $\epsilon$.
Item Metadata
Title |
Free Complementation of Certain MASAs in $L(\mathbb{F}_d)$ via Conditional Transport of Measure
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2019-10-01T16:30
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Description |
The free group factor $L(\mathbb{F}_d)$ can realized as the von Neumann
algebra generated by $d$ freely independent semicircular varables $S_1, \ldots, S_d$, the free probabilistic analogue of an independent family of
Gaussians. We consider non-commutative random variables $X_1, \ldots, X_m$
whose non-commutative distribution satisfy the integration-by-parts
relation $\tau(D_{X_j} V(X) p(X)) = \tau \otimes \tau(\partial_{X_j}
p(X))$, where $V$ is a suitably regular convex function. By studying
conditional transport of measure" for the associated $N \times N$
random matrix models in the large $N$ limit, we show that
there is an isomorphism $W^*(X_1, \ldots, X_d) \to W^*(S_1, \ldots, S_d)$ such
that the $W^*$-algebra of the first $k$ generators is mapped to the
$W^*$-algebra of the first $k$ generators for every $k$. In particular,
$W^*(X_1)$ is sent to the generator MASA in $L(F_d)$, so it is freely
complemented. As an application, we deduce that for every non-commutative
polynomial $p$, the algebra $W^*(S_1 + \epsilon p(S))$ is a freely
complemented MASA in $L(F_d)$ for sufficiently small $\epsilon$.
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Extent |
30.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: UCLA
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Series | |
Date Available |
2020-03-30
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0389675
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International