Subharmonicity in von Neumann algebras

Thomas Ransford; Michel Valley

Studia Mathematica (2005)

• Volume: 170, Issue: 3, page 219-226
• ISSN: 0039-3223

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Abstract

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Let ℳ be a von Neumann algebra with unit $1_{ℳ}$. Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by ${\mu }_t{\left(x\right)}_{t\ge 0}$ the generalized s-numbers of x, defined by ${\mu }_t\left(x\right)$ = inf||xe||: e is a projection in ℳ i with $\tau (1_{ℳ}-e)$ ≤ t (t ≥ 0). We prove that, if D is a complex domain and f:D → ℳ is a holomorphic function, then, for each t ≥ 0, $\lambda \mapsto {\int }_0^{t}log{\mu }_s\left(f\left(\lambda \right)\right)ds$ is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.

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Thomas Ransford, and Michel Valley. "Subharmonicity in von Neumann algebras." Studia Mathematica 170.3 (2005): 219-226. <http://eudml.org/doc/284793>.

@article{ThomasRansford2005,
abstract = {Let ℳ be a von Neumann algebra with unit $1_\{ℳ\}$. Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by $μ_\{t\}(x)_\{t≥0\}$ the generalized s-numbers of x, defined by $μ_\{t\}(x)$ = inf||xe||: e is a projection in ℳ i with $τ(1_\{ℳ\} - e)$ ≤ t (t ≥ 0). We prove that, if D is a complex domain and f:D → ℳ is a holomorphic function, then, for each t ≥ 0, $λ ↦ ∫_\{0\}^\{t\} log μ_\{s\} (f(λ))ds$ is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.},
author = {Thomas Ransford, Michel Valley},
journal = {Studia Mathematica},
keywords = {von Neumann algebra; singular value; trace; determinant; subharmonic function},
language = {eng},
number = {3},
pages = {219-226},
title = {Subharmonicity in von Neumann algebras},
url = {http://eudml.org/doc/284793},
volume = {170},
year = {2005},
}

TY - JOUR
AU - Thomas Ransford
AU - Michel Valley
TI - Subharmonicity in von Neumann algebras
JO - Studia Mathematica
PY - 2005
VL - 170
IS - 3
SP - 219
EP - 226
AB - Let ℳ be a von Neumann algebra with unit $1_{ℳ}$. Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by $μ_{t}(x)_{t≥0}$ the generalized s-numbers of x, defined by $μ_{t}(x)$ = inf||xe||: e is a projection in ℳ i with $τ(1_{ℳ} - e)$ ≤ t (t ≥ 0). We prove that, if D is a complex domain and f:D → ℳ is a holomorphic function, then, for each t ≥ 0, $λ ↦ ∫_{0}^{t} log μ_{s} (f(λ))ds$ is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.
LA - eng
KW - von Neumann algebra; singular value; trace; determinant; subharmonic function
UR - http://eudml.org/doc/284793
ER -

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