## Hypercontractivity in non-commutative holomorphic spaces (2005)

Venue: | Commun. Math. Phys |

Citations: | 8 - 6 self |

### BibTeX

@ARTICLE{Kemp05hypercontractivityin,

author = {Todd Kemp},

title = {Hypercontractivity in non-commutative holomorphic spaces},

journal = {Commun. Math. Phys},

year = {2005},

volume = {259},

pages = {615--637}

}

### OpenURL

### Abstract

ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of non-commutative “holomorphic ” algebras. Our setting is the q-Gaussian algebras Γq associated to the q-Fock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a q-Segal-Bargmann transform, and prove Janson’s strong hypercontractivity L 2 (Hq) → L r (Hq) for r an even integer. 1.

### Citations

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Citation Context ...o hold. He developed this further in [G3]. Over the subsequent three decades, Nelson’s hypercontractivity inequality (and its equivalent form, the logarithmic Sobolev inequality, invented by Gross in =-=[G2]-=-) found myriad applications in analysis, probability theory, differential geometry, statistical mechanics, and other areas of mathematics and physics. See, for example, the recent survey [G5]. The Fer... |

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Citation Context ...n this paper, non-commutative algebras Hq will be introduced, which are q-deformations of the algebra of holomorphic functions. The special cases q = ±1 and q = 0 are already known; H−1 is defined in =-=[BSZ]-=-, while H0 is isomorphic to the free Segal-Bargmann space of [B2]. We will construct a unitary isomorphism Sq from L 2 (Γq) to L 2 (Hq), which is a q-analog of the Segal-Bargmann transform. Hq itself ... |

65 |
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Citation Context ... generators. One known fact about the algebras Γq(H ) for −1 < q < 1 is that they are all type II1 factors. This is a consequence (in the dim H = ∞ case) of the following theorem, which was proved in =-=[BSp]-=-. Proposition 2.3 (Bozejko, Speicher). Let −1 < q < 1. The vacuum expectation state τq(A) = (AΩ, Ω)q on B(Fq(H )) restricts to a faithful, normal, finite trace on Γq(H ). 4sThe reader may wish to veri... |

64 | q-Gaussian processes: noncommutative and classical aspects
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Citation Context ...same constants.) Then, in 1997, Biane [B1] extended Carlen and Lieb’s work beyond the Fermionic (Clifford algebra) setting to the q-Gaussian von Neumann algebras Γq of Bozejko, Kümmerer, and Speicher =-=[BKS]-=-. His theorem may be stated as follows. Date: November 5, 2005. This is an author-generated copy of the paper [K], published in Communications in Mathematical Physics in November, 2005. It is also ava... |

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Citation Context ...onsistent theory of interacting quantum fields, Nelson proved his famous hypercontractivity inequality in its initial form [N1]; by 1973 it evolved into the following statement, which may be found in =-=[N2]-=-. Theorem 1.1 (Nelson, 1973). Let Aγ be the Dirichlet form operator for Gauss measure dγ(x) = (2π) −n/2 e −|x|2 /2 dx on R n . For 1 < p ≤ r < ∞ and f ∈ L p (R n , γ), �e −tAγ f�r ≤ �f�p, for t ≥ tN(p... |

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Citation Context ... R n . The initial purpose of such hypercontractive inequalities was to prove the semiboundedness of Hamiltonians in the theory of Boson quantum fields. (See, for example, [Gli], [N1], and [Se2].) In =-=[G1]-=-, Gross used this inequality (through an appropriate cut-off approximation) to show that the Boson energy operator in a model of 2-dimensional Euclidean quantum field theory has a unique ground state.... |

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Citation Context ... unproven in its sharp form until the early 1990s. Lindsay [L] and Meyer [LM] proved that it holds L2 → Lr for r = 2, 4, 6, . . . (and in the dual cases Lr′ → L2 as well). Soon after, Carlen and Lieb =-=[CL]-=- were able to complete Gross’ original argument with some clever non-commutative integration inequalities, thus proving the full result. (Precisely: they showed that the Clifford algebra analogs of th... |

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Citation Context ...ON As part of the work in the 1960s and 1970s to construct a mathematically consistent theory of interacting quantum fields, Nelson proved his famous hypercontractivity inequality in its initial form =-=[N1]-=-; by 1973 it evolved into the following statement, which may be found in [N2]. Theorem 1.1 (Nelson, 1973). Let Aγ be the Dirichlet form operator for Gauss measure dγ(x) = (2π) −n/2 e −|x|2 /2 dx on R ... |

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Citation Context ...bra, this family is not known to be complex interpolation scale. For example, in the q = 1 case, the family is not complex interpolation scale when H is infinite-dimensional (this is almost proven in =-=[JPR]-=-). Hence, once we have proved Theorem 1.4, it is not an easy matter to generalize to the case p > 2, r �= 2, 4, 6, . . . 3. MIXED SPIN AND STRONG HYPERCONTRACTIVITY We will consider the mixed-spin alg... |

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Citation Context ...th image spanned by ei1 ⊗· · ·⊗ein and kernel orthogonal to ej1 ⊗· · ·⊗ejm. It follows that W0 contains all finite rank operators, and hence is the full algebra B(F0(H )). For q �= 0, it is proved in =-=[DN]-=- that there is a unitary map Uq : F0 → Fq, which preserves the vacuum and satisfies UqC0U ∗ q ⊆ Cq, where Cq is the C ∗ -algebra generated by {cq(h) ; h ∈ H }. As Wq is the weak closure of Cq, it foll... |

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Citation Context ... original hypercontractive estimate (Theorem 1.1). Concurrent to the work on non-commutative hypercontractivity, a different sort of extension of Nelson’s theorem was being developed. In 1983 Janson, =-=[J]-=-, discovered that if one restricts the semigroup e −tAγ in Theorem 1.1 to holomorphic functions on R 2n ∼ = C n then the contractivity of Equation 1.1 is attained in a shorter time than tN. Writing HL... |

12 |
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Citation Context ...lying space R n . The initial purpose of such hypercontractive inequalities was to prove the semiboundedness of Hamiltonians in the theory of Boson quantum fields. (See, for example, [Gli], [N1], and =-=[Se2]-=-.) In [G1], Gross used this inequality (through an appropriate cut-off approximation) to show that the Boson energy operator in a model of 2-dimensional Euclidean quantum field theory has a unique gro... |

11 |
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Citation Context ... ∗-isomorphism, it preserves the operator norm; hence, it is an L∞ isometry. It is also easy to verify by calculation that α preserves the L2-norm. Therefore, by the complex interpolation method (see =-=[PX]-=- for an excellent discussion of interpolation in non-commutative Lp-spaces), α is an Lp contraction for 2 ≤ p ≤ ∞. The same arguments applied to α−1 show that α is an Lp isometry in this case. The cor... |

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Citation Context ...< p < r < ∞). Moreover, Janson’s result holds as p → 0, in a regime where the semigroup e −tAγ is not even well-defined in the full L p -space. These results have been further generalized by Gross in =-=[G4]-=- to the case of complex manifolds. In this paper, non-commutative algebras Hq will be introduced, which are q-deformations of the algebra of holomorphic functions. The special cases q = ±1 and q = 0 a... |

9 |
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Citation Context ...on n of the underlying space R n . The initial purpose of such hypercontractive inequalities was to prove the semiboundedness of Hamiltonians in the theory of Boson quantum fields. (See, for example, =-=[Gli]-=-, [N1], and [Se2].) In [G1], Gross used this inequality (through an appropriate cut-off approximation) to show that the Boson energy operator in a model of 2-dimensional Euclidean quantum field theory... |

8 |
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Citation Context ...paper he also showed that if one represents the Fock space for Fermions as the L 2 -space of a Clifford algebra (as in [Se1]), then inequalities similar to 1.1 also hold. He developed this further in =-=[G3]-=-. Over the subsequent three decades, Nelson’s hypercontractivity inequality (and its equivalent form, the logarithmic Sobolev inequality, invented by Gross in [G2]) found myriad applications in analys... |

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(Show Context)
Citation Context ...K ). Indeed, Γq can be construed as such a functor as well. The isomorphism classes of the von Neumann algebras Γq(H ) for q /∈ {±1, 0} are not yet understood. (For some partial results, however, see =-=[R]-=- and [´Sn].) The ±1 cases have been understood since antiquity: Γ1(H ) = L ∞ (M, γ) for a certain measure space M with a Gaussian measure γ , while Γ−1(H ) is a Clifford algebra modeled on H . These f... |

8 |
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Citation Context ... a system of mixed spins (in a von Neumann algebra generated by elements which satisfy some commutation and some anti-commutation relations), and then applying a central limit theorem due to Speicher =-=[S]-=-. The case q = −1 is Carlen and Lieb’s adaptation of Gross’ work, while the q = 1 case is Nelson’s original hypercontractive estimate (Theorem 1.1). Concurrent to the work on non-commutative hypercont... |

7 |
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(Show Context)
Citation Context ...i, j) = 1 for i �= j (i.e. when different generators commute), the generators of C (I, σ) may be modeled by |I| i.i.d. Bernoulli random variables, and so we reproduce the toy Fock space considered in =-=[M]-=-. In the general case, C (I, σ) has, as a vector space, a basis consisting of all xA with A = (i1, . . . , ik) increasing multi-indices in Ik , where xA = xi1 . . . xik , and x∅ denotes the identity 1... |

5 |
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Citation Context ...Gross in [G2]) found myriad applications in analysis, probability theory, differential geometry, statistical mechanics, and other areas of mathematics and physics. See, for example, the recent survey =-=[G5]-=-. The Fermion hypercontractivity inequality in [G3] remained unproven in its sharp form until the early 1990s. Lindsay [L] and Meyer [LM] proved that it holds L2 → Lr for r = 2, 4, 6, . . . (and in th... |

3 |
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(Show Context)
Citation Context ... other areas of mathematics and physics. See, for example, the recent survey [G5]. The Fermion hypercontractivity inequality in [G3] remained unproven in its sharp form until the early 1990s. Lindsay =-=[L]-=- and Meyer [LM] proved that it holds L2 → Lr for r = 2, 4, 6, . . . (and in the dual cases Lr′ → L2 as well). Soon after, Carlen and Lieb [CL] were able to complete Gross’ original argument with some ... |

2 | P.: A Gaussian Random Matrix Model for q–deformed Gaussian Variables - Sniady - 2001 |

1 | Contractivity properties of the Ornsetein-Uhlenbeck semigroup for general commutation realtions - Krolak |

1 |
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(Show Context)
Citation Context ...f mathematics and physics. See, for example, the recent survey [G5]. The Fermion hypercontractivity inequality in [G3] remained unproven in its sharp form until the early 1990s. Lindsay [L] and Meyer =-=[LM]-=- proved that it holds L2 → Lr for r = 2, 4, 6, . . . (and in the dual cases Lr′ → L2 as well). Soon after, Carlen and Lieb [CL] were able to complete Gross’ original argument with some clever non-comm... |

1 |
R.: Completely positice maps on Coxeter groups, deformed commutation relations, and operator spaces
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(Show Context)
Citation Context ...generators). One known fact about the algebras Γq(H ) for −1 < q < 1 is that they are all type II1 factors. This is a consequence (in the dim H = ∞ case) of the following theorem, which was proved in =-=[BSp]-=-. Proposition 2.2 (Bozejko, Speicher). Let −1 < q < 1. The vacuum expectation state τq(A) = (AΩ,Ω)q on B(Fq(H )) restricts to a faithful, normal, finite trace on Γq(H ). The reader may wish to verify ... |