scholarly journals MAXIMAL COMMUTATIVE SUBALGEBRAS INVARIANT FOR CP-MAPS: (COUNTER-)EXAMPLES

2008 ◽  
Vol 11 (04) ◽  
pp. 523-539 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
FRANCO FAGNOLA ◽  
MICHAEL SKEIDE
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann ◽  
Commutative Subalgebra ◽  
Commutative Subalgebras ◽  
Maximal Commutative Subalgebra

We solve, mainly by counterexamples, many natural questions regarding maximal commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a von Neumann algebra. In particular, we discuss the structure of the generators of norm continuous semigroups on [Formula: see text] leaving a maximal commutative subalgebra invariant and show that there exist Markov CP-semigroups on Md without invariant maximal commutative subalgebras for any d > 2.

scholarly journals Maximal commutative subalgebras of a Grassmann algebra

2019 ◽  
Vol 18 (07) ◽  
pp. 1950139
Author(s):  
Victor A. Bovdi ◽  
Ho-Hon Leung
Keyword(s):  
Grassmann Algebra ◽  
New Approach ◽  
Commutative Subalgebra ◽  
Intersecting Family ◽  
Grassmann Algebras ◽  
Commutative Subalgebras ◽  
Maximal Commutative Subalgebra

We provide a new approach to the investigation of maximal commutative subalgebras (with respect to inclusion) of Grassmann algebras. We show that finding a maximal commutative subalgebra in Grassmann algebras is equivalent to constructing an intersecting family of subsets of various odd sizes in [Formula: see text] which satisfies certain combinatorial conditions. Then we find new maximal commutative subalgebras in the Grassmann algebra of odd rank [Formula: see text] by constructing such combinatorial systems for odd [Formula: see text]. These constructions provide counterexamples to conjectures made by Domoskos and Zubor.

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

scholarly journals ON PRODUCT SYSTEMS ARISING FROM SUM SYSTEMS

2005 ◽  
Vol 08 (01) ◽  
pp. 1-31 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
R. SRINIVASAN
Keyword(s):  
Hilbert Spaces ◽  
Von Neumann Algebra ◽  
Type I ◽  
Product System ◽  
Von Neumann ◽  
Type Iii ◽  
Product Systems ◽  
Weyl Representation ◽  
Uncountable Family ◽  
Type Spaces

B. Tsirelson constructed an uncountable family of type III product systems of Hilbert spaces through the theory of Gaussian spaces, measure type spaces and "slightly colored noises", using techniques from probability theory. Here we take a purely functional analytic approach and try to have a better understanding of Tsireleson's construction and his examples. We prove an extension of Shale's theorem connecting symplectic group and Weyl representation. We show that the "Shale map" respects compositions (this settles an old conjecture of K. R. Parthasarathy8). Using this we associate a product system to a sum system. This construction includes the exponential product system of Arveson, as a trivial case, and the type III examples of Tsirelson. By associating a von Neumann algebra to every "elementary set" in [0, 1], in a much simpler and direct way, we arrive at the invariants of the product system introduced by Tsirelson, given in terms of the sum system. Then we introduce a notion of divisibility for a sum system, and prove that the examples of Tsirelson are divisible. It is shown that only type I and type III product systems arise out of divisible sum systems. Finally, we give a sufficient condition for a divisible sum system to give rise to a unitless (type III) product system.

QUASI-INVARIANCE FORMULAS FOR COMPONENTS OF QUANTUM LÉVY PROCESSES

2004 ◽  
Vol 07 (01) ◽  
pp. 131-145 ◽  
Author(s):  
UWE FRANZ ◽  
NICOLAS PRIVAULT
Keyword(s):  
Lie Algebras ◽  
Lévy Processes ◽  
Unitary Operator ◽  
Levy Processes ◽  
Absolutely Continuous ◽  
Commutative Subalgebra ◽  
Independent Increments ◽  
Commutative Subalgebras ◽  
Current Representation ◽  
General Method

A general method for deriving Girsanov or quasi-invariance formulas for classical stochastic processes with independent increments obtained as components of Lévy processes on real Lie algebras is presented. Letting a unitary operator arising from the associated factorizable current representation act on an appropriate commutative subalgebra, a second commutative subalgebra is obtained. Under certain conditions the two commutative subalgebras lead to two classical processes such that the law of the second process is absolutely continuous w.r.t. to the first. Examples include the Girsanov formula for Brownian motion as well as quasi-invariance formulas for the Poisson process, the Gamma process,15,16 and the Meixner process.

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