Wyniki wyszukiwania

1

On Q-independence, limit theorems and q-Gaussian distribution

100%

Marciniak M.

Studia Mathematica

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1998

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tom 129

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nr 2

113-135

EN

We formulate the notion of Q-independence which generalizes the classical independence of random variables and free independence introduced by Voiculescu. Here Q stands for a family of polynomials indexed by tiny partitions of finite sets. The analogs of the central limit theorem and Poisson limit theorem are proved. Moreover, it is shown that in some special cases this kind of independence leads to the q-probability theory of Bożejko and Speicher.

2

On extremal positive maps acting between type I factors

100%

Marciniak M.

Banach Center Publications

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2010

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tom 89

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nr 1

201-221

EN

The paper is devoted to the problem of classification of extremal positive linear maps acting between 𝔅(𝒦) and 𝔅(ℋ) where 𝒦 and ℋ are Hilbert spaces. It is shown that every positive map with the property that rank ϕ(P) ≤ 1 for any one-dimensional projection P is a rank 1 preserver. This allows us to characterize all decomposable extremal maps as those which satisfy the above condition. Further, we prove that every extremal positive map which is 2-positive turns out to be automatically completely positive. Finally, we get the same conclusion for extremal positive maps such that rank ϕ(P) ≤ 1 for some one-dimensional projection P and satisfy the condition of local complete positivity. This allows us to give a negative answer to Robertson's problem in some special cases.

3

Quantum symmetries in noncommutative C*-systems

100%

Marciniak M.

Banach Center Publications

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1998

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tom 43

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nr 1

297-307

EN

We introduce the notion of a completely quantum C*-system (A,G,α), i.e. a C*-algebra A with an action α of a compact quantum group G. Spectral properties of completely quantum systems are investigated. In particular, it is shown that G-finite elements form the dense *-subalgebra 𝐴 of A. Furthermore, properties of ergodic systems are studied. We prove that there exists a unique α-invariant state ω on A. Its properties are described by a family of modular operators ${σ_z}_{z∈ℂ}$ acting on 𝐴. It turns out that ω is a KMS state provided that ω is faithful.

4

On the structure of positive maps between matrix algebras

64%

Majewski W., Marciniak M.

Banach Center Publications

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2007

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tom 78

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nr 1

249-263

EN

The structure of the set of positive unital maps between M₂(ℂ) and Mₙ(ℂ) (n ≥ 3) is investigated. We proceed with the study of the "quantized" Choi matrix thus extending the methods of our previous paper [MM2]. In particular, we examine the quantized version of Størmer's extremality condition. Maps fulfilling this condition are characterized. To illustrate our approach, a careful analysis of Tang's maps is given.

5

Decomposability of extremal positive unital maps on M₂(ℂ)

64%

Majewski W., Marciniak M.

Banach Center Publications

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2006

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tom 73

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nr 1

347-356

EN

A map φ: Mₘ(ℂ) → Mₙ(ℂ) is decomposable if it is of the form φ = φ₁ + φ₂ where φ₁ is a CP map while φ₂ is a co-CP map. It is known that if m = n = 2 then every positive map is decomposable. Given an extremal unital positive map φ: M₂(ℂ) → M₂(ℂ) we construct concrete maps (not necessarily unital) φ₁ and φ₂ which give a decomposition of φ. We also show that in most cases this decomposition is unique.

Ograniczanie wyników

4 Banach Center Publications

1 Studia Mathematica

5 Marciniak M.

2 Majewski W.

1 2010

1 2007

1 2006

2 1998

On Q-independence, limit theorems and q-Gaussian distribution

On extremal positive maps acting between type I factors

Quantum symmetries in noncommutative C*-systems

On the structure of positive maps between matrix algebras

Decomposability of extremal positive unital maps on M₂(ℂ)