A. Ya. Helemskii’s research while affiliated with Lomonosov Moscow State University and other places

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Publications (34)


Free and projective generalized multinormed spaces
  • Article

September 2022

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11 Reads

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1 Citation

Journal of Mathematical Analysis and Applications

A.Ya. Helemskii

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The paper investigates free and projective L-spaces, where L is a given normed space. These spaces form a far-reaching generalization of known p-multinormed spaces; in particular, if L=Lp(X), the L-spaces can be considered as p-multinormed spaces, based on arbitrary σ-finite measure spaces X (for “canonical” p-multinormed spaces, X=N with the counting measure). We first describe a “naturally appearing” functor, based on paving L with contractively complemented finite dimensional subspaces. This finite dimensionality is essential; it permits us to describe a free L-space for this functor. As a corollary, we obtain a wide variety of projective L-spaces. For “nice” spaces L (such as the space of simple p-integrable functions on a measure space), we obtain a full characterization of projective L-spaces; as a particular case, we recover a description of projective p-multinormed spaces.


Free and projective generalized multinormed spaces
  • Preprint
  • File available

September 2021

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38 Reads

The paper investigates free and projective L{\bf L}-spaces, where L{\bf L} is a given normed space. These spaces form a far-reaching generalization of known p-multinormed spaces; in particular, if L=Lp(X){\bf L}=L_p(X), the L{\bf L}-spaces can be considered as p-multinormed spaces, based on arbitrary σ\sigma-finite measure spaces X (for "canonical" p-multinormed spaces, X=NX=\mathbb N with the counting measure). We first describe a "naturally appearing" functor, based on paving L{\bf L} with contractively complemented finite dimensional subspaces. This finite dimensionality is essential; it permits us to describe a free L{\bf L}-space for this functor. As a corollary, we obtain a wide variety of projective L{\bf L}-spaces. For "nice" L{\bf L} (such as the space of simple p-integrable functions on a measure space), we obtain a full description of projective L{\bf L}-spaces.

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The existence of p-convex tensor products of Lp(X)L_p(X)–spaces for the case of an arbitrary measure

Positivity

We obtain the existence theorem for the projective tensor product of p-convex, p∈[1,∞)p[1,)p\in [1,\infty ), Lp(X)Lp(X)L_p(X)-spaces, generalizing p-multinormed spaces of Dales et al. Earlier this result was known under additional assumptions on the measure space X. Now it is proved in full generality.



Projective Quantum Modules and Projective Ideals of C*-algebras

March 2018

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4 Reads

We introduce in non-coordinate presentation the notions of a quantum algebra and of a quantum module over such an algebra. Then we give the definition of a projective quantum module and of a free quantum module, the latter as a particular case of the notion of a free object in a rigged category. (Here we say “quantum” instead of frequently used protean adjective “operator”). After this we discuss the general connection between projectivity and freeness. Then we show that for a Banach quantum algebra A and a Banach quantum space E the Banach quantum A-module A⊗opE is free, where “⊗op” denotes the operator-projective tensor product of Banach quantum spaces. This is used in the proof of the following theorem: all closed left ideals in a separable C*-algebra, endowed with the standard quantization, are projective left quantum modules over this algebra. © Springer International Publishing AG, part of Springer Nature 2018.


Multi-normed spaces, based on non-discrete measures, and their tensor products

June 2017

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22 Reads

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10 Citations

Izvestiya Mathematics

It was A. Lambert who discovered a new type of structures, situated, in a sense, between normed spaces and (abstract) operator spaces. His definition was based on the notion of amplification a normed space by means of spaces 2n\ell_2^n. Afterwards several mathematicians investigated more general structure, "p-multi-normed space", introduced with the help of spaces pn\ell_p^n; 1p1\le p\le\infty. In the present paper we pass from p\ell_p to Lp(X,μ)L_p(X,\mu) with an arbitrary measure. This happened to be possible in the frame-work of the non-coordinate ("index-free") approach to the notion of amplification, equivalent in the case of a discrete counting measure to the approach in mentioned articles. Two categories arise. One consists of amplifications by means of an arbitrary normed space, and another one consists of p-convex amplifications by means of Lp(X,μ)L_p(X,\mu). Each of them has its own tensor product of its objects whose existence is proved by a respective explicit construction. As a final result, we show that the "p-convex" tensor product has especially transparent form for the so-called minimal LpL_p-amplifications of LqL_q-spaces, where q is the conjugate of p. Namely, tensoring Lq(Y,ν)L_q(Y,\nu) and Lq(Z,λ)L_q(Z,\lambda), we get Lq(Y×Z,ν×λ)L_q(Y\times Z,\nu\times\lambda).


Projective tensor product of proto-quantum spaces

June 2017

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13 Reads

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3 Citations

Colloquium Mathematicum

A proto-quantum space is a (general) matricially normed space in the sense of Effros and Ruan presented in a `matrix-free' language. We show that these spaces have a special (projective) tensor product possessing the universal property with respect to completely bounded bilinear operators. We study some general properties of this tensor product (among them a kind of adjoint associativity), and compute it for some tensor factors, notably for L1L_1 spaces. In particular, we obtain what could be called the proto-quantum version of the Grothendieck theorem about classical projective tensor products by L1L_1 spaces. At the end, we compare the new tensor product with the known projective tensor product of operator spaces, and show that the standard construction of the latter is not fit for general proto-quantum spaces.


Projective tensor product of protoquantum spaces

June 2017

A proto-quantum space is a (general) matricially normed space in the sense of Effros and Ruan presented in a `matrix-free' language. We show that these spaces have a special (projective) tensor product possessing the universal property with respect to completely bounded bilinear operators. We study some general properties of this tensor product (among them a kind of adjoint associativity), and compute it for some tensor factors, notably for L1L_1 spaces. In particular, we obtain what could be called the proto-quantum version of the Grothendieck theorem about classical projective tensor products by L1L_1 spaces. At the end, we compare the new tensor product with the known projective tensor product of operator spaces, and show that the standard construction of the latter is not fit for general proto-quantum spaces.


Multi-normed spaces, based on non-discrete measures, and their tensor products

June 2017

It was A. Lambert who discovered a new type of structures, situated, in a sense, between normed spaces and (abstract) operator spaces. His definition was based on the notion of amplification a normed space by means of spaces 2n\ell_2^n. Afterwards several mathematicians investigated more general structure, "p-multi-normed space", introduced with the help of spaces pn\ell_p^n; 1p1\le p\le\infty. In the present paper we pass from p\ell_p to Lp(X,μ)L_p(X,\mu) with an arbitrary measure. This happened to be possible in the frame-work of the non-coordinate ("index-free") approach to the notion of amplification, equivalent in the case of a discrete counting measure to the approach in mentioned articles. Two categories arise. One consists of amplifications by means of an arbitrary normed space, and another one consists of p-convex amplifications by means of Lp(X,μ)L_p(X,\mu). Each of them has its own tensor product of its objects whose existence is proved by a respective explicit construction. As a final result, we show that the "p-convex" tensor product has especially transparent form for the so-called minimal LpL_p-amplifications of LqL_q-spaces, where q is the conjugate of p. Namely, tensoring Lq(Y,ν)L_q(Y,\nu) and Lq(Z,λ)L_q(Z,\lambda), we get Lq(Y×Z,ν×λ)L_q(Y\times Z,\nu\times\lambda).


Structures on the way from classical to quantum spaces and their tensor products

June 2017

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14 Reads

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4 Citations

We study tensor products of two structures situated, in a sense, between normed spaces and (abstract) operator spaces. We call them Lambert and proto-Lambert spaces and pay more attention to the latter ones. The considered two tensor products lead to essentially different norms in the respective spaces. Moreover, the proto-Lambert tensor product is especially nice for spaces with the maximal proto-Lambert norm and in particular, for L1L_1-spaces. At the same time the Lambert tensor product is nice for Hilbert spaces with the minimal Lambert norm.


Citations (20)


... We also note that a more "axiomatic" approach to freeness has been pursued by A. Ya. Helemskii in, e.g., [75], [76], [77], [78], and [79] (see also [6] for a different take on the same approach). Specifically, suppose K and L are categories, and is a faithful covariant functor K → L (usually, a "forgetful functor"). ...

Reference:

Free Banach lattices
Free and projective generalized multinormed spaces
  • Citing Article
  • September 2022

Journal of Mathematical Analysis and Applications

... Subsequently, several authors considered more general p-multi-normed spaces defined in terms of ℓ n p -spaces, 1 ⩽ p ⩽ ∞ (see [3]- [6]). A further generalization of this construction was due to Helemskii (see [7] and [8]), who considered L p (X, µ)-spaces for arbitrary measurable X as base spaces. This advance was made by using the 'noncoordinate' ('index-free') approach to the definition of the amplification of a normed space (for a systematic treatment of this approach for operator spaces, see [9]). ...

Multi-normed spaces, based on non-discrete measures, and their tensor products
  • Citing Article
  • June 2017

Izvestiya Mathematics

... This approach has been applied to quite a few categories in functional analysis, and not only for the metric projectivity, but for the extreme projectivity as well (in the latter case one needs to replace retracts by the so-called near-retracts). For application to operator spaces and modules see [7,8]; general matricially normed spaces are treated in [9]. In Section 2 of the present paper we introduce a reasonable notion of a free Lspace. ...

Projective and free matricially normed spaces
  • Citing Article
  • June 2017

... Subsequently, several authors considered more general p-multi-normed spaces defined in terms of ℓ n p -spaces, 1 ⩽ p ⩽ ∞ (see [3]- [6]). A further generalization of this construction was due to Helemskii (see [7] and [8]), who considered L p (X, µ)-spaces for arbitrary measurable X as base spaces. This advance was made by using the 'noncoordinate' ('index-free') approach to the definition of the amplification of a normed space (for a systematic treatment of this approach for operator spaces, see [9]). ...

Structures on the way from classical to quantum spaces and their tensor products
  • Citing Article
  • June 2017

... By L op we will denote the category of the right B(Ω) modules of the form L p (Ω, µ). In [6] Helemskii gave a complete characterisation of morphisms of L , but only for for locally compact Ω, with Borel σ-algebra. Careful inspection of his proof shows that this characterization valid for all σ-finite measure spaces. ...

Tensor products and multipliers of modules L p on locally compact measure spaces
  • Citing Article
  • September 2014

Mathematical Notes

... Чтобы сохранить единый стиль обозначений, мы будем называть метрически плоскими A-модули статьи [18], где они назывались экстремально плоскими. Через ⊗ A мы будем обозначать проективное модульное тензорное произведение банаховых модулей. ...

Metric version of atness and Hahn-Banach type theorems for normed modules over sequence algebras
  • Citing Article
  • January 2011

Studia Mathematica

... Первая работа по этой теме была опубликована в 1978 году Гравеном [4]. Позже эквивалентные определения были даны Уайтом [5] и Хелемски [6,7]. ...

Metric freeness and projectivity for classical and quantum normed modules
  • Citing Article
  • July 2013

Russian Academy of Sciences Sbornik Mathematics

... Christensen and Sinclair used this cohomology as a means for computing the (usual simplicial) cohomology 7i n (A, A*) of a C*-algebra A with coefficients in its dual bimodule A*, and they have shown [1] that the vanishing of H™{A**, A*) implies the vanishing of H n (A, A*). In [5] one of us has asked the question: is it possible to express the cohomology in terms of some suitable Ext? Now we give the desired expression in terms of an Ext for some Banach 7^-bimodules. This, in particular, enables us to prove that the space H*(7l, 7£») in fact coincides, up to a topological isomorphism, with H n (7l, 71*); thus we obtain a "simplicial version" of a well-known result of Johnson, Kadison and Ringrose [6] on the coincidence of the "usual" and the normal cohomology. ...

Expression of the Normal Cohomology in Terms of “Ext”, and Some Applications
  • Citing Article
  • January 1990

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