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V. Manuilov

Publications and source records attributed to V. Manuilov.

At least 19 recordsLinked to original sources
arXiv2025

Paths in graphs: bounded geometry and property A

We expose a class of discrete metric spaces, for which bounded geometry is equivalent to the property A of G. Yu. This class includes the coarse disjoint union of $(\mathbb Z/2\mathbb Z)^n$, $n\in\mathbb N$, and consists of spaces of simple paths in a class of graphs that includes cactus graphs, with the metric defined as the number of edges in the symmetric difference of the paths. We also show that if a space in this class does not have bounded geometry then it contains a subspace of bounded geometry without property A.

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arXiv2025

Bounded geometry version of property A

For uniformly dicrete metric spaces without bounded geometry we suggest a modified version of property A based on metrics of bounded geometry greater than the given metric. We show that this version still implies coarse embeddability in Hilbert spaces, and that some examples of non-property A spaces of unbounded geometry satisfy this version. We also relate this version of property A to our version of uniform Roe algebras for spaces without bounded geometry and introduce an appropriate equivalence relation.

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arXiv2025

A continuous field of Roe algebras

Let $X$ be a metric measure space. A Delone subset $D\subset X$ is a uniformly discrete set coarsely equivalent to $X$. We consider the space $\mathcal D_F$ of controlled Delone subsets of $X$ with an appropriate metric, and show that it, together with $X$ itself, is a compact space. By assigning to each point $D$ of $\mathcal D_F$ (resp., to $X$) the uniform Roe algebra $C^*_u(D)$ (resp., the \u Spakula's version $C_k^*(X)$ of the Roe algebra of $X$) we get a tautological family of $C^*$-algebras. For a sequence $\{D_n\}_{n\in\mathbb N}$ of controlled Delone subsets convergent to $X$ we show that the corresponding uniform Roe algebras $C^*_u(D_n)$, together with $C^*_k(X)$, form a continuous field of $C^*$-algebras over $\mathbb N\cup\{\infty\}$ when $X$ is a proper metric measure space of bounded geometry with no isolated points.

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arXiv2024

On large submodules in Hilbert C*-modules

We consider several natural ways of expressing the idea that a one-sided ideal in a C*-algebra (or a submodule in a Hilbert C*-module) is large, and show that they differ, unlike the case of two-sided ideals in C*-algebras. We then show how these different notions, for ideals and for submodules, are related. We also study some permanence properties for these notions. Finally, we use essential right ideals to extend the inner product on a Hilbert C*-module to a part of the dual module.

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arXiv2024

Restricting operators to thick Hilbert C*-submodules

Given an essential ideal $J\subset A$ of a C*-algebra $A$, and a Hilbert C*-module $M$ over $A$, we place $M$ between two other Hilbert C*-modules over $A$, $M_J\subset M\subset M^J$, in such a way that each submodule here is thick, i.e. its orthogonal conmplement in the greater module is trivial. We introduce the class $\mathbb B_J(M)$ of $J$-adjointable operators on a Hilbert C*-module $M$ over $A$, and prove that this class isometrically embeds into the C*-algebras of all adjointable operators both of $M_J$ and of $M^J$.

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arXiv2024

Mapping graph homology to $K$-theory of Roe algebras

Given a graph $Γ$, one may conside the set $X$ of its vertices as a metric space by assuming that all edges have length one. We consider two versions of homology theory of $Γ$ and their $K$-theory counterparts -- the $K$-theory of the (uniform) Roe algebra of the metric space $X$ of vertices of $Γ$. We construct here a natural map from homology of $Γ$ to the $K$-theory of the Roe algebra of $X$, and its uniform version. We show that, when $Γ$ is the Cayley graph of $\mathbb Z$, the constructed maps are isomorphisms.

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arXiv2023

An example of a continuous field of Roe algebras

The Roe algebra $C^*(X)$ is a non-commutative $C^*$-algebra reflecting metric properties of a space $X$, and it is interesting to understand relation between the Roe algebra of $X$ and the (uniform) Roe algebra of its discretization. Here we do a minor step in this direction in the simplest non-trivial example $X=\mathbb R$ by constructing a continuous field of $C^*$-algebras over $[0,1]$ with the fibers over non-zero points the uniform $C^*$-algebra of the integers, and the fiber over 0 a $C^*$-algebra related to $\mathbb R$.

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arXiv2023

On topological obstructions to the existence of non-periodic Wannier bases

Recently, M. Ludewig and G. C. Thiang introduced a notion of a uniformly localized Wannier basis with localization centers in an arbitrary uniformly discrete subset $D$ in a complete Riemannian manifold $X$. They show that, under certain geometric conditions on $X$, the class of the orthogonal projection onto the span of such a Wannier basis in the $K$-theory of the Roe algebra $C^*(X)$ is trivial. In this paper, we clarify the geometric conditions on $X$, which guarantee triviality of the $K$-theory class of any Wannier projection. We show that this property is equivalent to triviality of the unit of the uniform Roe algebra of $D$ in the $K$-theory of its Roe algebra, and provide a geometric criterion for that. As a consequence, we prove triviality of the $K$-theory class of any Wannier projection on a connected proper measure space $X$ of bounded geometry with a uniformly discrete set of localization centers.

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arXiv2022

Inverse semigroups of metrics on doubles related to certain subsets

Recently we have shown that the equivalence classes of metrics on the double of a metric space $X$ form an inverse semigroup. Here we define an inverse subsemigroup related to a family of isometric subspaces of $X$, which is more computable. As a special case, we study this subsemigroup related to the family of geodesic rays starting from the basepoint, for Euclidean spaces and for trees.

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arXiv2022

On extendability of functionals on Hilbert $C^*$-modules

Let $M\subset N$ be Hilbert $C^*$-modules over a $C^*$-algebra $A$ with $M^\perp=0$. It was shown recently by J. Kaad and M. Skeide that there exists a non-zero $A$-valued functional on $N$ such that its restriction onto $M$ is zero. Here we show that this may happen even if $A$ is monotone complete. On the other hand, we show that for certain type I $W^*$-algebras this cannot happen.

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arXiv2022

On Hochschild homology of uniform Roe algebras with coefficients in uniform Roe bimodules

It was shown recently by M. Lorentz and R. Willett that all bounded derivations of the uniform Roe algebras of metric spaces of bounded geometry are inner. Here we calculate the space of outer derivations of the uniform Roe algebras with coefficients in uniform Roe bimodules related to various metrics on the two copies of the given space. We also give some results on the higher Hochschild cohomology with coefficients in uniform Roe bimodules.

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arXiv2021

Hilbert C*-modules related to discrete metric spaces

It is shown that the metric on the union of the sets $X$ and $Y$ defines a Hilbert $C^*$-module over the uniform Roe algebra of the space $X$ with a fixed metric $d_X$. A number of examples of such Hilbert $C^*$-modules are described.

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arXiv2021

Inverse semigroup from metrics on doubles III. Commutativity and (in)finiteness of idempotents

We have shown recently that, given a metric space $X$, the coarse equivalence classes of metrics on the two copies of $X$ form an inverse semigroup $M(X)$. Here we study the property of idempotents in $M(X)$ of being finite or infinite, which is similar to this property for projections in C*-algebras. We show that if $X$ is a free group then the unit of $M(X)$ is infinite, while if $X$ is a free abelian group then it is finite. As a by-product, we show that the inverse semigroup $M(X)$ is not a quasi-isometry invariant. More examples of finite and infinite idempotents are provided. We also give a geometric description of spaces, for which their inverse semigroup $M(X)$ is commutative.

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arXiv2019

Roe bimodules as morphisms of discrete metric spaces

For two discrete metric spaces, $X$ and $Y$ we consider metrics on $X\sqcup Y$ compatible with the metrics on $X$ and $Y$. As morphisms from $X$ to $Y$ we consider the Roe bimodules, i.e. the norm closures of bounded finite propagation operators from $l^2(X)$ to $l^2(Y)$. We study the corresponding category $\mathcal M$, which is also a 2-category. We show that almost isometries determine morphisms in $\mathcal M$. We also consider the case $Y=X$, when there is a richer algebraic structure on the set of morphisms of $\mathcal M$: it is a partially ordered semigroup with the neutral element, with involution, and with a lot of idempotents. We also give a condition when a morphism is a $C^*$-algebra.

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arXiv2016

A finitedimensional version of Fredholm representations

We consider pairs of maps from a discrete group to the unitary group. The deficiencies of these maps from being homomorphisms may be great, but if they are close to each other then we call such pairs balanced. We show that balanced pairs determine elements in the K-theory group of the classifying space of the discrete group. We also show that a Fredholm representation determines balanced pairs.

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arXiv2010

A reflexivity criterion for Hilbert C*-modules over commutative C*-algebras

A C*-algebra $A$ is C*-reflexive if any countably generated Hilbert C*-module $M$ over $A$ is C*-reflexive, i.e. the second dual module $M''$ coincides with $M$. We show that a commutative C*-algebra $A$ is C*-reflexive if and only if for any sequence $I_k$ of disjoint non-zero C*-subalgebras, the canonical inclusion $\oplus_k I_k\subset A$ doesn't extend to an inclusion of $\prod_k I_k$.

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arXiv2009

Hilbert C*-modules from group actions: beyond the finite orbits case

Continuous actions of topological groups on compact Hausdorff spaces $X$ are investigated which induce almost periodic functions in the corresponding commutative C*-algebra. The unique invariant mean on the group resulting from averaging allows to derive a C*-valued inner product and a Hilbert C*-module which serve as an environment to describe characteristics of the group action. For uniformly continuous, Lyapunov stable actions the derived invariant mean $M(ϕ_x)$ is continuous on $X$ for any element $ϕ\in C(X)$, and the induced C*-valued inner product corresponds to a conditional expectation from $C(X)$ onto the fixed point algebra of the action defined by averaging on orbits. In the case of selfduality of the Hilbert C*-module all orbits are shown to have the same cardinality. Stable actions on compact metric spaces give rise to C*-reflexive Hilbert C*-modules. The same is true if the cardinality of finite orbits is uniformly bounded and the number of closures of infinite orbits is finite. A number of examples illustrate typical situations appearing beyond the classified cases.

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