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Vladimir Manuilov

Publications and source records attributed to Vladimir Manuilov.

At least 19 recordsLinked to original sources
arXiv2026

On ideals in the semilattice of coarse equivalence classes of metrics

For a Hausdorff topology on the set of ideals of the semilattice $M(X)$ of coarse equivalence classes of metrics on a set $X$, the space $I(M(X))$ of ideals is the closure of the set of principal ideals, thus allowing to view non-principal ideals as generalizations of coarse equivalence classes of metrics. Some ideals arise from coarse structures on $X$. We define a map $Φ$ from $I(M(X))$ to the set $CS(X)$ of coarse structures on $X$, and a map $Ψ$ backwards, and show that $Ψ\circΦ$ is the identity map, thus allowing to identify coarse structures with some ideals of $M(X)$. We show that there are ideals that do not come from $CS(X)$. For any ideal $F$ we define the generalized uniform Roe algebra as the direct limit $C^*$-algebra of the uniform Roe algebras for the equivalence classes of metrics in the ideal, and show that it coincides with the uniform Roe algebra of $Φ(F)$.

math.MG↗
arXiv2026

A tautological continuous field of Roe bimodules

We generalize the notion of a continuous field of C*-algebras to that of Hilbert C*-bimodules. Given a partially ordered set $P$ and a monotonically non-decreasing family of ternary rings of operators (TROs) assigned to the points of $P$, we equip $P$ with a certain zero-dimensional Hausdorff topology and use a certain compactification $γP$ to get the base space for a continuous field of Hilbert C*-bimodules over $γP$. As a motivating example, we consider the set $D(X,Y)$ of coarse equivalence classes of metrics on the disjoint union of two metric spaces, $X$ and $Y$. Each such class gives rise to a uniform Roe bimodule, a TRO linking the uniform Roe algebras of $X$ and $Y$. The resulting family of TROs is non-decreasing with respect to the natural partial order on $D(X,Y)$ and thus yields a tautological continuous field of Hilbert C*-bimodules over $γD(X,Y)$.

math.OA↗
arXiv2023

On Hilbert C*-modules with Hilbert dual and C*-Fredholm operators

We study such Hilbert C*-modules over a C*-algebra $A$, that the Banach $A$-dual module carries a natural structure of Hilbert $A$-module. In this direction we prove that if $A$ is monotone complete, $M$ and $N$ are Hilbert $A$-modules, $M$ is self-dual, and both $T:M\to N$ and its Banach $A$-dual $T':N'\to M'$ have trivial kernels and cokernels then $M\cong N'$. With the help of this result, for a monotone complete $C^*$-algebra $A$, we prove that the index of any $A$-Fredholm operator can be calculated as the difference of its kernel and cokernel, as in the Hilbert space case.

math.OA↗
arXiv2021

On the inverse semigroup of bimodules over a C*-algebra

It was noticed recently that, given a metric space $(X,d_X)$, the equivalence classes of metrics on the disjoint union of the two copies of $X$ coinciding with $d_X$ on each copy form an inverse semigroup $M(X)$ with respect to concatenation of metrics. Now put this inverse semigroup construction in a more general context, namely, we define, for a C*-algebra $A$, an inverse semigroup $S(A)$ of Hilbert C*-$A$-$A$-bimodules. When $A$ is the uniform Roe algebra $C^*_u(X)$ of a metric space $X$, we construct a map $M(X)\to S(C^*_u(X))$ and show that this map is injective, but not surjective in general. This allows to define an analog of the inverse semigroup $M(X)$ that does not depend on the choice of a metric on $X$ within its coarse equivalence class.

math.OA↗
arXiv2020

Metrics on doubles as an inverse semigroup

For a metric space $X$ we study metrics on the two copies of $X$. We define composition of such metrics and show that the equivalence classes of metrics are a semigroup $M(X)$ Our main result is that $M(X)$ is an inverse semigroup, therefore, one can define the $C^*$-algebra of this inverse semigroup. We characterize the metrics that are idempotents, find a minimal projection in $M(X)$ and give examples of metric spaces, for which the semigroup $M(X)$ is commutative. We show that if the Gromov-Hausdorff distance between two metric spaces, $X$ and $Y$, is finite then $M(X)$ and $M(Y)$ are isomorphic. We also describe the class of metrics determined by subsets of $X$ in terms of the closures of the subsets in the Higson corona of $X$.

math.MG↗
arXiv2020

Metrics on doubles as an inverse semigroup II

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 give several descriptions of the set $E(M(X))$ of idempotents of this inverse semigroup and its Stone dual space $\widehat X$. We also construct $σ$-additive measures on $\widehat X$ from finitely additive probability measures on $X$ that vanish on bounded subsets.

math.MG↗
arXiv2020

B-spline interpolation problem in Hilbert C*-modules

We introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued sesquilinear form on a Hilbert $C^*$-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert $C^*$-module is self-dual. Extending a bounded $C^*$-valued sesquilinear form on a Hilbert $C^*$-module to a sesquilinear form on its second dual, we then provide some necessary and sufficient conditions for the $B$-spline interpolation problem to have a solution. Passing to the setting of Hilbert $W^*$-modules, we present our main result by characterizing when the spline interpolation problem for the extended $C^*$-valued sesquilinear to the dual $\mathscr{X}'$ of the Hilbert $W^*$-module $\mathscr{X}$ has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert $W^*$-module $\mathscr{X}$ is orthogonally complemented with respect to another $C^*$-inner product on $\mathscr{X}$. Finally, solutions of the $B$-spline interpolation problem for Hilbert $C^*$-modules over $C^*$-ideals of $W^*$-algebras are extensively discussed. Several examples are provided to illustrate the existence or lack of a solution for the problem.

math.OA↗
arXiv2020

On Kuiper type theorems for uniform Roe algebras

Generalizing the case of an infinite discrete metric space of finite diameter, we say that a discrete metric space $(X,d)$ is a Kuiper space, if the group of invertible elements of its uniform Roe algebra is norm-contractible. Various sufficient conditions on $(X,d)$ to be or not to be a Kuiper space are obtained.

math.OA↗
arXiv2019

Douglas factorization theorem revisited

Inspired by the Douglas lemma, we investigate the solvability of the operator equation $AX=C$ in the framework of Hilbert C*-modules. Utilizing partial isometries, we present its general solution when $A$ is a semi-regular operator. For such an operator $A$, we show that the equation $AX=C$ has a positive solution if and only if the range inclusion ${\mathcal R}(C) \subseteq {\mathcal R}(A)$ holds and $CC^*\le t\, CA^*$ for some $t>0$. In addition, we deal with the solvability of the operator equation $(P+Q)^{1/2}X=P$, where $P$ and $Q$ are projections. We provide a counterexample to show that there exists a $C^*$-algebra $\mathfrak{A}$, a Hilbert $\mathfrak{A}$-module $\mathscr{H}$ and projections $P$ and $Q$ on $\mathscr{H}$ such that the operator equation $(P+Q)^{1/2}X=P$ has no solution. Moreover, we give a perturbation result related to the latter equation.

math.OA↗
arXiv2019

Hypergraph partitions

We suggest a reduction of the combinatorial problem of hypergraph partitioning to a continuous optimization problem.

cs.DS↗
arXiv2018

On the C*-algebra of matrix-finite bounded operators

Let $H$ be a separable Hilbert space with a fixed orthonormal basis. Let $\mathbb B^{(k)}(H)$ denote the set of operators, whose matrices have no more than $k$ non-zero entries in each line and in each column. The closure of the union (over $k\in\mathbb N$) of $\mathbb B^{(k)}(H)$ is a C*-algebra. We study some properties of this C*-algebra. We show that this C*-algebra is not an AW*-algebra, has a proper closed ideal greater than compact operators, and its group of invertibles is contractible.

math.OA↗
arXiv2018

Approximately uniformly locally finite graphs

Let $Γ$ be a locally finite graph, $L$ the normalized Laplacian of $Γ$. If $Γ$ is uniformy locally finite, i.e. if each vertex has no more than $d$ adjacent vertices, then the matrix of $L$ (with respect to the standard basis) has no more than $d+1$ non-zero entries in each row and in each column. We consider the class of locally finite graphs, for which the Laplacian can be approximated by matrices of this type with arbitrary $d$. We provide examples of locally finite graphs which are or are not in this class, and show that the graphs from this class share certain regularity property: vertices of high degree cannot have too many adjacent vertices of low degree.

math.CO↗
arXiv2018

On a family of representations of residually finite groups

For a residually finite group $G$, its normal subgroups $G\supset G_1\supset G_2\cdots$ with $\cap_{n\in\mathbb N}G_n=\{e\}$ and for a growth function $γ$ we construct a unitary representation $π_γ$ of $G$. For the minimal growth, $π_γ$ is weakly equivalent to the regular representation, and for the maximal growth it is weakly equivalent to the direct sum of the quasiregular representations on the quotients $G/G_n$. In the case of intermediate growth we show two examples of different behaviour of $π_γ$.

math.GR↗
arXiv2017

A more symmetric picture for Kasparov's KK-bifunctor

For C*-algebras $A$ and $B$, we generalize the notion of a quasihomomorphism from $A$ to $B$, due to Cuntz, by considering quasihomomorphisms from some C*-algebra $C$ to $B$ such that $C$ surjects onto $A$, and the two maps forming a quasihomomorphism agree on the kernel of this surjection. Under an additional assumption, the group of homotopy classes of such generalized quasihomomorphisms coincides with $KK(A,B)$. This makes the definition of Kasparov's bifunctor slightly more symmetric and gives more flexibility for constructing elements of $KK$-groups. These generalized quasihomomorphisms can be viewed as pairs of maps directly from $A$ (instead of various $C$'s), but these maps need not be $*$-homomorphisms.

math.OA↗
arXiv2017

A KK-like picture for E-theory of C*-algebras

Let $A$, $B$ be separable C*-algebras, $B$ stable. Elements of the E-theory group $E(A,B)$ are represented by asymptotic homomorphisms from the second suspension of $A$ to $B$. Our aim is to represent these elements by (families of) maps from $A$ itself to $B$. We have to pay for that by allowing these maps to be even further from $*$-homomorphisms. We prove that $E(A,B)$ can be represented by pairs $(φ^+,φ^-)$ of maps from $A$ to $B$ that are not necessarily asymptotic homomorphisms, but have the same deficiency from being ones. Not surprisingly, such pairs of maps can be viewed as pairs of asymptotic homomorphisms from some C*-algebra $C$ that surjects onto $A$, and the two maps in a pair should agree on the kernel of this surjection. We give examples of full surjections $C\to A$, i.e. those, for which all classes in $E(A,B)$ can be obtained from pairs of asymptotic homomorphisms from $C$.

math.OA↗
arXiv2017

A noncommutative version of Farber's topological complexity

Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative C*-algebras. Topological complexity for spaces is closely related to the Lusternik--Schnirelmann category, for which we do not know any noncommutative extension, so there is no hope to generalize the known estimation methods, but we are able to evaluate the topological complexity for some very simple examples of noncommutative C*-algebras.

math.OA↗