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V. M. Manuilov

Publications and source records attributed to V. M. Manuilov.

9 recordsLinked to original sources
arXiv2021

Hilbert $C^*$-module independence

We introduce the notion of Hilbert $C^*$-module independence: Let $\mathscr{A}$ be a unital $C^*$-algebra and let $\mathscr{E}_i\subseteq \mathscr{E},\,\,i=1, 2$, be ternary subspaces of a Hilbert $\mathscr{A}$-module $\mathscr{E}$. Then $\mathscr{E}_1$ and $\mathscr{E}_2$ are said to be Hilbert $C^*$-module independent if there are positive constants $m$ and $M$ such that for every state $φ_i$ on $\langle \mathscr{E}_i,\mathscr{E}_i\rangle,\,\,i=1, 2$, there exists a state $φ$ on $\mathscr{A}$ such that \begin{align*} mφ_i(|x|)\leq φ(|x|) \leq Mφ_i(|x|^2)^{\frac{1}{2}},\qquad \mbox{for all~}x\in \mathscr{E}_i, i=1, 2. \end{align*} We show that it is a natural generalization of the notion of $C^*$-independence of $C^*$-algebras. Moreover, we demonstrate that even in case of $C^*$-algebras this concept of independence is new and has a nice characterization in terms of extensions. This enriches the theory of independence of $C^*$-algebras. We show that if $\langle \mathscr{E}_1,\mathscr{E}_1\rangle $ has the quasi extension property and $z\in \mathscr{E}_1\cap \mathscr{E}_2$ with $\|z\|=1$, then $|z|=1$. Several characterizations of Hilbert $C^*$-module independence and a new characterization of $C^*$-independence are given. One of characterizations states that if $z_0\in \mathscr{E}_1\cap \mathscr{E}_2$ is such that $\langle z_0,z_0\rangle=1$, then $\mathscr{E}_1$ and $\mathscr{E}_2$ are Hilbert $C^*$-module independent if and only if $\|\langle x,z_0\rangle\langle y,z_0\rangle\|=\|\langle x,z_0\rangle\|\,\|\langle y,z_0\rangle\|$ for all $x\in \mathscr{E}_1$ and $y\in \mathscr{E}_2$. We also provide some technical examples and counterexamples to illustrate our results.

math.OA↗
arXiv2021

Left multipliers of reproducing kernel Hilbert $C^*$-modules and the Papadakis theorem

We give a modified definition of a reproducing kernel Hilbert $C^*$-module (shortly, $RKHC^*M$) without using the condition of self-duality and discuss some related aspects; in particular, an interpolation theorem is presented. We investigate the exterior tensor product of $RKHC^*M$s and find their reproducing kernel. In addition, we deal with left multipliers of $RKHC^*M$s. Under some mild conditions, it is shown that one can make a new $RKHC^*M$ via a left multiplier. Moreover, we introduce the Berezin transform of an operator in the context of $RKHC^*M$s and construct a unital subalgebra of the unital $C^*$-algebra consisting of adjointable maps on an $RKHC^*M$ and show that it is closed with respect to a certain topology. Finally, the Papadakis theorem is extended to the setting of $RKHC^*M$, and in order for the multiplication of two specific functions to be in the Papadakis $RKHC^*M$, some conditions are explored.

math.OA↗
arXiv2019

Extensions of the Lax-Milgram theorem to Hilbert C*-modules

We present three versions of the Lax-Milgram theorem in the framework of Hilbert C*-modules, two for those over W*-algebras and one for those over C*-algebras of compact operators. It is remarkable that while the Riesz theorem is not valid for certain Hilbert C*-modules over C*-algebras of compact operators, our Lax-Milgram theorem turns out to be valid for all of them. We also give several examples to illustrate our results, in particular, we show that the main theorem is not true for Hilbert modules over arbitrary C*-algebras.

math.OA↗
arXiv1997

Relations between asymptotic and Fredholm representations

We prove that for matrix algebras $M_n$ there exists a monomorphism $(\prod_n M_n/\oplus_n M_n)\otimes C(S^1) \to {\cal Q} $ into the Calkin algebra which induces an isomorphism of the $K_1$-groups. As a consequence we show that every vector bundle over a classifying space $Bπ$ which can be obtained from an asymptotic representation of a discrete group $π$ can be obtained also from a representation of the group $π\times Z$ into the Calkin algebra. We give also a generalization of the notion of Fredholm representation and show that asymptotic representations can be viewed as asymptotic Fredholm representations.

funct-an↗
arXiv1996

Diagonalizing operators over continuous fields of C*-algebras

It is well known that in the commutative case, i.e. for $A=C(X)$ being a commutative C*-algebra, compact selfadjoint operators acting on the Hilbert C*-module $H_A$ (= continuous families of such operators $K(x)$, $x\in X$) can be diagonalized if we pass to a bigger W*-algebra $L^\infty(X)={\bf A} \supset A$ which can be obtained from $A$ by completing it with respect to the weak topology. Unlike the "eigenvectors", which have coordinates from $\bf A$, the "eigenvalues" are continuous, i.e. lie in the C*-algebra $A$. We discuss here the non-commutative analog of this well-known fact. Here the "eigenvalues" are defined not uniquely but in some cases they can also be taken from the initial C*-algebra instead of the bigger W*-algebra. We prove here that such is the case for some continuous fields of real rank zero C*-algebras over a one-dimensional manifold and give an example of a C*-algebra $A$ for which the "eigenvalues" cannot be chosen from $A$, i.e. are discontinuous. The main point of the proof is connected with a problem on almost commuting operators. We prove that for some C*-algebras if $h\in A$ is a selfadjoint, $u\in A$ is a unitary and if the norm of their commutant $[u,h]$ is small enough then one can connect $u$ with the unity by a path $u(t)$ so that the norm of $[u(t),h]$ would be also small along this path.

funct-an↗
arXiv1996

Diagonalization of compact operators in Hilbert modules over C*-algebras of real rank zero

It is known that the classical Hilbert--Schmidt theorem can be generalized to the case of compact operators in Hilbert $A$-modules $H_A^*$ over a $W^*$-algebra of finite type, i.e. compact operators in $H_A^*$ under slight restrictions can be diagonalized over $A$. We show that if $B$ is a weakly dense $C^*$-subalgebra of real rank zero in $A$ with some additional property then the natural extension of a compact operator from $H_B$ to $H_A^*\supset H_B$ can be diagonalized with diagonal entries being from the $C^*$-algebra $B$.

funct-an↗
arXiv1995

Lusin's C-property is not valid for functional Hilbert modules

We show that elements of Hilbert $A$-module obtained by completion of the space of square-integrable functions on a space with measure $X$ taking values in a $C^*$-algebra $A$ cannot be viewed as $A$-valued functions on $X$ defined almost everywhere

funct-an↗
arXiv1995

Diagonalization of compact operators in Hilbert modules over finite W*-algebras

It is known that a continuous family of compact operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators in Hilbert modules over a commutative W*-algebra. The aim of the present paper is to generalize this fact for a finite W*-algebra $A$ not necessarily commutative. We prove that for a compact operator $K$ acting in the right Hilbert $A$-module $H^*_A$ dual to $H_A$ under slight restrictions one can find a set of "eigenvectors" $x_i\in H^*_A$ and a non-increasing sequence of "eigenvalues" $λ_i\in A$ such that $K\,x_i = x_i\,λ_i$ and the autodual Hilbert $A$-module generated by these "eigenvectors" is the whole $H_A^*$. As an application we consider the Schrödinger operator in magnetic field with irrational magnetic flow as an operator acting in a Hilbert module over the irrational rotation algebra $A_θ$ and discuss the possibility of its diagonalization.

funct-an↗