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We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group $U(\mathcal{H})$ in a Hilbert space $\mathcal{H}$ with $U(\mathcal{H})$ equipped with the strong operator…

We prove that if $G$ and $H$ are $\alpha$-back-and-forth equivalent groups (in the sense of computable structure theory) for some ordinal $\alpha \geq \omega$, then their group von Neumann algebras $L(G)$ and $L(H)$ are also…

Given a second-order, holomorphic, linear differential equation $Lf=0$ on a punctured Riemann surface, we say that its monodromy group $G\subset\operatorname{GL}(2,\mathbb{C})$ is `unitary' if it preserves a non-degenerate Hermitian form…

Let $G$ be a locally compact group, $L(G)$ be its group von Neumann algebra equipped with the Plancherel weight $\varphi_G$. In this paper, we consider the following two questions. (1) When is the restriction of $\varphi_G$ to the…

Let ${\mathcal B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 3$, and let $\mathcal A$ and $\mathcal B$ be subsets of ${\mathcal B}(H)$ which contain all rank one operators.…

We show that the unitary group of any SOT-separable $\mathrm{II}_1$ factor $M$, with the strong operator topology, is contractible. Combined with several old results, this implies that the same is true for any SOT-separable von Neumann…

Let $G$ be a discrete countable group, and let $\Gamma$ be an almost normal subgroup. In this paper we investigate the classification of (projective) unitary representations $\pi$ of $G$ into the unitary group of the Hilbert space…

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group $\mathbb{G}$ and 1-injectivity of…

In this paper we suggest an approach for constructing an L1-type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we intro- duce a seminorm, and prove that it is a norm if and only if the…

A classical theorem of von Neumann asserts that every unbounded self-adjoint operator $A$ in a separable Hilbert space $H$ is unitarily equivalent to an operator $B$ in $H$ such that $D(A)\cap D(B)=\{0\}$. Equivalently this can be…

A unital $C^*$-algebra is called $N$-subhomogeneous if its irreducible representations are finite dimensional with dimension at most $N$. We extend this notion to operator systems, replacing irreducible representations by boundary…

A hereditarily atomic von Neumann algebra $A$ is a $W^*$ product of matrix algebras, regarded as the underlying function algebra of a quantum set. Projections in $A\overline{\otimes}A^{\circ}$ are interpreted as quantum binary relations on…

This paper addresses a conjecture of Kadison and Kastler that a von Neumann algebra M on a Hilbert space H should be unitarily equivalent to each sufficiently close von Neumann algebra N and, moreover, the implementing unitary can be chosen…

We present several operator and norm inequalities for Hilbert space operators. In particular, we prove that if $A_{1},A_{2},...,A_{n}\in {\mathbb B}({\mathscr H})$, then…

Suppose $\mathcal{A}$ is a compact normal operator on a Hilbert space $H$ with certain lacunarity condition on the spectrum (which means, in particular, that its eigenvalues go to zero exponentially fast), and let $\mathcal{L}$ be its rank…

Let $\A$ be a unital separable nuclear $C^*$--algebra which belongs to the bootstrap category $\N$ and $\B$ be a separable stable $C^*$--algebra. In this paper, we consider the group $\Ext_u(\A,\B)$ consisting of the unitary equivalence…

Let ${\mathcal O} \subset {\mathbb R}^d$ be a bounded domain with the boundary of class $C^{1,1}$. In $L_2({\mathcal O};{\mathbb C}^n)$, a matrix elliptic second order differential operator ${\mathcal A}_{N,\varepsilon}$ with the Neumann…

Let $u_1,\ldots,u_n$ be unitary operators on a Hilbert space $H$. We study the norm $$\left\|\sum^{i=n}_{i=1} u_i \otimes \bar u_i\right\|\leqno (1)$$ of the operator $\sum u_i \otimes \bar u_i$ acting on the Hilbertian tensor product…

Let $A$ be a unital operator algebra. Let us assume that every {\it bounded\/} unital homomorphism $u\colon \ A\to B(H)$ is similar to a {\it contractive\/} one. Let $\text{\rm Sim}(u) = \inf\{\|S\|\, \|S^{-1}\|\}$ where the infimum runs…

Given an irreducible lattice $\Gamma$ in the product of higher rank simple Lie groups, we prove a co-finiteness result for the $\Gamma$-invariant von Neumann subalgebras of the group von Neumann algebra $\mathcal{L}(\Gamma)$, and for the…