CyberSec Research

Let $\Gamma$ be a discrete subgroup of a unimodular locally compact group $G$. In Math. Ann. 388, 4251-4305 (2024), it was shown that the $L_p$ norm of a Fourier multiplier $m$ on $\Gamma$ can be bounded locally by its $L_p$-norm on $G$, modulo a constant $c(A)$ which depends on the support $A$ of $m$. In the context where $G$ is a connected Lie group with Lie algebra $\mathfrak{g}$, we develop tools to find explicit bounds on $c(A)$. We show that the problem reduces to: 1) The adjoint representation of the semisimple quotient $\mathfrak{s} = \mathfrak{g}/\mathfrak{r}$ of $\mathfrak{g}$ by the radical $\mathfrak{r}$ of $\mathfrak{g}$ (which was handled in the paper mentioned above). 2) The action of $\mathfrak{s}$ on a set of real irreducible representations that arise from quotients of the commutator series of $\mathfrak{r}$. In particular, we show that $c(G) = 1$ for unimodular connected solvable Lie groups.

A quasi-projection pair consists of two operators $P$ and $Q$ acting on a Hilbert $C^*$-module $H$, where $P$ is a projection and $Q$ is an idempotent satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ denotes the adjoint operator of $Q$, and $I$ is the identity operator on $H$. Such a pair is said to be harmonious if both $P(I-Q)$ and $(I-P)Q$ admit polar decompositions. The primary goal of this paper is to present the block matrix representations for a harmonious quasi-projection pair $(P,Q)$ on a Hilbert $C^*$-module, and additionally to derive new block matrix representations for the matched projection, the range projection, and the null space projection of $Q$. Several applications of these newly obtained block matrix representations are also explored.

We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, we use this information on optimal solutions to give an analytical proof of the triangle inequality for the induced quantum Wasserstein divergences.

Given a locally compact second countable group $G$ with a 2-cocycle $\omega$, we show that the restriction of the twisted Plancherel weight $\varphi^\omega_G$ to the subalgebra generated by a closed subgroup $H$ in the twisted group von Neumann algebra $L_\omega(G)$ is semifinite if and only if $H$ is open. When $G$ is almost unimodular, i.e. $\ker\Delta_G$ is open, we show that $L_\omega(G)$ can be represented as a cocycle action of the $\Delta_G(G)$ on $L_\omega(\ker\Delta_G)$ and the basic construction of the inclusion $L_\omega(\ker\Delta_G)\leq L_\omega(G)$ can be realized as a twisted group von Neumann algebra of $\Delta_G(G)\hat{\ } \times G$, where $\Delta_G$ is the modular function. Furthermore, when $G$ has a sufficiently large non-unimodular part, we give a characterization of $L_\omega(G)$ being a factor and provide a formula for the modular spectrum of $L_\omega(G)$.

We study the representation theory of various convolution algebras attached to the $q$-deformation of $\mathrm{SL}(2,\mathbb{R})$ from an algebraic perspective and beyond the unitary case. We show that many aspects of the classical representation theory of real semisimple groups can be transposed to this context. In particular, we prove an analogue of the Harish-Chandra isomorphism and we introduce an analogue of parabolic induction. We use these tools to to classify the (non-unitary) irreducible admissible representations of $q$-deformed $\mathrm{SL}(2,\mathbb{R})$. Moreover, we explicitly show how these irreducible representations converge to the classical admissible dual of $\mathrm{SL}(2,\mathbb{R})$. For that purpose, we define a version of the quantized universal enveloping algebra defined over the ring of analytic functions on $\mathbb{R}_+^*$, which specializes at $q = 1$ to the enveloping $\ast$-algebra of $\mathfrak{sl}(2,\mathbb{R})$ .

In this paper, we employ operator system techniques to investigate structural properties of C*-algebras. In particular, we provide more direct proofs of results concerning exactness and the local lifting property (LLP) of group C$^*$-algebras that avoid relying on the traditional heavy machinery of C$^*$-algebra theory. Briefly, these methods allow us to deduce that any C*-algebra containing $n$ unitaries whose $\textit{ joint numerical radius}$, in the sense defined by \cite{FKP}, is strictly less than $n$, must fail certain of these properties.

We aim to give a self-contained and detailed yet simplified account of the foundations of the theory of double operator integrals, in order to provide an accessible entry point to the theory. We make two new contributions to these foundations: (1) a new proof of the existence of the product of two projection-valued measures, which allows for the definition of the double operator integral for Hilbert-Schmidt operators, and (2) a variant approach to the integral projective tensor product on arbitrary (not necessarily separable) Hilbert spaces using a somewhat more explicit norm than has previously been given. We prove the Daletskii-Krein formula for strongly differentiable perturbations of a densely-defined self-adjoint operator and conclude by reviewing an application of the theory to quantum statistical mechanics.

Under mild assumptions, we prove that reduced free products and reduced graph products of $\mathrm{C}^\ast$-algebras are selfless in the sense of L. Robert, without assuming the rapid decay property. In particular, our main theorems yield numerous new examples of simple, monotracial $\mathrm{C}^\ast$-algebras with strict comparison, stable rank one, and admitting a unique unital embedding of the Jiang-Su algebra $\mathcal{Z}$ up to approximate unitary equivalence.

Inspired by the classical Bohman-Korovkin-Wulbert (BKW) operators, we initiate a study of noncommutative BKW-operators. Let $A$ be a unital $C^*$-algebra, and $S$ be a set of generators of $A$. A unital completely positive (UCP)-map $\phi: A\rightarrow B(H)$ is said to be a \textit{noncommutative BKW-operator} for $S$ with respect to norm or weak operator topology (WOT) or strong operator topology (SOT) if for any sequence of UCP-maps $\phi_n:A\rightarrow B(H)$, $n=1,2,...,$ $\lim_{n\rightarrow \infty}\phi_n(s)=\phi(s),\forall ~s\in S$ in norm (or WOT or SOT) $\Rightarrow \lim_{n\rightarrow \infty}\phi_n(a)=\phi(a), \forall ~a\in A$ in norm (or WOT or SOT, respectively). We identify a connection between noncommutative BKW-operators and the unique CP-extension of UCP-maps. We have discussed several examples and explored different notions of noncommutative BKW-operators and their interconnections. Additionally, we introduce the concept of hyperrigidity with respect to a UCP-map and characterize it along the lines of Arveson. Although independent yet related to noncommutative BKW-operators, we provide a noncommutative version of operator version of the Korovkin theorem recently proposed by D. Popa.

The injective envelope of a $C^*$-algebra, introduced by Hamana in his foundational 1979 papers, has become a central tool in noncommutative analysis. For a compact Hausdorff $X$, injectivity on the commutative side amounts to extremal disconnectedness: $I(C(X))\cong C(K)$ with $K$ extremally disconnected. Gleason's classical construction \cite{Gleason} of $K$ is intricate; B{\l}aszczyk \cite{Blaszczyk} later gave a strikingly concise route: first \emph{maximise} the regular topology under an irreducibility constraint, then \emph{compactify} to obtain $G(X)$. Indeed, in B{\l}aszczyk argument, maximality is the driver and extremal disconnectedness is the consequence.\smallskip Our aim is to transpose this to the noncommutative setting. The multiplier algebra $M(E)$ is the natural analogue of the \v{C}ech--Stone compactification: $M(C_0(Y))\cong C(\beta Y)$. We introduce \emph{$A$-multiplier covers} $(E,\iota)$ and a rigidity notion paralleling Hamana's. The punchline is that a~\emph{maximal rigid} cover forces $M(E)$ to be a rigid essential extension of $A$, hence identifies canonically with $I(A)$.

We investigate when a map on a selfadjoint operator space $E$ is an embedding, i.e., when its unitisation in the sense of Werner is completely isometric. Combining with results of Russell, of Ng, and of Dessi, the second and the last author, it is shown that this is equivalent to: (a) extending bounded positive functionals on each matrix level with the same norm; (b) extending quasistates to quasistates in each matrix level; (c) extending completely bounded completely positive maps with the same cb-norm; and (d) the map being a gauge maximal isometry in the sense of Russell. If $E$ is approximately positively generated and $\mathrm{C}^*(E)$ is unital, or if $E_{sa}$ is singly generated, then completely positive maps on $E\subseteq\mathcal{B}(H)$ have completely positive extensions on $\mathrm{C}^*(E)$, but possibly not with the same cb-norm; and this is not enough for the inclusion $E \subseteq \mathrm{C}^*(E)$ to be an embedding. We show that the inclusion $E \subseteq \mathrm{C}^*(E)$ is always an embedding when $E$ is completely approximately 1-generated, and we fully resolve the case when $E_{sa}$ is singly generated. Combining with the works of Salomon, Humeniuk--Kennedy--Manor, and previous work of the third author, we show that if the inclusion $E \subseteq \mathrm{C}^*(E)$ is an embedding, then rigidity at zero, in the sense of Salomon, coincides with $E$ being approximately positively generated. Consequently, we show that $E$ is approximately positively generated if and only if $M_n(E)$ is approximately positively generated for all $n\in \mathbb{N}$, thus extending a previous result of Humeniuk--Kennedy--Manor to the approximation setting. As an application we show that hyperrigidity of $E$ in $\mathrm{C}^*(E)$ allows to identify $\mathrm{C}^*(E)$ as the C*-envelope of $E$ in several (non-unital) contexts.

It is shown that every linear strong Birkhoff-James isomorphism between unital $C^*$-algebras is a $*$-isomorphism followed by a unitary multiplication. Moreover, as a partial extension of this result to the non-unital case, the form of (possibly nonlinear) strong Birkhoff-James isomorphisms between compact $C^*$-algebras are determined. A nonlinear characterization of compact $C^*$-algebras in terms of strong Birkhoff-James orthogonality is also given.

We propose a novel method for deforming the algebra of smooth functions on a compact Riemannian manifold based on spectral data rather than group actions or Poisson structures. The deformation is defined through spectral coefficients of the Laplacian and a weight function satisfying a fusion-type cocycle condition, producing a noncommutative product that depends intrinsically on Laplacian spectrum of the manifold. We develop the analytic and algebraic foundations of this construction, establishing associativity, continuity, and the existence of a group structure on admissible weights. We show that the construction is functorial with respect to spectrum-preserving maps of manifolds. This spectral-geometric deformation quantization approach provides a fully intrinsic analytic model of noncommutative geometry based solely on spectral data of the manifold.

A recent development in random matrix theory, the intrinsic freeness principle, establishes that the spectrum of very general random matrices behaves as that of an associated free operator. This reduces the study of such random matrices to the deterministic problem of computing spectral statistics of the free operator. In the self-adjoint case, the spectral edges of the free operator can be computed exactly by means of a variational formula due to Lehner. In this note, we provide variational formulas for the largest and smallest singular values in the non-self-adjoint case.

We give some natural conditions on actions of discrete countable groups on abelian locally compact groups of Lie type that imply factoriality of the group von Neumann algebras of their semidirect products. This allows us to give a fairly large class of examples of locally compact groups whose group von Neumann algebras are factors.

Bessel duality of regular Gabor systems states that a Gabor system over a lattice is a Bessel sequence if and only if the corresponding Gabor system over the adjoint lattice is a Bessel sequence. We show that this fundamental result of time-frequency analysis can be deduced from a theorem in the theory of bimodules over von Neumann algebras, namely that under certain conditions, their left and right bounded vectors coincide.

In this paper I study properties of the generators $\triangle_\gamma$ of non-local Dirichlet forms $\mathcal{E}^\mu_\gamma$ on ultrametric spaces which are the path space of simple stationary Bratteli diagrams. The measures used to define the Dirichlet forms are taken to be the Gibbs measures $\mu_\psi$ associated to H\"older continuous potentials $\psi$ for one-sided shifts. I also define a cohomology $H_{lc}(X_B)$ for $X_B$ which can be seen as dual to the homology of Bowen and Franks. Besides studying spectral properties of $\triangle_\gamma$, I show that for $\gamma$ large enough (with sharp bounds depending on the diagram and the measure theoretic entropy $h_{\mu_\psi}$ of $\mu_\psi$) there is a unique harmonic representative of any class $c\in H_{lc}(X_B)$.

We show that Mumford's Approximately Macroscopically Unique (AMU) states exist for quantum systems consisting of unbounded self-adjoint operators when the commutators are small. In particular, AMU states always exist in position and momentum systems when the Planck constant $|\hbar|$ is sufficiently small. However, we show that these standard quantum mechanical systems are far away from classical mechanical (commutative) systems even when $|\hbar|\to 0.$