Narrow Results

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation.

We consider, for which subgroup G′ of G, the restriction π|_G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p₁) × U(p₂) × U(q) for p₁, p₂ ≥ 1 with p₁ + p₂ = p is multiplicity-free, whereas the restriction to U(p₁) × U(p₂) × U(q₁) × U(q₂) for q₁, q₂ ≥ 1 with q₁ + q₂ = q has infinite multiplicities.

In this paper, we study the isospectrality problem for a free quantum particle confined in a ring with a junction, analyzing all the self-adjoint realizations of the corresponding Hamiltonian in terms of a boundary condition at the junction. In particular, by characterizing the energy spectrum in terms of a spectral function, we classify the self-adjoint realizations in two classes, identifying all the families of isospectral Hamiltonians. These two classes turn out to be discerned by the action of parity (i.e. space reflection), which plays a central role in our discussion.

We study hermitian forms and unitary groups defined over a local ring, not necessarily commutative, equipped with an involution. When the ring is finite we obtain formulae for the order of the unitary groups as well as their point stabilizers, and use these to compute the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring.

We recover the holomorphic discrete series representations of $SU(1,n)$ as well as some unitary irreducible representations of $SU(n+1)$ by deformation of a minimal realization of $sl(n+1,ℂ)$.

Let A be a unital C*-algebra, n ∈ N ∪ {∞}. It is proved that the isomorphism is isometric for some suitable distances. As an application, the author has the split exact sequence with i_A contractive (and isometric if n = ∞) under certain condition of A.

Let H_n(q) denote the Heisenberg group defined over the field with q elements. Using the geometry of the finite symplectic group, we describe all the ordinary irreducible characters of H_n(q). A special group of the same type appears as a subgroup of the finite unitary group whose ordinary irreducible characters are described using the geometry of the unitary group.

Let F be a field with charF ≠ 2 and |F| > 9, and let B_2n(F) be the standard Borel subgroup of the unitary group U_2n(F) over F. For n ≥ 3, we obtain a complete description of all bijective maps preserving commutators on B_2n(F).

The subgroups XU(2), YU(2), and ZU(2) of the unitary group U(2) allow to find twenty-four different decompositions of an arbitrary $2\times 2$ unitary matrix. Because the group YU(2) has not been considered before, 20 out of the 24 decompositions are new. Introducing this YU(2) group allows to highlight, within quantum circuit design, a symmetry similar to the threefold symmetry of the Pauli matrices of spin systems. Similar subgroups of the unitary group U($n$) lead to various new decompositions of an arbitrary $n\times n$ unitary matrix. Whereas for $n$ equal 2, this leads to the synthesis of single-qubit quantum gates, for $n$ equal a power of 2, this leads to new design methods for multiple-qubit circuits.

In this paper, we define a $\gamma$-factor for generic representations of $\mathrm{\text{U}}(1,1)\times {\mathrm{\text{Res}}}_{E/F}({\mathrm{\text{GL}}}_1)$ and prove a local converse theorem for $\mathrm{\text{U}}(1,1)$ using the $\gamma$-factor we defined. We also give a new proof of the local converse theorem for ${\mathrm{\text{GL}}}_2$ using a $\gamma$-factor of ${\mathrm{\text{GL}}}_2\times {\mathrm{\text{GL}}}_2$ type which was originally defined by Jacquet in [Automorphic Forms on$\mathrm{\text{GL}}(2)$, Part 2, Springer Lecture Notes in Mathematics, Vol. 278 (Springer, 1972)].

We study the homotopy type of the space of the unitary group ${\mathrm{\text{U}}}_1(C_u^{*}(\left|{\mathbb{Z}}^n\right|))$ of the uniform Roe algebra $C_u^{*}(|{\mathbb{Z}}^n|)$ of ${\mathbb{Z}}^n$. We show that the stabilizing map ${\mathrm{\text{U}}}_1(C_u^{*}(\left|{\mathbb{Z}}^n\right|))\to {\mathrm{\text{U}}}_{\infty }(C_u^{*}(\left|{\mathbb{Z}}^n\right|))$ is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy type of ${\mathrm{\text{U}}}_1(C_u^{*}(\left|{\mathbb{Z}}^n\right|))$, which is the product of the unitary group ${\mathrm{\text{U}}}_1(C^{*}(\left|{\mathbb{Z}}^n\right|))$ (having the homotopy type of ${\mathrm{\text{U}}}_{\infty }(ℂ)$ or $\mathbb{Z}\times B{\mathrm{\text{U}}}_{\infty }(ℂ)$ depending on the parity of $n$) of the Roe algebra $C^{*}(|{\mathbb{Z}}^n|)$ and rational Eilenberg–MacLane spaces.

We study properties of solutions of the operator equation , , where a closable linear operator on a Hilbert space , such that there exists a self-adjoint operator on , with the resolution of identity E(·), which commutes with . We are interested in the question of regular admissibility of the subspace , i.e. when for every there exists a unique (mild) solution u in of this equation. We introduce the notion of equation spectrum Σ associated with Eq. (*), and prove that if Λ ⊂ ℝ is a compact subset such that Λ ⋂ Σ = ∅, then is regularly admissible. If Λ ⊂ ℝ is an arbitrary Borel subset such that Λ ⋂ Σ = ∅, then, in general, needs not be regularly admissible, but we derive necessary and sufficient conditions, in terms of some inequalities, for the regular admissibility of . Our results are generalizations of the well-known spectral mapping theorem of Gearhart-Herbst-Howland-Prüss [4], [5], [6], [9], as well as of the recent results of Cioranescu-Lizama [3], Schüler [10] and Vu [11], [12].

We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups $\mathrm{\text{O(N)}}$, $\mathrm{\text{U(N)}}$ and $\mathrm{\text{Sp(2N)}}$. For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal and symplectic cases. Asymptotic expansions in strong and weak non-unitarity regimes are obtained. Using the connection to matrix hypergeometric functions, we establish limit theorems for the log-modulus of the characteristic polynomial evaluated on the unit circle.

We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n + 1 complex variables of fixed degrees will find all the roots of the system. We also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context. Further complexity theory is developed.

This article is a summary of our talk in QBIC2010. We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group in a Hilbert space with equipped with the strong operator topology. More precisely, for any strongly closed subgroup G of the unitary group in a finite von Neumann algebra , we show that the set of all generators of strongly continuous one-parameter subgroups of G forms a complete topological Lie algebra with respect to the strong resolvent topology. We also characterize the algebra of all densely defined closed operators affiliated with from the viewpoint of a tensor category.