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Let A be a C∗-algebra with a faithful state ϕ. It is proved thatthe projective unitary group P UA of A,P UA = UA/T.1,(UA denotes the unitary group of A) is a C∞-submanifold of the Banach spaceBs(A) of bounded operators acting in A, which are symmetric for the ϕ-innerproduct, and are usually called symmetrizable linear operators in A.A quotient Finsler metric is introduced in P UA, following the theory ofhomogeneous spaces of the unitary group of a C∗-algebra. Curves of minimallength with any given initial conditions are exhibited. Also it is proved that ifA is a von Neumann algebra (or more generally, an algebra where the unitarygroup is exponential) two elements in P UA can be joined by a minimal curve.In the case when A is a von Neumann algebra with a finite trace, theseminimality results hold for the quotient of the metric induced by the p-normof the trace (p ≥ 2), which metrizes the strong operator topology of P UA. Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina

1. Introduction. The group U(H) of unitary transformations of a complex Hilbert space H, endowed with its strong operator topology, is of interest in the study of unitary representations of a topological group. The unitary transformations of H induce a group U(Ĥ) of transformations of the associated projective space Ĥ. The projective unitary group U(Ĥ) with its strong operator topology is used in the study of projective (ray) representations. U(Ĥ) is, as a group, the quotient of U(H) by the subgroup S1 of scalar multiples of the identity. In this paper we prove that the strong operator toplogy of U(Ĥ) is in fact the quotient of the strong operator topology on U(H). This is related to the fact that U(H) is a principal bundle over U(Ĥ) with fibre S.

The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d .

We present in this paper a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction., Final version to appear in Discrete & Computational Geometry

Let G be a finite group. Let X 1(G) be the first column of the ordinary character table of G. We will show that if X 1(G) = X1(PGU3(q 2)), then G ≅ PGU3(q 2). As a consequence, we show that the projective general unitary groups PGU3(q 2) are uniquely determined by the structure of their complex group algebras.

We consider the action of p-toral subgroups of U(n) on the unitary partition complex Ln. We show that if H ⊆ U(n) has noncontractible fixed points on Ln, then the image of H in the projective unitary group U(n)/S 1 is an elementary abelian p- group.

We define the semi-classical quantum automatic complexity \(Q_{s}(x)\) of a word x as the infimum in lexicographic order of those pairs of nonnegative integers (n, q) such that there is a subgroup G of the projective unitary group \({{\mathrm{PU}}}(n)\) with \(|G|\le q\) and with \(U_0,U_1\in G\) such that, in terms of a standard basis \(\{e_k\}\) and with \(U_z=\prod _k U_{z(k)}\), we have \(U_x e_1=e_2\) and \(U_y e_1 \ne e_2\) for all \(y\ne x\) with \(|y|=|x|\). We show that \(Q_s\) is unbounded and not constant for strings of a given length. In particular, $$\begin{aligned} Q_{s}(0^21^2)\le (2,12) < (3,1) \le Q_{s}(0^{60}1^{60}) \end{aligned}$$ and \(Q_s(0^{120})\le (2,121)\).

It is well known that a Gabor representation on L 2 ( R d ) L^{2}(\mathbb {R}^{d}) admits a frame generator h ∈ L 2 ( R d ) h\in L^{2}(\mathbb {R}^{d}) if and only if the associated lattice satisfies the Beurling density condition, which in turn can be characterized as the “trace condition” for the associated von Neumann algebra. It happens that this trace condition is also necessary for any projective unitary representation of a countable group to admit a frame vector. However, it is no longer sufficient for general representations, and in particular not sufficient for Gabor representations when they are restricted to proper time-frequency invariant subspaces. In this short note we show that the condition is also sufficient for a large class of projective unitary representations, which implies that the Gabor density theorem is valid for subspace representations in the case of irrational types of lattices.

Two combinatorial characterizations of the Hermitian surface of the finite 3-dimensional projective space PG ( 3 , q 2 ) in terms of quasi-Hermitian varieties are given.

In this paper we determine the structure of all so-called generalized isometries of the special unitary group which are transformations that respect any member of a large collection of generalized distance measures.

Many applications of finite tight frames (e.g., the use of SICs and mutually unbiased bases (MUBs) in quantum information theory and harmonic frames for the analysis of signals subject to erasures) depend only on the vectors up to projective unitary equivalence. It is well known that two finite sequences of vectors in inner product spaces are unitarily equivalent if and only if their respective inner products (Gramian matrices) are equal. Here we present a corresponding result for the projective unitary equivalence of two sequences of vectors (lines) in inner product spaces, i.e., that a finite number of (Bargmann) projective (unitary) invariants are equal. This result is equivalent to finding a rank-one matrix completion of a certain matrix. We give an algorithm to recover the sequence of vectors (up to projective unitary equivalence) from a small subset of these projective invariants and apply it to SICs, MUBs, and harmonic frames. We also extend our results to the projective similarity of vectors.

We study the question how quickly products of a fixed conjugacy class in the projective unitary group of a II${}_1$-factor von Neumann algebra cover the entire group. Our result is that the number of factors that are needed is essentially as small as permitted by the $1$-norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups. As an application of the techniques, we prove that every homomorphism from the projective unitary group of a II${}_1$-factor to a polish SIN group is continuous. Moreover, we show that the projective unitary group of a II${}_1$-factor carries a unique polish group topology., Comment: v2 minor changes, 42 pages

We study the mod p p cohomology of the classifying space of the projective unitary group P U ( p ) PU(p) . We first prove that conjectures due to J.F. Adams and Kono and Yagita (1993) about the structure of the mod p p cohomology of the classifying space of connected compact Lie groups hold in the case of P U ( p ) PU(p) . Finally, we prove that the classifying space of the projective unitary group P U ( p ) PU(p) is determined by its mod p p cohomology as an unstable algebra over the Steenrod algebra for p > 3 p>3 , completing previous work by Dwyer, Miller and Wilkerson (1992) and Broto and Viruel (1998) for the cases p = 2 , 3 p=2,3 .

In the paper, the automorphism group of the first subconstituent of the unitary graph , when over a finite field has been determined. Moreover, we have , where is the stabilizer group of the group of automorphisms of unitary graphs fixing .

Matrices U in unitary group U4 that satisfy the implication ∀Aℱℋ4 → B =U*AU ∈ ℱℋ4 are examined. Here, ℱℋ4 is a set of order four (T+H)-matrices. Such matrices U can be identified with unitary automorphisms of the space ℱℋ4.

A procedure of calculation of two-valued space group representations and two-valued projective representations of point groups is considered. A method of construction of factor systems w2(r2, r1), which reflect the transformations of half-integer spin quantum wave functions and are required in order to find the two-valued irreducible projective representations of the point groups, is presented. This method is based on the introduction of an operation q, firstly used by Bethe, as an additional symmetry element. The pathway of introducing the relations, which permit to make a one-valued algebra of double groups and, particularly, their multiplication tables, is shown by the examples of the 222 (D2) and 32 (D3) groups. The construction of a standard factor-system w′(1)(r2, r1) of the projective class K1 for the group 222 on the base of the discussed relations is presented for the first time. The whole role and the possibilities of Bethe’s method and its modifications for the construction of two-valued representations of the point and space groups are discussed., Розглянуто методику побудови двозначних представлень просторових та двозначних проективних представлень точкових груп. Представлено метод побудови фактор-систем w2(r2, r1), якi вiдображають перетворення хвильових функцiй квантових систем з напiвцiлим спiном, i якi є необхiдними для знаходження двозначних незвiдних проективних представлень точкових груп. Цей метод ґрунтується на введеннi в ролi додаткового елемента симетрiї операцiї q, вперше використаної Бете. На прикладi груп 222 (D2) та 32 (D3) показано, яким чином вводяться спiввiдношення, що дозволяють зробити однозначними алгебру подвiйних груп та, зокрема, їх таблицi множення. Показано, яким чином на основi спiввiдношень, що обговорюються, будується вперше представлена для групи 222 стандартна фактор-система класу K1 – фактор-система w′(1)(r2, r1). Обговорюються також в цiлому роль та можливостi методу Бете та його модифiкацiй в побудовi двозначних представлень точкових та просторових груп.

We prove that a periodic group, saturated with projective linear groups of dimension 2 over finite fields, is isomorphic to the projective linear group of dimension 2 over a locally finite field.

We introduce the notion of the projective biquandle (an object related to links in projective space). The paper is devoted to the proof that for any link in projective space the number of admissible colorings by projective biquandle of its diagram is invariant.

For a finite group we introduce a particular central extension, the unitary cover, having minimal exponent among those satisfying the projective lifting property. We obtain new bounds for the exponent of the Schur multiplier relating to subnormal series, and we discover new families for which the bound is the exponent of the group. Finally, we show that unitary covers are controlled by the Zel'manov solution of the restricted Burnside problem for 2-generator groups., Comment: In memory of David Chillag. The author is indebted to his PhD menthors Prof. Eli Aljadeff (Technion) and Dr. Yuval Ginosar (University of Haifa). A preliminary version has been presented at the conference "Groups St Andrews 2013", University of St Andrews, Scotland. Slides available at

In this paper, we construct a character theory for projective representations of finite groups. Consequently, we compute the number of distinct irreducible projective representations (up to isomorphism) of a finite group with a given associated Schur multiplier and deduce properties on the degrees of such projective representations.

We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM- and POVM-dynamical (quantum) entropies. They quantify the randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator. We study the class of chaotic unitaries, i.e., the ones of maximal entropy or, equivalently, such that they can be represented by suitably rescaled complex Hadamard matrices in some orthonormal bases. We provide necessary conditions for a unitary operator to be chaotic, which become also sufficient for qubits and qutrits. These conditions are expressed in terms of the relation between the trace and the determinant of the operator. We also compute the volume of the set of chaotic unitaries in dimensions two and three, and the average PVM-dynamical entropy over the unitary group in dimension two. We prove that this mean value behaves as the logarithm of the dimension of the Hilbert space, which implies that the probability that the dynamical entropy of a unitary is almost as large as possible approaches unity as the dimension tends to infinity., 10 double column pages, 4 figures, accepted for publication in IEEE Transactions on Information Theory

Let $\mathbf{B}PU_{n}$ be the classifying space of $PU_n$, the projective unitary group of order $n$, for $n>1$. We use the Serre spectral sequence associated to a fiber sequence $\mathbf{B}U_n\rightarrow\mathbf{B}PU_n\rightarrow K(\mathbb{Z},3)$ to determine the ring structure of $H^{*}(\mathbf{B}PU_{n}; \mathbb{Z})$ up to degree $10$, as well as a family of distinguished elements of $H^{2p+2}(\mathbf{B}PU_{n}; \mathbb{Z})$, for each prime divisor $p$ of $n$. We also study the primitive elements of $H^*(\mathbf{B}U_n;\mathbb{Z})$ as a comodule over $H^*(K(\mathbb{Z},2);\mathbb{Z})$, where the comodule structure is given by an action of $K(\mathbb{Z},2)\simeq\mathbf{B}S^1$ on $BU_n$ corresponding to the action of taking the tensor product of a complex line bundle and an $n$ dimensional complex vector bundle., 43 pages, 3 figures. Published version plus an appendix. Accepted by the Journal of Topology and Analysis

The topological classification of gerbes, as principal bundles with the structure group the projective unitary group of a complex Hilbert space, over a topological space $H$ is given by the third cohomology $\text{H}^3(H, \Bbb Z)$. When $H$ is a topological group the integral cohomology is often related to a locally continuous (or in the case of a Lie group, locally smooth) third group cohomology of $H$. We shall study in more detail this relation in the case of a group extension $1\to N \to G \to H \to 1$ when the gerbe is defined by an abelian extension $1\to A \to \hat N \to N \to 1$ of $N$. In particular, when $\text{H}_s^1(N,A)$ vanishes we shall construct a transgression map $\text{H}^2_s(N, A) \to \text{H}^3_s(H, A^N)$, where $A^N$ is the subgroup of $N$-invariants in $A$ and the subscript $s$ denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper., in J. Geom. Phys. (2016)

In this paper, we study the essential dimension of classes of central simple algebras with involutions of index less or equal to 4. Using structural theorems for simple algebras with involutions, we obtain the essential dimension of projective and symplectic groups of small degree.

A complex projective tower, or simply a ℂP-tower, is an iterated complex projective fibration starting from a point. In this paper we classify all six-dimensional ℂP-towers up to diffeomorphism, and as a consequence we show that all such manifolds are cohomologically rigid, i.e., they are completely determined up to diffeomorphism by their cohomology rings.

The renormalization group dynamics is studied in the four-component fermionic hierarchical model in the space of coefficients that determine the Grassmann-valued density of the free measure. This space is treated as a two-dimensional projective space. If the renormalization group parameter is greater than 1, then the only attracting fixed point of the renormalization group transformation is defined by the density of the Grassmann δ-function. Two different invariant neighborhoods of this fixed point are described, and an algorithm is constructed that allows one to classify the points on the plane according to the way they tend to the fixed point.

Let be the set of Toeplitz matrices. We describe the matrices in the unitary group such that

Let Γ be a finite abelian group and let H be an infinite-dimensional separable complex Hilbert space. We prove that the set of realizable by an action unitary representations of Γ on H is comeager in the space of all unitary representations of Γ on H .

We describe a construction method of transitive 1-designs from a finite group. We use this method to construct 2-designs and strongly regular graphs on conjugacy classes of maximal and second maximal subgroups under the action of the unitary group U(3, 3), or its maximal subgroups. Keywords: block design, strongly regular graph, unitary group, transitive group

In this paper, all the binary codes from the primitive permutation group P S U 2 ( 16 ) are constructed. It is shown that the groups P S U 2 ( 16 ) : 4 and S 17 are the automorphism groups of the constructed codes.

We characterize geodesic hyperspheres of radius r such that cot2(r)=\frac{1}{2} as the unique real hypersurfaces in complex projective space whose structure Jacobi operator satisfies a pair of conditions.