Topologically complete representations of inverse semigroups

A representation of an inverse semigroup by means of partial open homeomorphisms of a topological T₀-space is called topologically complete if the domains of these partial homeomorphisms form a base of the topology. It is shown how to construct topologically complete representations on the base of a ternary relation satisfying some elementary axioms. This result makes it possible to obtain a pseudo-elementary axiomatization for inverse semigroups that have faithful topologically complete representations in T₁, T₂ and T₃-spaces. A topology is introduced on any antigroup; this topology is a concomitant of the algebraic structure and every topologically complete representation is continuous with respect to this topology.     

Operator and matrix representation for the generalized inverse AT,S(2)A_{T,S}^{(2)}

This paper presents the matrix representation for extension of inverse of restriction of a linear operator to a subspace, on the basis of which we establish useful representations in operator and matrix form for the generalized inverse A_T,S⁽²⁾A_{T,S}^{(2)} and give some of their applications.     

On complete partially ordered sets and compatible topologies

We describe two complete partially ordered sets which are the intersection of complete linear orderings but which have no compatible Hausdorff topology. One is two-dimensional, while the second is countable, and leads to an example of a countable, compact, T ₁ space with a countable base which is not the continuous image of any compact Hausdorff space.     

Complete,Recursively Enumerable Relations in Arithmetic

Using only propositional connectives and the provability predicate of a Σ₁-sound theory T containing Peano Arithmetic we define recursively enumerable relations that are complete for specific natural classes of relations, as the class of all r. e. relations, and the class of all strict partial orders. We apply these results to give representations of these classes in T by means of formulas.     

Duality in Function Spaces

In this paper we use Bartle’s technique to study duality between a topological space and a function space. Normally such a duality forms an essential part of Functional Analysis. We introduce several new topologies such as the topology of even convergence T_e, the closed-cocompact topology T_{k ′}, the (strong) local proximal convergence. We explore the topological groups of self-homeomorphisms of a topological space and shed light on the earlier work of Arens, Dieudonné, Di Concilio. We also study the concepts such as evenly equidistant, functionally equicontinuous, due to Bouziad-Troallic and topologically equicontinuous due to Royden. In memory of Professor Enrico Meccariello who made a considerable contribution to this work and who suddenly passed away before his time     

Disjointness of representations arising in harmonic analysis on the infinite-dimensional unitary group

We prove the pairwise disjointness of representations T _z,w of the infinite-dimensional unitary group. These representations are a natural generalization of the regular representation to the “big” group U(∞). They were introduced and studied by G. Olshanski and A. Borodin. The disjointness of these representations reduces to that of certain probability measures on the space of paths in the Gelfand-Tsetlin graph. We prove the latter disjointness using probabilistic and combinatorial methods.     

Transversals in non-discrete groups

The concept of ‘topological right transversal’ is introduced to study right transversals in topological groups. Given any right quasigroupS with a Tychonoff topologyT, it is proved that there exists a Hausdorff topological group in whichS can be embedded algebraically and topologically as a right transversal of a subgroup (not necessarily closed). It is also proved that if a topological right transversal(S, T _S ,T ^S , o) is such thatT _S =T ^S is a locally compact Hausdorff topology onS, thenS can be embedded as a right transversal of a closed subgroup in a Hausdorff topological group which is universal in some sense.     

Any two irreducible Markov chains are finitarily orbit equivalent

Two invertible dynamical systems (X, gA, μ, T) and (Y, ℬ, ν, S), where X, Y are metrizable spaces and T, S are homeomorphisms on X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X ₀ of X of full measure to a subset Y ₀ of Y of full measure such that ϕ|x ₀ is continuous in the relative topology on X ₀, ϕ ⁻¹|Y ₀ is continuous in the relative topology on Y ₀ and ϕ(Orb _T (x)) = Orb _Sϕ (x) for μ-a.e. x ∈ X. In this article a finitary orbit equivalence mapping is shown to exist between any two irreducible Markov chains.     

Actions of GLq (2,C) on C(1,3) and its four dimensional representations

A complete classification is given of all inner actions on the Clifford algebra C(l,3) defined by representations of the quantum group GL_q (2,C)q^m ≠1, which are not reduced to representations of two commuting “q-spinors”. As a consequence of this classification it is shown that the space of invariants of every GL_q (2,C)-action of this type, which is not an action of SL_q (2,C), is generatedby 1 and the value of the quantum determinant for the given representation.     

Algebraic Topological Closure Operators

For any closure operator c there is a T_o-closure operator whose lattice of closed subsets are isomorphic to that of c. A correspondence between algebraic topological (T_o) closure operators on a nonempty set X and pre-orderes (partial orders) on X is established. Equivalent conditions are obtained for a T_o-lattice to be a complete atomic Boolean algebra and for the lattice of closed subsets of an algebraic topological closure operator to be a complete atomic Boolean algebra. Further it is proved that a complete lattice is an algebraic T_o-lattice if and only if it is isomorphic to the lattice of closed subsets of some algebraic topological closure operator on a suitable set.AMS Subject Classification (1991): 06A23, 54D65.     

Inactivity times of coherent systems with dependent components under periodical inspections

If we know that a coherent system has failed before a time t, the inactivity time is the period (from t) in which the system has been broken. If T is the system lifetime, the inactivity time at t is (t?T|T<t). Under periodical inspections, we may typically know that the system was working at a time t₁, but that is broken at another time t₂>t₁. Under this assumption, we obtain representations for the reliability function of the system inactivity time (t₂?T|t₁<T<t₂). We consider both the cases of independent and dependent components. Similar representations are obtained under other assumptions with partial information about component failures at times t₁ and t₂. These representations are used to compare stochastically the inactivity times under different assumptions. Some illustrative examples are provided.     

Reducible polar representations

In this paper the reducible polar representations of the compact connected Lie groups are classified. It turns out that there only exist “interesting” reducible polar representations of Lie groups of the types A ₃, A ₃×T ¹, B ₃, B ₃×T ¹, D ₄, D ₄×T ¹ and D ₄×A ₁. Up to equivalence, there is just one such representation of the first four Lie groups, there are three reducible polar representations of D ₄ and six of D ₄×T ¹ and D ₄×A ₁, respectively. From this follows immediately the classification of the compact connected subgroups of SO(n) which act transitively on products of spheres. Received: 28 April 2000     

Representations of Hardy Algebras: Absolute Continuity,Intertwiners, and Superharmonic Operators

Suppose T₊(E){\mathcal{T}_{+}(E)} is the tensor algebra of a W*-correspondence E and H ^∞(E) is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of T₊(E){\mathcal{T}_{+}(E)} on a Hilbert space to ultra-weakly continuous completely contractive representations of H ^∞(E) on the same Hilbert space. Our work extends the classical Sz.-Nagy–Foiaş functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu’s noncommutative disc algebra.     

Comparison of Some Classical Topologies on Hyperspaces within the Framework of Point Spaces

S. Dolecki, G. Greco and A. Lechicki call a space X consonant if the co-compact topology and the upper Kuratowski topology on the set of closed subsets of X coincide. We call a space X hyperconsonant if Fell's topology and the (Kuratowski) convergence topology coincide. Recently, we proved that a first countable, locally paracompact, T ₃-space is hyperconsonant if and only if the space possesses at most one point without a compact neighbourhood, extending the same result of D. Fremlin obtained for metrizable spaces. In this paper, we pursue the study of hyperconsonance within the framework of point spaces (countable T ₁-spaces with exactly one accumulation point) and we compare consonance and hyperconsonance in such spaces. In particular, we answer a question of T. Nogura and D. Shakhmatov: does there exist a nonconsonant point space? We provide a Fréchet, -point space which is not consonant. Moreover, this example proves that the consonance is not preserved by continuous closed compact-covering maps of separable complete metrizable spaces onto Hausdorff spaces.     

On determinantal representation for the generalized inverse A and its applications

In this paper, first we establish a determinantal representation for the group inverse A_g of a square matrix A. Based on this, a determinantal representation for the generalized inverse A is presented. As an application, we give a determinantal formula for the unique solution of the general restricted linear system: Ax=b(x ∈ T, b ∈ AT and dim(AT)=dim(T)), which reduces to the common Cramer rule if A is non‐singular. These results extend our earlier work. Copyright © 2006 John Wiley & Sons, Ltd.     

A theorem of Riesz type with Pettis integrals in topological vector spaces

Let X be a locally convex Hausdorff space and let C₀(S,X) be the space of all continuous functions f:S→X, with compact support on the locally compact space S. In this paper we prove a Riesz representation theorem for a class of bounded operators T:C₀(S,X)→X, where the representing integrals are X-valued Pettis integrals with respect to bounded signed measures on S. Under the additional assumption that X is a locally convex space, having the convex compactness property, or either, X is a locally convex space whose dual X^′ is a barrelled space for an appropriate topology, we obtain a complete identification between all X-valued Pettis integrals on S and the bounded operators T:C₀(S,X)→X they represent. Finally we give two illustrations of the representation theorem proved, in the particular case when X is the topological dual of a locally convex space.     

Exact calculability,semigroup of representations,and the stability property for representations of the algebra of functions on the quantum groupSU q(2)

An operation of the coproduct of representations of a bialgebra is defined. The coproduct operation of representations for the Hopf algebra of functions on the quantum groupSU _q (2) is investigated. For this Hopf algebra a representation II called the stable representation is constructed. This representation is invariant with respect to the coproduct with an arbitrary representation. An expression for the trace in the representation II is derived. The invariant Voronovich integal inSU _q (2) takes the form fdµ=tr(fcc ^*).V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 163–178, November, 1994.