Nonlinear Preserver Problems on B（H）

Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤* B by A*A = A*B and AA* = AB*. Then (B(H), “≤*”) is a partially ordered set and the relation “≤*” is called the star order on B(H). Denote by Bs(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B _s (H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.     

Quantum Spaces Associated to Multipermutation Solutions of Level Two

We study finite set-theoretic solutions (X,r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over ℂ with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra A(\mathbbC,X,r)\mathcal{A}(\mathbb{C},X,r) having a q-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of the finite abelian group G\mathcal{G} of left actions on X. We study the structure of A(\mathbbC,X,r)\mathcal{A}(\mathbb{C},X,r) and show that they have a ∙-product form ‘quantizing’ the commutative algebra of polynomials in |X| variables. We obtain the ∙-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite product for a certain crossed G\mathcal{G}-module (over any field k). We provide first steps in the noncommutative differential geometry of A(k,X,r)\mathcal{A}(k,X,r) arising from these results. As a byproduct of our work we find that every such level 2 solution (X,r) factorises as r = f ∘ τ ∘ f ^− 1 where τ is the flip map and (X,f) is another solution coming from X as a crossed G\mathcal{G}-set.     

Piecewise Hereditary Nakayama Algebras

Let k be an algebraically closed field. Let Λ be the path algebra over k of the linearly oriented quiver \mathbb A_n\mathbb A_n for n ≥ 3. For r ≥ 2 and n > r we consider the finite dimensional k −algebra Λ(n,r) which is defined as the quotient algebra of Λ by the two sided ideal generated by all paths of length r. We will determine for which pairs (n,r) the algebra Λ(n,r) is piecewise hereditary, so the bounded derived category D ^b (Λ(n,r)) is equivalent to the bounded derived category of a hereditary abelian category H\mathcal H as triangulated category.     

The Lattice of Compatible Quasiorders of Acyclic Monounary Algebras

Acyclic monounary algebras are characterized by the property that any compatible partial order r can be extended to a compatible linear order. In the case of rooted monounary algebras A=(A,f){\cal A}=(A,f) we characterize the intersection of compatible linear extensions of r by several equivalent conditions and generalize these results to compatible quasiorders of A{\cal A}. We show that the lattice QuordA{\rm{Quord}}{\cal A} of compatible quasiorders is a disjoint union of semi-intervals whose maximal elements equal the intersection of their compatible quasilinear extensions. We also investigate algebraic properties of the lattices QuordA{\rm{Quord}}{\cal A} and ConA{\rm{Con}}{\cal A}.     

Algebras whose equivalence relations are congruences

It is proved that all the equivalence relations of a universal algebra A are its congruences if and only if either |A| ≤ 2 or every operation f of the signature is a constant (i.e., f(a ₁ , . . . , a _n ) = c for some c ∈ A and all the a ₁ , . . . , a _n ∈ A) or a projection (i.e., f(a ₁ , . . . , a _n ) = a _i for some i and all the a ₁ , . . . , a _n ∈ A). All the equivalence relations of a groupoid G are its right congruences if and only if either |G| ≤ 2 or every element a ∈ G is a right unit or a generalized right zero (i.e., x _a  = y _a for all x, y ∈ G). All the equivalence relations of a semigroup S are right congruences if and only if either |S| ≤ 2 or S can be represented as S = A∪B, where A is an inflation of a right zero semigroup, and B is the empty set or a left zero semigroup, and ab = a, ba = a ² for a ∈ A, b ∈ B. If G is a groupoid of 4 or more elements and all the equivalence relations of it are right or left congruences, then either all the equivalence relations of the groupoid G are left congruences, or all of them are right congruences. A similar assertion for semigroups is valid without the restriction on the number of elements.     

Geometric computation of the numerical radius of a matrix

We use methods of geometric computing combined with hermitean matrix eigenvalue/eigenvector evaluations to find the numerical radius w(A) of a real or complex square matrix A simply, quickly, and accurately. The numerical radius w(A) is defined as the maximal distance of points in the field of values F(A) = { x^* A x | ||x||₂ = 1 }F(A) = \{ x^* A x \mid \|x\|_2 = 1 \} from zero in ℂ. Its value is an indicator of the transient behavior of the discrete dynamical system f _k + 1 = Af _k . We describe and test a MATLAB code for solving this optimization problem that can have multiple and even infinitely many solutions with maximal distance.     

On Defining Ideals or Subrings of Hall Algebras

Let A be a finitary algebra over a finite field k, and A- \textmod\text{mod} the category of finite dimensional left A-modules. Let H(A)\mathcal{H}(A) be the corresponding Hall algebra, and for a positive integer r let D _r (A) be the subspace of H(A)\mathcal{H}(A) which has a basis consisting of isomorphism classes of modules in A- \textmod\text{mod} with at least r + 1 indecomposable direct summands. If A is the path algebra of the quiver of type A _n with linear orientation, then D _r (A) is known to be the kernel of the map from the twisted Hall algebra to the quantized Schur algebra indexed by n + 1 and r. For any A, we determine necessary and sufficient conditions for D _r (A) to be an ideal and some conditions for D _r (A) to be a subring of H(A)\mathcal{H}(A). For A the path algebra of a quiver, we also determine necessary and sufficient conditions for D _r (A) to be a subring of H(A)\mathcal{H}(A).     

Unitarily covariant maps in approximately finite-dimensional <Emphasis Type="Italic">C</Emphasis>*-algebras

We consider maps defined on a real space Asa of all self-adjoint elements of a C*-algebra A commuting with the conjugation by unitaries: F(u* au) = u* F(a)u for any a ∈ A _sa, u ∈ (A). In the case where A is a full matrix algebra, there is a functional realization of these maps (in terms of multivariable functions) and analytical properties of these maps can be expressed in terms of corresponding functions. In the present work, these results are generalized to the class of uniformly hyperfinite C*-algebras and to the algebra of all compact operators in a Hilbert space. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 213–227, 2007.     

Sur l'extension dans le domaine complémentaire d'une variété

Résumé étant donnés une variétéM, un ensemble compactN enM, des entiersr, n avec 1≤r≤n, un espace euclidienE àn dimensions, uner-variétéA enE et une transformation continuef: N→E−A, alors c'est un problème bien connu s'il existe une extensionf′ def satisfaisantf′(M)∈E−A. Ce problème est discuté.

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Riassunto SianoM una varietà,N un insieme compatto inM, 1≤r≤n degli interi,E uno spazio euclideo adn dimensioni,A unar-varietà inE edf:N→E−A una trasformazione continua. Allora è trattato il problema ben noto dell'esistenza di una estensionef′ dif conf′ (M)∈E−A.

     

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G = GL _N or SL _N as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2  ^N : Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H ^*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H ⁱ (G, M) are Noetherian A ^G -modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.     

A distortion theorem for the class of typically real functions

The paper continues the studies of the well-known class T of typically real functions f(z) in the disk U = {z:|z| < 1}. The region of values of the system {f(z ₀), f(z ₀), f(r ₁), f(r ₂),…, f(r _n )} in the class T is investigated. Here, z ₀ ∈ U, Im z ₀ ≠ 0, 0 < r _j  < 1 for j = 1,…, n, n ≥ 2. As a corollary, the region of values of f′(z ₀) in the class of functions f ∈ T with fixed values f(z ₀) and f(r _j ) (j = 1,…, n) is determined. The proof is based on the criterion of solvability of the power problem of moments. Bibliography: 10 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 33–45.     

Maximal extensions of partial orders preserving antimonotonicity

Let (A, ?? _r ) be a partially ordered set and f : A ?? A be an order reversing (antimonotone) map. We characterize the maximal partial order extensions of r preserving the antimonotonicity of f.     

On ideal extensions of partial monounary algebras

In the present paper we introduce the notion of an ideal of a partial monounary algebra. Further, for an ideal (I, f _I ) of a partial monounary algebra (A, f _A ) we define the quotient partial monounary algebra (A, f _A )/(I, f _I ). Let (X, f _X ), (Y, f _Y ) be partial monounary algebras. We describe all partial monounary algebras (P, f _P ) such that (X, f _X ) is an ideal of (P, f _P ) and (P, f _P )/(X, f _X ) is isomorphic to (Y, f _Y ). This work was supported by the Slovak VEGA Grant No. 1/3003/06 and by the Science and Technology Assistance Agency under the contract No. APVT-20-004104.     

On the structure of inner ideals of nest algebras

IfA is a nest algebra andA _s=A ∩ A* , whereA* is the set of the adjoints of the operators lying inA, then the pair (A, A _s) forms a partial Jordan *-triple. Important tools when investigating the structure of a partial Jordan *-triple are its tripotents. In particular, given an orthogonal family of tripotents of the partial Jordan *-triple (A, A _s), the nest algebraA splits into a direct sum of subspaces known as the Peirce decomposition relative to that family. In this paper, the Peirce decomposition relative to an orthogonal family of minimal tripotents is used to investigate the structure of the inner ideals of (A, A _s), whereA is a nest algebra associated with an atomic nest. A property enjoyed by inner ideals of the partial Jordan *-triple (A, A _s) is presented as the main theorem. This result is then applied in the final part of the paper to provide examples of inner ideals. A characterization of the minimal tripotents as a certain class of rank one operators is also obtained as a means to deduce the principal theorem.     

Inversion of matrices over a pseudocomplemented lattice

We compute the greatest solutions of systems of linear equations over a lattice (P, ≤). We also present some applications of the results obtained to lattice matrix theory. Let (P, ≤) be a pseudocomplemented lattice with and and let A = ‖a _ij ‖ _n×n , where a _ij ∈ P for i, j = 1,..., n. Let A* = ‖a _ij ^′ ‖ _n×n and for i, j = 1,..., n, where a* is the pseudocomplement of a ∈ P in (P, ≤). A matrix A has a right inverse over (P, ≤) if and only if A · A* = E over (P, ≤). If A has a right inverse over (P, ≤), then A* is the greatest right inverse of A over (P, ≤). The matrix A has a right inverse over (P, ≤) if and only if A is a column orthogonal over (P, ≤). The matrix D = A · A* is the greatest diagonal such that A is a left divisor of D over (P, ≤). Invertible matrices over a distributive lattice (P, ≤) form the general linear group GL _n (P, ≤) under multiplication. Let (P, ≤) be a finite distributive lattice and let k be the number of components of the covering graph Γ(join(P,≤) − , ≤), where join(P, ≤) is the set of join irreducible elements of (P, ≤). Then GL _a (P, ≤) ≅ = S _n ^k . We give some further results concerning inversion of matrices over a pseudocomplemented lattice. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 139–154, 2005.     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

Polynomially dependent homomorphisms and frobenius n-homomorphisms

We introduce and study polynomially dependent homomorphisms, which are special linear maps between associative algebras with identity. The multiplicative structure is much involved in the definition of such homomorphisms (we consider only the case of maps f: A → B with commutative B). The most important particular case of these maps are the Frobenius n-homomorphisms, which were introduced by V.M. Buchstaber and E.G. Rees in 1996–1997. A 1-homomorphism f: A → B is just an algebra homomorphism (the algebra B is commutative). A typical example of an n-homomorphism is given by the sum of n algebra homomorphisms, f = f ₁ + ... + f _n , f _i : A → B, 1 ≤ i ≤ n. Another example is the trace of n × n matrices over a field R of characteristic zero, tr: M _n (R) → R, and, more generally, the character of any n-dimensional representation, tr ρ: A → R, ρ: A → M _n (R). The properties of n-homomorphisms (some of which were proved by Buchstaber and Rees under additional conditions) are derived, and a general theory of polynomially dependent homomorphisms is developed. One of the main results of the paper is a uniqueness theorem, which distinguishes the classes of n-homomorphisms among all polynomially dependent homomorphisms by a single natural completeness condition. As a topological application of n-homomorphisms, we consider the theory of n-homomorphisms between commutative C*-algebras with identity. We prove that the norm of any such n-homomorphism is equal to n and describe the structure of all such n-homomorphisms, which generalizes the classical Gelfand transform (the case of n = 1). An interesting fact discovered is that the Gelfand transform, which is a functorial bijection between appropriate spaces of maps, becomes a homeomorphism after considering natural topologies on these spaces.     

On the filter of computably enumerable supersets of an r-maximal set

We study the filter ℒ^*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ^*(A) is still Σ⁰ ₃-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ^*(A). Received: 6 November 1999 / Revised version: 10 March 2000 /?Published online: 18 May 2001