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.     

Functorial Properties of Star Operations

We define a universal star operation to be an assignment *: A ? * _A of a star operation * _A on A to every integral domain A. Prime examples of universal star operations include the divisorial closure star operation v, the t-closure star operation t, and the star operation w = F _∞ of Hedstrom and Houston. For any universal star operation *, we say that an extension B ? A of integral domains is *-ideal class linked if there is a group homomorphism Cl_{* A} (A) → Cl_{* B} (B) of star class groups induced by the map I ? (IB)^{* _B} on the set of * _A -ideals I of A. We study several natural subclasses of the class of *-ideal class linked extensions.     

Characterizations of *-Cancellation Ideals of an Integral Domain

Let D be an integral domain and * a star-operation on D. For a nonzero ideal I of D, let I ^{* _f} = ?{J* | (0) ≠ J ? I is finitely generated} and I ^{* _w} = ? _{P∈* f -Max(D)} ID _P . A nonzero ideal I of D is called a *-cancellation ideal if (IA)* = (IB)* for nonzero ideals A and B of D implies A* =B*. Let X be an indeterminate over D and N _* = {f ∈ D[X] | (c(f))* =D}. We show that I is a * _w -cancellation ideal if and only if I is * _f -locally principal, if and only if ID[X] _{N *} is a cancellation ideal. As a corollary, we have that each nonzero ideal of D is a * _w -cancellation ideal if and only if D _P is a principal ideal domain for all P ∈ * _f -Max(D), if and only if D[X] _{N *} is an almost Dedekind domain. We also show that if I is a * _w -cancellation ideal of D, then I ^{* _w}  = I ^{* _f}  = I _t , and I is * _w -invertible if and only if I ^{* _w}  = J _v for a nonzero finitely generated ideal J of D.     

Localization of Monoid Objects and Hochschild Homology

Let A be a (not necessarily commutative) monoid object in an abelian symmetric monoidal category (C, ?,1) satisfying certain conditions. In this paper, we continue our study of the localization M _S of any A-module M with respect to a subset S ? Hom _A?Bimod (A, A) that is closed under composition. In particular, we prove the following theorem: if P is an A-bimodule such that P is symmetric as a bimodule over the center Z(A) of A, we have isomorphisms HH _*(A, P) _S  ? HH _*(A, P _S ) ? HH _*(A _S , P _S ) of Hochschild homology groups.     

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.     

A presentation of the group Sl * (2, A), A a simple artinian ring with involution

Let (A,*) be an involutive ring. Then the groups Sl _*(2, A), are a non commutative version of the special linear groups Sl(2, F) defined over a field F. In particular, if A = M(n, F) and * is transposition, then Sl _*(2, M _n (F)) = Sp(2n, F). The above groups were defined by Pantoja and Soto-Andrade, and a set of generators for the group SSl _*(2, A) (which is either Sl _*(2, A) or a index 2 subgroup of Sl _*(2, A)) was given in the case when A is an artinian ring. In this paper, we prove that the mentioned generators provide a presentation of the mentioned groups in the case of simple artinian rings.Partially supported by FONDECYT project 1030907 and Pontificia Universidad Católica de Valparaíso     

Linear maps between C*-algebras that are *-homomorphisms at a fixed point

Let A and B be C*-algebras. A linear map T : A → B is said to be a *-homomorphism at an element z ∈ A if ab* = z in A implies T (ab*) = T (a)T (b)* = T (z), and c*d = z in A gives T (c*d) = T (c)*T (d) = T (z). Assuming that A is unital, we prove that every linear map T : A → B which is a *-homomorphism at the unit of A is a Jordan *-homomorphism. If A is simple and infinite, then we establish that a linear map T : A → B is a *-homomorphism if and only if T is a *-homomorphism at the unit of A. For a general unital C*-algebra A and a linear map T : A → B, we prove that T is a *-homomorphism if, and only if, T is a *-homomorphism at 0 and at 1. Actually if p is a non-zero projection in A, and T is a ^?-homomorphism at p and at 1 ? p, then we prove that T is a Jordan *-homomorphism. We also study bounded linear maps that are *-homomorphisms at a unitary element in A.     

Collinear Systems and Normal Contractive Projections on JBW*-Triples

Given a family {x _k }_k∈K of elements x _k in the predual A _* of a JBW^*-triple A, such that the support tripotents e _k of x _k form a collinear system in the sense of [31], necessary and sufficient criteria for the existence of a contractive projection from A _*. onto the subspace are provided. Preparatory to these results, and interesting in itself, is a set of necessary and sufficient algebraic conditions upon a contractive projection P on A for its range PA to be a subtriple. The results also provide criteria for the range of a normal contractive projection on A to be a Hilbert space. Supported by the Irish Research Council for Science, Engineering and Technology, Grant No. R 9854.     

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.     

On generalised Putnam—Fuglede theorems

LetB(H) denote the algebra of operators on the Hilbert spaceH, and letP denote the class ofAB(H) which are such that the restriction ofA to an invariant subspace is inP wheneverAP and which satisfy the property, henceforth called property (P 2), that if the restriction ofA to an invariant subspace is normal, then the subspace reducesA. GivenP-classesP ₁ andP ₂, the pair (P ₁,P ₂) is said to satisfy the (PF)-property if givenAP ₁ andB ^* P ₂ such thatAB=XB for someXB(H), we haveA ^* X=XB ^* . Generalising the (classical) Putnam—Fuglede theorem, it is shown here that the pair (P ₁,P ₂) has the (PF)-property if and only if, givenAP ₁ andB ^*P ₂ such thatAX=XB for some quasi-affinityXB(H), the following conditions hold: (i)B ^* is normal impliesA is normal; (ii)A has a normal direct summand impliesB ^* has a normal direct summand; (iii)A andB ^* pure impliesX is non-existent. An interestingP-class is the classC ₀ of contractions withC ₀ completely non-unitary parts which satisfy property (P 2). AssumingH to be separable, it is shown that ifC ₁ denotes thoseA C ₀ for which the defect operatorsD _A =(1–A^*A)^1/2 is of Hilbert—Schmidt class and for which either the pure part ofA has empty point spectrum or the eigen-values ofA are all simple, then the pair (C ₀,C ₁) has the (PF)-property. The classC ₁ defines aP-class; a crucial role in the proof of this statement is played by the interesting result that aC ₀ contraction with spectrum on the unit circle can not satisfy property (P 2). Applications of these results are considered, amongst them that ifA andB are quasi-similar hyponormal contractions such that the pure part ofA has finite multiplicity andD _B is of Hilbert —Schmidt class, then their normal parts are unitarily equivalent and their pure parts are quasi-similar.     

CertainC*-algebras with real rank zero and their corona and multiplier algebras: II

We first determine the homotopy classes of nontrivial projections in a purely infinite simpleC*-algebraA, in the associated multiplier algebraM(A) and the corona algebraM A/A in terms ofK _*(A). Then we describe the generalized Fredholm indices as the group of homotopy classes of non-trivial projections ofA; consequently, we determine theK _*-groups of all hereditaryC*-subalgebras of certain corona algebras. Secondly, we consider a group structure of *-isomorphism classes of hereditaryC*-subalgebras of purely infinite simpleC*-algebras. In addition, we prove that ifA is aC*-algebra of real rank zero, then each unitary ofA, in caseA it unital, each unitary ofM(A) and ofM(A)/A, in caseA is nonunital but -unital, can be factored into a product of a unitary homotopic to the identity and a unitary matrix whose entries are all partial isometries (with respect to a decomposition of the identity).Partially supported by a grant from the National Science Foundation.     

Straight Rings

A (commutative integral) domain is called a straight domain if A ? B is a prime morphism for each overring B of A; a (commutative unital) ring A is called a straight ring if A/P is a straight domain for all P ∈ Spec(A). A domain is a straight ring if and only if it is a straight domain. The class of straight rings sits properly between the class of locally divided rings and the class of going-down rings. An example is given of a two-dimensional going-down domain that is not a straight domain. The classes of straight rings, of locally divided rings, and of going-down rings coincide within the universe of seminormal weak Baer rings (for instance, seminormal domains). The class of straight rings is stable under formation of homomorphic images, rings of fractions, and direct limits. The “straight domain" property passes between domains having the same prime spectrum. Straight domains are characterized within the universe of conducive domains. If A is a domain with a nonzero ideal I and quotient field K, characterizations are given for A ? (I: _K I) to be a prime morphism. If A is a domain and P ∈ Spec(A) such that A _P is a valuation domain, then the CPI-extension C(P) := A + PA _P is a straight domain if and only if A/P is a straight domain. If A is a going-down domain and P ∈ Spec(A), characterizations are given for A ? C(P) to be a prime morphism. Consequences include divided domain-like behavior of arbitrary straight domains.     

Banach spaces of operators that are complemented in their biduals

Let [A, a] be a normed operator ideal. We say that [A, a] is boundedly weak*-closed if the following property holds: for all Banach spaces X and Y, if T: X → Y** is an operator such that there exists a bounded net (T _i ) _i∈I in A(X, Y) satisfying lim _i 〈y*, T _i x y*〉 for every x ∈ X and y* ∈ Y*, then T belongs to A(X, Y**). Our main result proves that, when [A, a] is a normed operator ideal with that property, A(X, Y) is complemented in its bidual if and only if there exists a continuous projection from Y** onto Y, regardless of the Banach space X. We also have proved that maximal normed operator ideals are boundedly weak*-closed but, in general, both concepts are different.      

The Effros–Ruan conjecture for bilinear forms on C<Superscript>*</Superscript>-algebras

In 1991 Effros and Ruan conjectured that a certain Grothendieck-type inequality for a bilinear form on C^*-algebras holds if (and only if) the bilinear form is jointly completely bounded. In 2002 Pisier and Shlyakhtenko proved that this inequality holds in the more general setting of operator spaces, provided that the operator spaces in question are exact. Moreover, they proved that the conjecture of Effros and Ruan holds for pairs of C^*-algebras, of which at least one is exact. In this paper we prove that the Effros–Ruan conjecture holds for general C^*-algebras, with constant one. More precisely, we show that for every jointly completely bounded (for short, j.c.b.) bilinear form on a pair of C^*-algebras A and B, there exist states f ₁, f ₂ on A and g ₁, g ₂ on B such that for all a∈A and b∈B,

PRIME IDEALS IN RINGS WITH INVOLUTION

Abstract

Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is essential in X and X is essential in R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R. Further characterizations of *-primeness are provided.     

Nonlinear Maps Preserving the Jordan Triple *-Product on Factor von Neumann Algebras

Let A and B be two factor von Neumann algebras. For A, B ∈ A, define by [A, B]_*= AB-BA~*the skew Lie product of A and B. In this article, it is proved that a bijective map Φ : A → B satisfies Φ([[A, B]_*, C]_*) = [[Φ(A), Φ(B)]_*, Φ(C)]_*for all A, B, C ∈ A if and only if Φ is a linear *-isomorphism, or a conjugate linear *-isomorphism, or the negative of a linear *-isomorphism, or the negative of a conjugate linear *-isomorphism.     

A Characterization of ＊-Automorphism on B（H）

Let H be a Hilbert space and A be a standard ＊-subalgebra of B（H）. We show that a bijective map Ф ： A →A preserves the Lie-skew product AB - BA＊ if and only if there is a unitary or conjugate unitary operator U ∈A（H） such that Ф（A） = UAU＊ for all A ∈ A, that is, Фis a linear ＊ -isomorphism or a conjugate linear ＊-isomorphism.     

The Facial and Inner Ideal Structure of a Real JBW*–Triple

Let B be a real JBW*–triple with predual B_* and canonical hermitification the JBW*–triple A It is shown that the set 𝒰(B)^∼ consisting of the partially ordered set 𝒰(B) of tripotents in B with a greatest element adjoined forms a sub–complete lattice of the complete lattice 𝒰(A)^∼ of tripotents in A with the same greatest element adjoined. The complete lattice 𝒰(B)^∼ is shown to be order isomorphic to the complete lattice ℱ_n(B_*1 of norm–closed faces of the unit ball B_*1 in B_* and anti–order isomorphic to the complete lattice ℱ_w*(B₁) of weak*–closed faces of the unit ball B₁ in B. Consequently, every proper norm–closed face of B_*1 is norm–exposed (by a tripotent) and has the property that it is also a norm–closed face of the closed unit ball in the predual of the hermitification of B. Furthermore, every weak*–closed face of B₁ is weak*–semi–exposed, and, if non–empty, of the form u + B₀(u)₁ where u is a tripotent in B and B₀(u)₁ is the closed unit ball in the zero Peirce space B₀(u) corresponding to u. A structural projection on B is a real linear projection R on B such that, for all elements a and b in B, {Ra b Ra}_B is equal to R{a Rb a}_B. A subspace J of B is said to be an inner ideal if {J B J}_B is contained in J and J is said to be complemented if B is the direct sum of J and the subspace Ker(J) defined to be the set of elements b in B such that, for all elements a in J, {a b a}_B is equal to zero. It is shown that every weak*–closed inner ideal in B is complemented or, equivalently, the range of a structural projection. The results are applied to JBW–algebras, real W*–algebras and certain real Cartan factors.     

Modules and comodules for corings

A coring C over a ring A is an (A, A)-bimodule with a comultiplication Δ: C → C _⨂A C and a counit ε: C → A, both being left and right A-linear mappings satisfying additional conditions. The dual spaces C* = Hom _A (C, A) and *C = _A Hom(C, A) allow the ring structure, and the right (left) comodules over C can be considered as left (right) modules over *C (respectively, C*). In fact, under weak restrictions on the A-module properties of C, the category of right C-comodules can be identified with the subcategory σ[_*C C] of *C-Mod, i.e., the category subgenerated by the left *C-module C. This point of view allows one to apply results from module theory to the investigation of coalgebras and comodules. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 51–72, 2005.     

Partitioned matrices and Seidel convergence

Summary Without using spectral resolution, an elementary proof of convergence of Seidel iteration. The proof is based on the lemma (generalizing a lemma of P. Stein): If (A+A ^*)–B ^*(A+A ^*)B>0, whereB=–(P+L) ^–1 R,A=P+L (Lower)+R (upper), then Seidel iteration ofAX=Y ₀ converges if and only ifA+A ^*>0. This lemma has as corollaries not only the well-known results of E. Reich and Stein, but also applications to a matrix that can be far from symmetric, e.g.M=[A _ij ] ₁ ² , whereA ₂₁=–A ₁₂ ^* ,A ₁₁,A ₂₂ are invertible;A ₁₁ +A ₁₁ ^* =A₂₂+A ₂₂ ^* ; and the proper values ofA ₁₂ ^–1 A ₁₁,A ₁₂ ^*–1 A ₂₂ are in the interior of the unit disk.Supported under NSF GP 32527.Supported under NSF GP 8758.