Canonical Homomorphism Research Articles

On tensor products of certain group C∗-algebras

If A and B are C ∗-algebras there is, in general, a multiplicity of C ∗ -norms on their algebraic tensor product A ⊙ B, including maximal and minimal norms ν and α, respectively. A is said to be nuclear if α and ν coincide, for arbitrary B. The earliest example, due to Takesaki [11], of a nonnuclear C ∗-algebra was C l ∗(F 2) , the C ∗-algebra generated by the left regular representation of the free group on two generators F 2. It is shown here that W ∗-algebras, with the exception of certain finite type I's, are nonnuclear. If C ∗(F 2) is the group C ∗-algebra of F 2, there is a canonical homomorphism λ l of C ∗(F 2) onto C l ∗(F 2) . The principal result of this paper is that there is a norm ζ on C l ∗(F 2) ⊙ C l ∗(F 2) , distinct from α, relative to which the homomorphism λ ⊙ λ l: C ∗(F 2) ⊙ C ∗(F 2) → C l ∗(F 2) ⊙ C l ∗(F 2) is bounded ( C ∗(F 2) ⊙ C ∗(F 2) being endowed with the norm α). Thus quotients do not, in general, respect the norm α; a consequence of this is that the set of ideals of the α-tensor product of C ∗-algebras A and B may properly contain the set of product ideals { I ⊗ B + A ⊗ J: I ◁ A, J ◁ B}. Let A and B be C ∗-algebras. If A or B is a W ∗-algebra there are on A ⊙ B certain C ∗-norms, defined recently by Effros and Lance [3], the definitions of which take account of normality. In the final section of the paper it is shown by example that these norms, with α and ν, can be mutually distinct.

Quotient groups and realization of tight Riesz groups

Let (G, ≼) be an l-group having a compatible tight Riesz order ≦ with open-interval topology U, and H a normal subgroup. The first part of the paper concerns the question: Under what conditions on H is the structure of (G, ≼, ∧, ∨, ≦, U) carried over satisfactorily to by the canonical homomorphism; and its answer (Theorem 8°): H should be an l-ideal of (G, ≼) closed and not open in (G, U). Such a normal subgroup is here called a tangent. An essential step is to show that ≼′ is the associated order of ≦′.

On Factorization of Polynomials Modulo n

Let A be an ideal of the commutative ring R with identity. There is a canonical homomorphism ϕA from the polynomial ring R[X] onto (R/A)[X], obtained by reducing all coefficients modulo A. If fR[X], then we say that f is reducible (irreducible) modulo A if ϕA(f) is reducible (irreducible) in (R/A)[X].

Sums of lengtht in Abelian groups

LetG be an Abelian group written additively,B a finite subset ofG, and lett be a positive integer. Fort≦|B|, letB t denote the set of sums oft distinct elements overB. Furthermore, letK be a subgroup ofG and let σ denote the canonical homomorphism σ:G→G/K. WriteB t (modB t) forB tσ and writeB t (modK) forBσ. The following addition theorem in groups is proved. LetG be an Abelian group with no 2-torsion and letB a be finite subset ofG. Ift is a positive integer such thatt<|B| then |B t (modK)|≧|B (modK)| for any finite subgroupK ofG.

On Inflation and Torsion of Amitsur Cohomology

Studies of torsion [2] and inflation [11] of Amitsur cohomology have primarily been concerned with module-finite faithful projective algebras. In this paper, our goal is to consider these topics for more general algebras. The fundamental tool, in case R is a domain with quotient field K, is the functor UK/U (defined in § 1), together with the monomorphic connecting map of Amitsur cohomology H1(S/R, UK/U) → H2(S/R, U) arising from a (commutative) flat R-algebra S and the unit functor U. This map was first considered in [10, Chapter IV, Theorem 1.6], where it was proved to be an isomorphism for certain étale faithfully flat algebras S in case R is an algebraic number ring with trivial Brauer group. In Corollary 1.5 it is shown, in the case of a module-finite faithful projective S over a regular domain R, that the connecting map is the kernel of the canonical homomorphism [8] from H2(S/R, U) to the split Brauer group B(S/R). As the latter map is often injective [8, Corollary 7.7], one might expect instances of vanishing of H1(S/R, UK/U) without assuming both R Noetherian and S module-finite R-projective. Much of § 1 (viz. (1.7)-(1.9)) is devoted to such examples.

Split and minimal abelian extensions of finite groups

Criteria for an abelian extension of a group to split are given in terms of a Sylow decomposition of the kernel and of normal series for the Sylow subgroups. An extension is minimal if only the entire extension is carried onto the given group by the canonical homomorphism. Various basic results on minimal extensions are given, and the structure question is related to the case of irreducible kernels of prime exponent. It is proved that an irreducible modular representation of SL(2, p) or PSL(2, p) for p prime and > 5 afford a minimal extension with kernel of exponent p only when the representation has degree 3, 3 i.e., when the kernel has order p

Embedding problems for modules and rings with application to Model-Companions

We study embedding problems for modules and rings. The first of these problems is the well-known problem of extension of scalars. Any ring can be embedded in a ring A such that for any ring A′ containing A and any A-module M the canonical homomorphism: M → A′ (⊗) A M is one-one. The second one concerns the embedding of couples ( A, M) where M is an A-module into couples of the form ( B, B) where B is a ring. We relate this problem to the existence of a model-companion for theories of modules over theories of rings and give some applications.

The quotient algebra of a finite von Neumann algebra

We will prove the following: Let M be a finite von Neumann algebra with center Z and A a von Neumann subalgebra of Z. Let Ω be the spectrum space of A and identify A with C(Ω). Let e be a σ-weakly continuous linear map of M onto A such that e(x*x) = e(a%c*) ^ 0 for every xeM, e(ax) — ae(x) for every a e A and xeM, e(l) = 1 and e(x*x) Φ 0 for every nonzero xeM. For each ωeΩ, let mω denote the set of all xeM with e(x*x)(ω) = 0. Then mω is a closed ideal and the quotient C*-algebla M/xnω is a finite von Neumann algebra. Furthermore, if πω denote the canonical homomorphism of M onto Λf/m», then πω(N) is a von Neumann subalgebra of M/mω for every von Neumann subalgebra N containing A.

On Trace for Modules

Classically, trace was defined as the sum of the diagonal entries of a square matrix with entries in a field. This notion played an important role in classical mathematics, e.g. in the theory of algebras over a field of characteristic zero, and in the theory of group characters (as in [1]). A generalization to endomorphisms of a finitely generated projective module over any ring R with unit is well-known. For such a module P the canonical homomorphism Ψ: P* ⊗ P→ EndR(P) is an isomorphism. Then the composite ε˚Ψ-1: EndR(P) → P* ↗ P→ R, where e denotes “evaluation”, is a homomorphism which coincides with the classical trace whenever P is free. This version of trace has been used by Hattori [3] and others to study projective modules. However, this approach to trace is limited to the finitely generated projective modules, since it can be shown that Ψ is an isomorphism if and only if P is finitely generated and projective.

A characterization of unitary duality

The concept of unitary duality for topological groups was introduced by H. Chu. All mapping spaces are given the compact-open topology. Let G and H be locally compact groups. Gx is the space of continuous finite-dimensional unitary of G. Let Hom (Gx, Hx) denote the space of all continuous maps from Gx to Hx which preserve degree, direct sum, tensor product and equivalence. We prove that if H satisfies unitary duality, then Hom (G, H) and Hom (Hx, G x) are naturally homeomorphic. Conversely, if Hom (Z, H) and Hom (Hx, Zx) are homeomorphic by the natural map, where Z denotes the integers, then H satisfies unitary duality. In different contexts, results similar to the first half of this theorem have been obtained by Suzuki and by Ernest. The proof relies heavily on another result in this paper which gives an explicit characterization of the topology on Hom (G x, Hx ). In addition, we give another necessary condition for locally compact groups to satisfy unitary duality and use this condition to present an example of a maximally almost periodic discrete group which does not satisfy unitary duality. Introduction. Let G be a locally compact topological group. Let GX denote the space of continuous finite-dimensional unitary of G with the compact-open topology. G x x is the space of continuous unitary representations of G x (see Definition 2 below), again with the compact-open topology. G x x becomes a topological group in a natural way and there is a canonical homomorphism from G to GX x. If this canonical map is a topological isomorphism we say that G satisfies unitary duality. This definition was introduced by Chu [2]. It presents in a more general context the duality theorems of Pontryagin [9, p. 94 ff.], Tannaka [7], and Takahashi [6]. In ?1 of this paper we give the important definitions related to unitary duality. Hom (G x, H X) for two locally compact groups G and H is defined in a natural way and given the compact-open topology also. In ?2 we give an explicit subbase for the neighborhoods of each point in Hom (G X, H x). In ?3 we use the result of ?2 to prove the following characterization of unitary duality: if G and H are locally Presented in part to the Society, January 22, 1970 under the title Unitary duality of locally compact groups. II; received by the editors July 9, 1969. AMS Subject Classifications. Primary 2220; Secondary 2260.

Commutators modulo the center in a properly infinite von Neumann algebra

1. Introduction. An element in a von Neumann algebra stf is said to be a commutator in stf if there are elements A and B in stf such that C=AB — BA. For finite homogeneous discrete algebras and for properly infinite factor algebras the set of commutators has been completely described [l]-[5], [10]. In each of these special cases any element is a commutator modulo a central element depending on C. In this paper we show that given any element in a properly infinite von Neumann algebra stf there is an element C0 in the center of stf depending on such that C— C0 is a commutator in stf. The element C0 is an arbitrary element in the intersection criTc of the center with the uniform closure of the convex hull of {U*CU | U unitary in ?tf} [6, III, §5]. We then present a few facts about those elements such that 0 e Xc or what is the same as far as determining commutators is concerned about those elements such that OeJf^-ics for some invertible S in stf. 2. Commutators. Let stf be a C*-algebra with identity and let / be a closed two-sided ideal in stf. The image of the element A e stf in the factor algebra stf(T) = stf/1 under the canonical homomorphism of stf onto stf/I will be denoted by A(T). If £ is a maximal ideal of the center of stf, the smallest closed two-sided ideal in stf containing £ is denoted by [{]. For simplicity we write A([Q) as A(t). The set of maximal (respectively, primitive) ideals of stf with the hull-kernel topology is called the strong structure space (respectively, structure space) of stf. If stf is a von Neumann algebra, then the strong structure space M(stf) of stf is homeomorphic with the spectrum of the center S of stf under the map M —> M C J? [13]. This means M(stf) is extremely disconnected. Proposition 1. Let stf be a properly infinite von Neumann algebra and let A be a fixed element of stf. The function M -> ||^(M)|| of the strong structure space M (stf) of stf into the real numbers is continuous. Proof. For every «?Owe know that the set X={M e M (stf) \\A(M) ^«} is closed. If /=p| X, then \\A(I)\\^a [8, Lemma 1.9] and so M(M)|| ^a for every MeM(stf) containing /. Thus X={M e M (stf) | I<=M}.

Galois cohomology and class number in constant extension of algebraic function fields

Let F be a field of algebraic functions of one variable having the finite field K as exact field of constants. The class number of F is defined as the order of the finite group, Co(F), of divisor classes of degree zero. Let L be the unique cyclic extension of K of degree n, E = F* L the corresponding constant extension with galois group G. Since K is perfect, the canonical homomorphism of the group of divisor classes of F in the group of divisor classes of E is an injection [2, p. 477]. If hE, hF denote the class numbers of E and F, respectively, we have hE=hF. k, for some integer k. The purpose of this note is to prove the following two theorems:

A direct limit representation for Abelian groups with an application to tensor sequences

Each group may be regarded as a direct limit of its finitely generated subgroups. We develop in w 1 a slightly different direct limit representation valid for all abelian groups. The advantage of this alternate representation lies in the fact that some properties of abetian groups are preserved under direct limits but are not inherited by arbitrary finitely generated subgroups. We will be concerned with the following such property which an abelian group G may have: for a particular subgroup A of a particular group B, the canonical homomorphism G| ~GQB is a monomorphism. In w we apply the direct limit representation of w 1 to discuss the exactness of tensor sequences. Since w l and w allow a quick description of those classes of abelian torsion groups which are closed under direct sums and direct limits, we insert this description as w In w we discuss some details of the splitting question for tensor sequences. We use Z(n) to denote the group of integers modulo n and Z(p ~) to denote the p-primary component of the group of rationals modulo I. Otherwise our general reference for terms and notations is [1]. 1. A direct limit representation. Let S, (~ E A) be the family of those subgroups of G which are isomorphic to the direct sum of a finitely generated free abelian group and a finite number of indecomposable primary groups each isomorphic to a direct summand of G. This family is directed by set inclusion. It is a consequence of our first theorem that G is isomorphic to the direct limit of this directed system of groups S~ (~EA). THEOREM 1. Each finitely generated subgroup A of each abelian group G is contained in a subgroup S of G which is the direct sum of a finitely generated free abelian group and a finite number of indecomposable primary groups where each of the indecomposable primary groups is isomorphic to a summand of G. PROOF. Case I: Let G be a p-primary group. Then A is finite and we may use finite induction on the order of A. The theorem is trivial for A = 0. Suppose g is an element of order p in A. (i) If g is in the maximal divisible subgroup D of G, let H (~-Z(p=)) be a divisible hull of {g} in D. Then A§ = AI~3HcPOH=G for some summands A1 of A and P of G for which PDA~. (ii) Ifg is of finite height n in G, let hEG be such that g=p"h and let H={h}. Then A+H = Aa| cP| = G for some summands A 1 of A and P of G for which P~A~. (iii) If g is of infinite height but is not an element of D then the basic subgroup of G must be unbounded. These conditions imply the existence of an integer n and an element

A Class of Homomorphism Theories for Groupoids

In a well-behaved homomorphism theory for a class of algebraic systems certain “closed objects” relative to a given G ∈ are distinguished which act as kernels of homomorphisms. For example, if is the class of groups then the closed objects relative to a given group G are the normal subgroups of G; if is the class of semigroups with zero element then one can devise a homomorphism theory in which the closed objects relative to a given S ∈ are the ideals of S[cf. Rees (3)]; in the class of groupoids one may define the closed objects relative to a given groupoid G to be the congruence relations on G, that is, subsets π⊆G×G which are equivalence relations having the property that (x1y1, x2y2) ∈ π whenever (x1, x2), (y1, y2) ∈ π. Given such a closed object N relative to G there exists a “factor” system G/N and a (canonical) homomorphism η: G→G/N characterised by the property: If σ G→H is a homomorphism with kernel N then there is a unique homomorphism : G/N→H such that . η = σ and the kernel of is trivial in the sense that the kernel of is the unique smallest closed object relative to G/N.

The universal representation kernel of a Lie group

Let G be a connected real Lie group. The universal representation kernel, Ko, oi G is defined as the intersection of all kernels of continuous finite dimensional representations of G. Evidently, Ko is a closed normal subgroup of G, and it is known from a theorem due to Goto (cf. [l, Theorem 7.1]) that G/Ko has a faithful continuous finite dimensional representation. Thus Kg is the smallest normal closed subgroup P of G such that G/P is isomorphic with a real analytic subgroup of a full linear group. The known criteria for the existence of a faithful representation lead to a determination oi Ko which we wish to record here. Suppose first that G is semisimple. Let @ denote the Lie algebra of G. Let C stand for the field of the complex numbers, and denote by @c the complexification of ®, i.e., the semisimple Lie algebra over C that is obtained by forming the tensor product, over the real field, of ® with C. Denote by 5(@) and 5(©c) the simply connected Lie groups whose Lie algebras are @ and ©c, respectively. The injection ©—>@c is the differential of a uniquely determined continuous homomorphism y of 5(@) into S(d$c). The kernel P of y is a discrete central subgroup of 5(@). Let (P), i.e., the universal representation kernel of the semisimple connected Lie group G is the image, under the universal covering epimorphism, of the kernel of the canonical homomorphism S(®)->S(®C).

Neron models for semiabelian varieties: Congruence and change of base field

Let O be a henselian discrete valuation ring with perfect residue field. Denote by K the fraction field of O = OK , and by p = pK the maximal ideal of O. Then every abelian variety A over K has a Neron model A over O. The Neron model A of A is a smooth group scheme of finite type over O, characterized by the property that for every finite unramified extension L of K, every L-valued point of AK extends uniquely to an OL-valued point of A. We refer to the book [BLR] for a thorough exposition of the construction and basic properties of Neron models. In general, the formation of Neron models does not commute with base change. Rather, for every finite extension field M of K, we have a canonical homomorphism