Discontinuous Homomorphisms from Non-Commutative Banach Algebras

In the 1970s, a question of Kaplansky about discontinuous homomorphismsfrom certain commutative Banach algebras was resolved. Let Abe the commutative C*-algebra C(), where is an infinite compactspace. Then, if the continuum hypothesis (CH) be assumed, thereis a discontinuous homomorphism from C() into a Banach algebra[2, 7]. In fact, let A be a commutative Banach algebra. Then(with (CH)) there is a discontinuous homomorphism from A intoa Banach algebra whenever the character space _A of A is infinite[3, Theorem 3] and also whenever there is a non-maximal, primeideal P in A such that |A/P|=2^₀ [4, 8]. (It is an open questionwhether or not every infinite-dimensional, commutative Banachalgebra A satisfies this latter condition.) 1991 MathematicsSubject Classification 46H40.     

Spectral Synthesis and Operator Synthesis for Compact Groups

Let G be a compact group and C(G) be the C^*-algebra of continuouscomplex-valued functions on G. The paper constructs an imbeddingof the Fourier algebra A(G) of G into the algebra V(G) = C(G)^hC(G)(Haagerup tensor product) and deduces results about parallelspectral synthesis, generalizing a result of Varopoulos. Itthen characterizes which diagonal sets in G x G are sets ofoperator synthesis with respect to the Haar measure, using thedefinition of operator synthesis due to Arveson. This resultis applied to obtain an analogue of a result of Froelich: atensor formula for the algebras associated with the pre-ordersdefined by closed unital subsemigroups of G.     

Topological Hochschild Homology

In the appendix to [20] Waldhausen discussed a trace map tr:K(R)HH(R),from the algebraic K-theory of a ring to its Hochschild homology,which can be used to obtain information about K(R) from HH(R).In [1] Bökstedt described a factorization of this tracemap. The intermediate functor THH(HR) is called the topologicalHochschild homology of the Eilenberg–MacLane spectrumHR associated with R, because it is constructed similarly toHochschild homology with the tensor product replaced by thesmash product of spectra.     

Rigged modules I: Modules over dual operator algebras and the Picard group

In a previous paper we generalized the theory of $W^{?}$-modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak^?-rigged modules. At that time we promised a forthcoming paper devoted to other aspects of the theory. We fulfill this promise in the present work and its sequel “Rigged modules II”, giving many new results about weak^?-rigged modules and their tensor products. We also discuss the Picard group of weak* closed subalgebras of a commutative algebra. For example, we compute the weak Picard group of $H^{\infty }(\mathbb{D})$, and prove that for a weak* closed function algebra A, the weak Picard group is a semidirect product of the automorphism group of A, and the subgroup consisting of symmetric equivalence bimodules.     

Tensor Products of C*-Algebras Over Abelian Subalgebras

Suppose that A is a C*-algebra and C is a unital abelian C*-subalgebrawhich is isomorphic to a unital subalgebra of the centre ofM(A), the multiplier algebra of A. Letting = , so that we maywrite C = C(), we call A a C()-algebra (following Blanchard[7]). Suppose that B is another C()-algebra, then we form A_CB, the algebraic tensor product of A with B over C as follows:A B is the algebraic tensor product over C, I_C = {ⁿ_i–1(f_i 1–1f_i)x|f_iC, xAB} is the ideal in AB generated by f1–1f|fC,and A _CB = AB/I_C. Then A_CB is an involutive algebra over C,and we shall be interested in deciding when A_CB is a pre-C*-algebra;that is, when is there a C*-norm on A_C B? There is a C*-semi-norm,which we denote by ||·||_C-min, which is minimal in thesense that it is dominated by any semi-norm whose kernel containsthe kernel of ||·||_C-min. Moreover, if A _C B has a C*-norm,then ||·||_C-min is a C*-norm on A_C B. The problem isto decide when ||·||_C-min is a norm. It was shown byBlanchard [7, Proposition 3.1] that when A and B are continuousfields and C is separable, then ||·||_C-min is a norm.In this paper we show that ||·||_C-min is a norm whenC is a von Neumann algebra, and then we examine some consequences.     

Structural Projections on JBW*-Triples

A linear projection R on a Jordan*-triple A is said to be structuralprovided that, for all elements a, b and c in A, the equality{Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to becomplemented if A = B + Ker(B), where Ker(B) = {aA: {B a B}= 0}. It is shown that a subtriple of a JBW*-triple is complementedif and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space Eis said to be an N*-ideal if every weak* continuous linear functionalon B has a norm preserving extension to a weak* continuous linearfunctional on E* and the set of elements in E which attain theirnorm on the unit ball in B is a subspace of E. It is shown thata subtriple of a JBW*-triple A is complemented if and only ifit is an N*-ideal, from which it follows that complemented subtriplesof A are weak* closed, and structural projections on A are weak*continuous and norm non-increasing. It is also shown that everyN*-ideal in A possesses a triple product with respect to whichit is a JBW*-triple which is isomorphic to a complemented subtripleof A.     

Spectral Inclusion and Analytic Continuation

Let a be an element in a complex Banach algebra with unit, andlet r be a nonnegative number. The Gelfand spectral radius formulaimplies that the spectrum of a is included in the disk {z C:|z| r} if and only if 1991Mathematics Subject Classification 46H99, 47A10, 30B40.     

Cyclic Artinian Modules without a Composition Series

Let R be a ring (always understood to be associative with aunit element 1). It is well known that an R-module is Noetherianif and only if all its submodules are finitely generated andthat it has a finite composition series if and only if it isNoetherian and Artinian. This raises the question whether everyfinitely generated Artinian module is Noetherian; here it isenough to consider cyclic Artinian modules, by an inductionon the length. This question has been answered (negatively)by Brian Hartley [5], who gives a construction of an Artinianuniserial module of uncountable composition-length over thegroup algebra of a free group of countable rank. If we are justinterested in finding cyclic modules that are Artinian but notNoetherian, there is a very simple construction based on thefact that over a free algebra every countably generated Artinianmodule can be embedded in a cyclic module which is again Artinian.This is described in 2 below.     

Are All Uniform Algebras Amnm?

A Banach algebra a is AMNM if whenever a linear functional on a and a positive number satisfy |(ab)–(a)(b)|||a||·||b||for all a, b a, there is a multiplicative linear functional on a such that ||–||=o(1) as 0. K. Jarosz [1] asked whetherevery Banach algebra, or every uniform algebra, is AMNM. B.E. Johnson [3] studied the AMNM property and constructed a commutativesemisimple Banach algebra that is not AMNM. In this note weconstruct uniform algebras that are not AMNM. 1991 MathematicsSubject Classification 46J10.     

Ideals in MOD-R and the {omega}-radical

Let R be an artin algebra, and let mod-R denote the categoryof finitely presented right R-modules. The radical rad = rad(mod-R)of this category and its finite powers play a major role inthe representation theory of R. The intersection of these finitepowers is denoted rad, and the nilpotence of this ideal hasbeen investigated, in [6, 13] for instance. In [17], arbitrarytransfinite powers, rad, of rad were defined and linked to theextent to which morphisms in mod-R may be factorised. In particular,it has been shown that if R is an artin algebra, then the transfiniteradical, rad, the intersection of all ordinal powers of rad,is non-zero if and only if there is a ‘factorisable system’of morphisms in rad and, in that case, the Krull–Gabrieldimension of mod-R equals (that is, is undefined). More preciseresults on the index of nilpotence of rad for artin algebraswere proved in [14, 20, 24–26].     

Mappings Preserving Submodules of Hilbert C*-Modules

A Hilbert module over a C*-algebra B is a right B-module X,equipped with an inner product ·, · which is linearover B in the second factor, such that X is a Banach space withthe norm ||x||:=||x, x||^1/2. (We refer to [8] for the basictheory of Hilbert modules; the basic example for us will beX=B with the inner product x, y=x*y.) We denote by B(X) thealgebra of all bounded linear operators on X, and we denoteby L(X) the C*-algebra of all adjointable operators. (In thebasic example X=B, L(X) is just the multiplier algebra of B.)Let A be a C*-subalgebra of L(X), so that X is an A-B-bimodule.We always assume that A is nondegenerate in the sense that [AX]=X,where [AX] denotes the closed linear span of AX. Denote by A_X the algebra of all mappings on X of the form (1.1) where m is an integer and a_iA, b_iB for all i. Mappings of form(1.1) will be called elementary, and this paper is concernedwith the question of which mappings on X can be approximatedby elementary mappings in the point norm topology.     

Products of Commuting Boolean Algebras of Projections and Banach Space Geometry

New criteria and Banach spaces are presented (for example, GL-spacesand Banach spaces with property () that ensure that the Booleanalgebra generated by a pair of bounded, commuting Boolean algebrasof projections is itself bounded. The notion of R-boundednessplays a fundamental role. It is shown that the strong operatorclosure of any R-bounded Boolean algebra of projections is necessarilyBade complete. Also, for a Dedekind -complete Banach latticeE, the Boolean algebra consisting of all band projections inE is R-bounded if and only if E has finite cotype. In this situation,every bounded Boolean algebra of projections in E is R-boundedand has a Bade complete strong closure. 2000 Mathematics SubjectClassification 46B20, 47L10 (primary), 46B42, 47B40, 47B60 (secondary).     

The Structure of Biserial Algebras

By an algebra we mean an associative k-algebra with identity,where k is an algebraically closed field. All algebras are assumedto be finite dimensional over k (except the path algebra kQ).An algebra is said to be biserial if every indecomposable projectiveleft or right -module P contains uniserial submodules U andV such that U+V=Rad(P) and UV is either zero or simple. (Recallthat a module is uniserial if it has a unique composition series,and the radical Rad(M) of a module M is the intersection ofits maximal submodules.) Biserial algebras arose as a naturalgeneralization of Nakayama's generalized uniserial algebras[2]. The condition first appeared in the work of Tachikawa [6,Proposition 2.7], and it was formalized by Fuller [1]. Examplesinclude blocks of group algebras with cyclic defect group; finitedimensional quotients of the algebras (1)–(4) and (7)–(9)in Ringel's list of tame local algebras [4]; the special biserialalgebras of [5, 8] and the regularly biserial algebras of [3].An algebra is basic if /Rad() is a product of copies of k.This paper contains a natural alternative characterization ofbasic biserial algebras, the concept of a bisected presentation.Using this characterization we can prove a number of resultsabout biserial algebras which were inaccessible before. In particularwe can describe basic biserial algebras by means of quiverswith relations.     

Continuity of Derivations, Intertwining Maps, and Cocycles from Banach Algebras

Let A be a Banach algebra, and let E be a Banach A-bimodule.A linear map S:AE is intertwining if the bilinear map is continuous, and a linear map D:AE is a derivation if ¹D=0,so that a derivation is an intertwining map. Derivations fromA to E are not necessarily continuous. The purpose of the present paper is to prove that the continuityof all intertwining maps from a Banach algebra A into each BanachA-bimodule follows from the fact that all derivations from Ainto each such bimodule are continuous; this resolves a questionleft open in [1, p. 36]. Indeed, we prove a somewhat strongerresult involving left- (or right-) intertwining maps.     

An Algebra of Pseudodifferential Operators with Slowly Oscillating Symbols

Let VR denote the Banach algebra of absolutely continuous functionsof bounded total variation on R, and let B_p be the Banach algebraof bounded linear operators acting on the Lebesgue space L^pRfor 1 < p < . We study the Banach algebra A B_p generatedby the pseudodifferential operators of zero order with slowlyoscillating VR-valued symbols on R. Boundedness and compactnessconditions for pseudodifferential operators with symbols inL R, VR are obtained. A symbol calculus for the non-closed algebraof pseudodifferential operators with slowly oscillating VR-valuedsymbols is constructed on the basis of an appropriate approximationof symbols by infinitely differentiable ones and by use of thetechniques of oscillatory integrals. As a result, the quotientBanach algebra A = A K, where K is the ideal of compact operatorsin B_p, is commutative and involutive. An isomorphism betweenthe quotient Banach algebra A of pseudodifferential operatorsand the Banach algebra of their Fredholm symbols is established. A Fredholm criterionand an index formula for the pseudodifferential operators A A are obtained in terms of their Fredholm symbols. 2000 MathematicsSubject Classification 47G30, 47L15 (primary), 47A53, 47G10(secondary).     

Indecomposable flat cotorsion modules

An additive functor from the category of flat right R-modulesto the category of abelian groups is continuous if it is isomorphicto a functor of the form–_R M, where M is a left R-module.It is shown that for any simple subfunctor A of– M thereis a unique indecomposable flat cotorsion module U_R for whichA(U)0. It is also proved that every subfunctor of a continuousfunctor contains a simple subfunctor. This implies that everyflat right R-module may be purely embedded into a product ofindecomposable flat cotorsion modules. If CE(R) is the cotorsion envelope of R_R and S= End;_R CE(R),then a local ring monomorphism is constructed from R/J(R) toS/J(S). This local morphism of rings is used to associate asemiperfect ring to any semilocal ring. It also proved thatif R is a semilocal ring and M a simple left R-module, thenthe functor–_R M on the category of flat right R-modulesis uniform, and therefore contains a unique simple subfunctor.     

Zel'Manov's Theorem for Primitive Jordan-Banach Algebras

The following result is well known and easy to prove (see [14,Theorem 2.2.6]). Theorem 0. If A is a primitive associative Banach algebra, thenthere exists a Banach space X such that A can be seen as a subalgebraof the Banach algebra BL(X) of all bounded linear operatorson X in such a way that A acts irreducibly on X and the inclusionABL(X) is continuous. In fact, if X is any vector space on which the primitive Banachalgebra A acts faithfully and irreducibly, then X can be convertedin a Banach space in such a way that the requirements in Theorem0 are satisfied and even the inclusion ABL(X) is contractive. Roughly speaking, the aim of this paper is to prove the appropriateJordan variant of Theorem 0.     

Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on Rⁿ (n2) is the coarsest topology for whichall superharmonic functions on Rⁿ are continuous. We refer toDoob [11, 1.XI] for its basic properties and its relationshipto the notion of thinness. This paper presents several theoremsrelating the fine topology to limits of functions along parallellines. (Results of this nature for the minimal fine topologyhave been given by Doob – see [10, Theorem 3.1] or [11,1.XII.23] – and the second author [15].) In particular,we will establish improvements and generalizations of resultsof Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihland Seidel [6], Schneider [21], Berman [7], and Armitage andNelson [4], and will also solve a problem posed by the latterauthors. An early version of our first result is due to Evans [12, p.234], who proved that, if u is a superharmonic function on R³,then there is a set ER²x{0}, of two-dimensional measure 0, suchthat u(x, y,·) is continuous on R whenever (x, y, 0)E.We denote a typical point of Rⁿ by X=(X' x), where X'R^n–1and xR. Let :RⁿR^n–1x{0} denote the projection map givenby (X', x) = (X', 0). For any function f:Rⁿ[–, +] andpoint X we define the vertical and fine cluster sets of f atX respectively by C_V(f;X)={l[–, +]: there is a sequence (t_m) of numbersin R\{x} such that t_mx and f(X', t_m)l}| and C_F(f;X)={l[–, +]: for each neighbourhood N of l in [–,+], the set f^–1(N) is non-thin at X}. Sets which are open in the fine topology will be called finelyopen, and functions which are continuous with respect to thefine topology will be called finely continuous. Corollary 1(ii)below is an improvement of Evans' result.     

On Borel Sets in Function Spaces with the Weak Topology

It is proved that the duality map ,:(, weak)x(()*, weak*)R isnot Borel. More generally, the evaluation e:(C)(K),x KR, e(f,x) = f(x), is not Borel for any function space C(K) on a compactF-space. It is also shown that a non-coincidence of norm-Boreland weak-Borel sets in a function space does not imply thatthe duality map is non-Borel.     

Intertwining Maps from Certain Group Algebras

In [17, 18, 19], we began to investigate the continuity propertiesof homomorphisms from (non-abelian) group algebras. Alreadyin [19], we worked with general intertwining maps [3, 12]. Thesemaps not only provide a unified approach to both homomorphismsand derivations, but also have some significance in their ownright in connection with the cohomology comparison problem [4]. The present paper is a continuation of [17, 18, 19]; this timewe focus on groups which are connected or factorizable in thesense of [26]. In [26], G. A. Willis showed that if G is a connectedor factorizable, locally compact group, then every derivationfrom L¹(G) into a Banach L¹(G)-module is automatically continuous.For general intertwining maps from L¹(G), this conclusion isfalse: if G is connected and, for some nN, has an infinite numberof inequivalent, n-dimensional, irreducible unitary representations,then there is a discontinuous homomorphism from L¹(G into aBanach algebra by [18, Theorem 2.2] (provided that the continuumhypothesis is assumed). Hence, for an arbitrary intertwiningmap from L¹(G), the best we can reasonably hope for is a resultasserting the continuity of on a ‘large’, preferablydense subspace of L¹(G). Even if the target space of is a Banachmodule (which implies that the continuity ideal I() of is closed),it is not a priori evident that is automatically continuous:the proofs of the automatic continuity theorems in [26] relyon the fact that we can always confine ourselves to restrictionsto L¹(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4].It is not clear if this strategy still works for an arbitraryintertwining map from L¹(G) into a Banach L¹(G)-module.