On the decomposition of cyclic algebras

A cyclic algebra (K/F, σ, a) of degreen hasproperty D(f) if it decomposes as a tensor product of a cyclic algebra of degreee=n/f containingL (the fixed subfield underσ ^e) and a cyclic subalgebra of degreef containing af-th root ofa. AlthoughD(2) holds for every cyclic algebra of degree 4 and exponent 2,D(p) fails for Brauer algebras of degreep ² and exponentp, andD(2) fails for Brauer algebras of degree 8 and exponent 2. Using this, one fills the gap in [6, Theorem 4] and [7, Theorem 7.3.28], to show that the example given there is indeed tensor indecomposable of degreep ² and exponentp. An easy ultraproduct argument provides an example containing allp ^k roots of 1, for allk.     

Inverse closedness ofC *-algebras in Banach algebras

LetA denote a unital Banach algebra, and letB denote aC ^*-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC ^*-subalgebras inC ^*-algebras.     

Elementary abelian operator groups

Letp be a prime,n a positive integer. Suppose thatG is a finite solvablep'-group acted on by an elementary abelianp-groupA. We prove that ifC _G (ϕ) is of nilpotent length at mostn for every nontrivial element ϕ ofA and |A|≥p ⁿ⁺¹ thenG is of nilpotent length at mostn+1.     

On Tate-Shafarevich groups over galois extensions

LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L) ^G ]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(A ^x/K)] whereA x is the twist ofA by the non-trivial characterχ ofG.     

On Codimension Growth of Finitely Generated Associative Algebras

LetAbe a PI-algebra over a fieldF. We study the asymptotic behavior of the sequence of codimensionsc_n(A) ofA. We show that ifAis finitely generated overFthenInv(A)=lim_n→∞ always exists and is an integer. We also obtain the following characterization of simple algebras:Ais finite dimensional central simple overFif and only ifInv(A)=dim=A.     

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.     

The Brauer group of a curve over a strictly local discrete valuation ring

LetK be the field of fractions of a curve overR whereR is the henselization of a regular local ring on an algebraic curve over a field which is algebraically closed and has characteristic 0. ThenK has the exponent=degree property for division algebras. In fact every central finite dimensionalK-division algebra with exponentn is a cyclic algebra of degreen. In memory of Professor S. A. Amitsur     

On the Schur multiplier of augmented algebras

Letk be a field, andA a finitely generatedk-algebra, with augmentation. Suppose there is a presentation ofA 0→I→R→A→0 whereR is a finitely generated freek-algebra andI is non-zero. IfA is infinite dimensional overk, Lewin proved thatR/I ² is not finitely presented. A stronger statement would be that the ‘Schur multiplier’ ofR/I ² is not finite dimensional. In the case thatA is an augmented domain, we prove this stronger statement, and some related statements.     

Tensor products and dimensions of simple modules for symmetric groups

LetK be an algebraically closed field of characteristic,p>0 and letD ^λ be the simple modules of the symmetric groupS _r overK where λ is a p-regular partition ofr. The dimensions ofD ^λ for λ with at mostn parts are the same as the multiplicities of direct summands ofD ^⊗r whereE is the natural module for the groupGL _n (K). Whenn=2 we determine generating functions for these multiplicities and hence for the dimensions ofD ^λ for all partitions λ with two parts. These can be expressed as rational functions of Chebyshev polynomials; and we obtain explicit formulae for the coefficients.     

Remarks on a paper of Bachelis and Gilbert

Several formulas for translation-invariant operator ideals overL ^p-spaces of compact groups are proved by tensor product methods.     

Construction of non-linear semi-groups using product formulas

Under certain circumstances, the Trotter-Lie formulaW _t=lim(U _t/nV_t/n) ⁿ is used to construct a non-linear semi-groupW _t on closed subsets ofL ^P, 1≦p<∞. In particular we consider the situation whereU _t=e ^tA is a positivity preservingC ₀ (linear) semi-group andV _t is generated by a (non-linear) functionF with certain monotonicity properties. In general,A andF are “singular” onL ^p and no requirement is made that one of them be “relatively bounded” with respect to the other. The generator of the resulting semi-groupW _t turns out to be an extension ofA +F restricted to a suitable domain. Research supported by a Danforth Graduate Fellowship and a Weizmann Postdoctoral Fellowship.     

Groups with few class sizes and the centraliser equality subgroup

LetG be ap-group whose conjugacy classes have at mostk sizes. We prove thatG is abelian-by-(exponentp ^k−1) (ifp=2, exponent 2 ^k−2). It follows that a 2-group with three class sizes is metabelian. Various other results on class sizes are proved, and some conjectures are formulated.     

Symmetric topological*-algebras

V. Pták's inequality is valid for every hermitian completeQ locallym-convex (:l.m.c.) algebra. Every algebra of the last kind is, in particular, symmetric. Besides, a (Hausdorff) locallyC ^*-algebra (being always symmetric) with the propertyQ is, within a topological algebraic isomorphism, aC ^*-algebra. Furthermore, a type of Raikov's criterion for symmetry is also valid for non-normed topological^*-algebras. Concerning topological tensor products, one gets that symmetry of the-completed tensor product of two unital Fréchet l.m.c.^*-algebrasE, F ( denotes the projective tensorial topology) is always passed toE, F, while the converse occurs when moreover either ofE, F is commutative.     

Complete positivity,tensor products and C*-nuclearity for inverse limits of C*-algebras

The paper aims at developing a theory of nuclear (in the topological algebraic sense) pro-C^*-algebras (which are inverse limits of C^*-algebras) by investigating completely positive maps and tensor products. By using the structure of matrix algebras over a pro-C^*-algebra, it is shown that a unital continuous linear map between pro-C^*-algebrasA andB is completely positive iff by restriction, it defines a completely positive map between the C^*-algebrasb(A) andb(B) consisting of all bounded elements ofA andB. In the metrizable case,A andB are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C^*-topology α and the projective pro-C^*-topology v on A⊗B are shown to be minimal and maximal pro-C^*-topologies; and α coincides with the topology of biequicontinous convergence iff eitherA orB is abelian. A nuclear pro-C^*-algebraA is one that satisfies, for any pro-C^*-algebra (or a C^*-algebra)B, any of the equivalent requirements; (i) α =v onA ⊗B (ii)A is inverse limit of nuclear C^*-algebras (iii) there is only one admissible pro-C^*-topologyon A⊗B (iv) the bounded partb(A) ofA is a nuclear C⊗-algebra (v) any continuous complete state map A→B^* can be approximated in simple weak^* convergence by certain finite rank complete state maps. This is used to investigate permanence properties of nuclear pro-C^*-algebras pertaining to subalgebras, quotients and projective and inductive limits. A nuclearity criterion for multiplier algebras (in particular, the multiplier algebra of Pedersen ideal of a C^*-algebra) is developed and the connection of this C^*-algebraic nuclearity with Grothendieck’s linear topological nuclearity is examined. A σ-C^*-algebraA is a nuclear space iff it is an inverse limit of finite dimensional C^*-algebras; and if abelian, thenA is isomorphic to the algebra (pointwise operations) of all scalar sequences.     

IndecomposableA-module summands inH * V which are unstable algebras

Letp be a prime and denote byA the modp Steenrod algebra. We determine the indecomposableA-module summands ofH*((ℤ/p)) ^d ;F _p which admit the structure of an unstableA-algebra. In fact, it turns out that this is equivalent to the problem of determining those indecomposableA-module summands which arise as the modp cohomology of a space (or even a classifying space of a finite group). We reduce this problem to one in modular representation theory, namely for whichd andp is the projective cover of the trivial one dimensional GL(d,F _p ) representationF _p a permutation module. Our solution of this latter problem makes use of the classification of subgroups of GL(d,F _p ) acting transitively on (F _p ) ^d \{0} and hence depends on the classification of finite simple groups (on Feit-Thompson's odd order theorem ifp=2).     

On Galois Coverings of the Enveloping Algebras of Self-Injective Nakayama Algebras

Abstract

There is constructed a Galois covering F of the enveloping K-algebra A ^e of a self-injective Nakayama K-algebra A such that the right A ^e -module A is of the first kind with respect to F. Then, with the help of the constructed Galois covering, the Auslander-Reiten translation period of A is computed.     

On the Witt Vectors of Perfect Rings in Positive Characteristic

The purpose of this article is to prove some results on the Witt vectors of perfect F _p -algebras. Let A be a perfect F _p -algebra for a prime integer p, and assume that A has the property P. Then does the ring of Witt vectors of A also have P? A main theorem gives an affirmative answer for P = ″integrally closed” under a very mild condition.     

The structure of a kind of connected components with oriented cycles and semi-stable vertices

LetF be an algebraically closed field. A be a finite-dimensional algebra overF,Γ _A be the Auslander-Reiten quiver ofA,Γ be a connected component of Γ _A with oriented cycles and semi-stable vertices and each non-stable vertex In Γ be a projective-injective vertex. The structure of Γ is studied. Projcct supported by Chinese Postdoctor Fund.     

NonstableK-Theory for operator algebras

We show that the homotopy groups of the group of quasi-unitaries inC ^*-algebras form a homology theory on the category of allC ^*-algebras which becomes topologicalK-theory when stabilized. We then show how this functorial setting, in particular the half-exactness of the involved functors, helps to calculate the homotopy groups of the group of unitaries in a series ofC ^*-algebras. The calculations include the case of all AbelianC ^*-algebras and allC ^*-algebras of the formAB, whereA is one of the Cuntz algebras O_n n=2, 3, ..., an infinite dimensional simpleAF-algebra, the stable multiplier or corona algebra of a-unitalC ^*-algebra, a properly infinite von Neumann algebra, or one of the projectionless simpleC ^*-algebras constructed by Blackadar.