Constructions of stable equivalences of Morita type for finite-dimensional algebras III

In this paper, we provide a new method to produce stable equivalencesof Morita type. Our main results can be stated as follows. LetA and B be two finite-dimensional k-algebras over a field k.Suppose that two bimodules _AM_B and _BN_A define a stable equivalenceof Morita type between A and B and that R is a generator forA-modules. Then there is a stable equivalence of Morita typedefined by X and Y between the endomorphism algebra End_A(R)of the module R and the endomorphism algebra End_B(N_AR) of themodule N_AR. If M and N satisfy the property that both (N_A–,M_B–) and (M_B–, N_A–) are adjoint pairs of functors,then so do the modules X and Y. Moreover, we show that the self-injectivedimension and the Gorenstein property are invariant under stableequivalences of Morita type with the above-mentioned adjointproperty.     

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.     

On the Discrepancy for Cartesian Products

Let B₂ denote the family of all circular discs in the plane.It is proved that the discrepancy for the family {B₁ x B₂ :B₁, B₂ B₂} in R⁴ is O(n^1/4+) for an arbitrarily small constant > 0, that is, it is essentially the same as that for thefamily B₂ itself. The result is established for the combinatorialdiscrepancy, and consequently it holds for the discrepancy withrespect to the Lebesgue measure as well. This answers a questionof Beck and Chen. More generally, we prove an upper bound forthe discrepancy for a family {^k_i=1A_i:A_iA_i, i = 1, 2, ..., k},where each A_i is a family in R^d_i, each of whose sets is describedby a bounded number of polynomial inequalities of bounded degree.The resulting discrepancy bound is determined by the ‘worst’of the families A_i, and it depends on the existence of certaindecompositions into constant-complexity cells for arrangementsof surfaces bounding the sets of A_i. The proof uses Beck's partialcoloring method and decomposition techniques developed for therange-searching problem in computational geometry.     

The Glauberman Correspondence and Subgroups of Operator Groups: a Counterexample

Let G and A be finite groups with coprime orders, and supposethat A acts on G by automorphisms. Let (G, A):Irr_A(G)Irr(C_G(A))be the Glauberman–Isaacs correspondence. Let B A andIrr_A(G). We exhibit a counterexample to the conjecture that(G, A) is an irreducible constituent of the restriction of (G,B) to C_G(A). 1991 Mathematics Subject Classification 20C15.     

Diophantine analysis and torsion on elliptic curves

In a recent paper of Bennett and the author, it was shown thatthe elliptic curve defined by y² = x³ + Ax + B, where A andB are integers, has no rational points of finite order if Ais sufficiently large relative to B (at least if one assumesthe abc Conjecture of Masser and Oesterlé). In the presentarticle we show, perhaps surprisingly, that the rational torsionon the above curve is also quite restricted if B is sufficientlylarge relative to A. In particular, we demonstrate that forany > 0 there is a constant c such that if A and B are integerssatisfying |B| > c |A|⁶⁺, then the elliptic curve definedabove has no rational torsion points, other than a possiblepoint of order 2 (again making use of the abc Conjecture insome cases). We then extend this by proving similar resultsfor elliptic curves admitting non-trivial -isogenies, ellipticcurves written in other forms, and elliptic curves over certainnumber fields. Curiously, the results on isogenies lead to twounexpected irrationality measures for certain algebraic numbers.     

An Oscillation Theorem for Self-Adjoint Differential Systems and the Rayleigh Principle for Quadratic Functionals

In this note an oscillation theorem on self-adjoint differentialsystems of the form = Ax + Bu, = (C – C₀)x – A^Tu is obtained,complementing, in particular, results of M. Morse. The applicationof this oscillation result yields the Rayleigh principle forquadratic functionals, respectively, the existence theorem forcorresponding self-adjoint eigenvalue problems, under the centralassumptions that the pair (A, B) is controllable (or identicallynormal) and the triple (A, B, C₀) is strongly observable.     

Residual Finiteness of Quasi-Positive One-Relator Groups

A criterion is given for showing that certain one-relator groupsare residually finite. This is applied to a one-relator groupwith torsion G = a₁,...,a_r|Wⁿ. It is shown that G is residuallyfinite provided that W is outside the commutator subgroup andn is sufficiently large. An important ingredient in the proofis a criterion which implies that a subgroup of a group is malnormal.A graded small-cancellation criterion is developed which detectswhether a map A B between graphs induces a ₁-injection, andwhether ₁A maps to a malnormal subgroup of ₁B.     

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.     

Norm Estimates for Commutators of Operators

Given two self-adjoint Hilbert space operators A, B, and a continuousfunction f, we prove several inequalities of the form ||f(A)X–Xf(B)||C(f)||AX–XB|| involving the L_p-norm of the derivative f' and the Besov B^r_,1-normof f.     

The Marica-Schonheim Inequality in Lattices

The Marica-Schönheim Inequality says that if A is a finitefamily of sets, then |A–||A| where A–A=[A₁\A₂:A₁,A₂A]. For a finite lattice L and AL, we define a–b=(J_a\J_b)where J_a=[jL:ja and j is join-irreducible], and if AL then welet A–A=[a₁–a₂: a₁, a₂A]. Then the analogue of theMarica-Schöonheim Inequality is |A–A|A| for all AL.We prove that this is true if L is distributive or complementedand modular or L is a partition lattice.     

Projective Modules over the Non-Commutative Sphere

The positive cone of the K₀-group of the non-commutative sphereB is explicitly determined by means of the four basic unboundedtrace functionals discovered by Bratteli, Elliott, Evans andKishimoto. The C*-algebra B is the crossed product A x Z₂ ofthe irrational rotation algebra A by the flip automorphism defined on the canonical unitary generators U, V by (U) = U*,(V) = V*, where VU = e²ⁱ UV and is an irrational real number.This result combined with Rieffel's cancellation techniquesis used to show that cancellation holds for all finitely generatedprojective modules over B. Subsequently, these modules are determinedup to isomorphism as finite direct sums of basic modules. Italso follows that two projections p and q in a matrix algebraover B are unitarily equivalent if, and only if, their vectortraces are equal: [p] = [q]. These results will have the following ramifications. They areused (elsewhere) to show that the flip automorphism on A isan inductive limit automorphism with respect to the basic buildingblock construction of Elliott and Evans for the irrational rotationalgebra. This will, in turn, yield a two-tower proof of thefact that B is approximately finite dimensional, first provedby Bratteli and Kishimoto.     

On the Magnitude of the Sum of Bases of the Natural Numbers

We construct two bases of the natural numbers B₁ and B₂, eachof order two, such that (B₁ + B₂ (n) <n^{+c/(log n)}. For alower estimate, it is proved that if B₂ and are two bases, eachof order two, then (B₁+B₂)(n) > n. Generalisations to sumsof bases of order h > 2 are also given.     

A reverse Denjoy theorem

Suppose that C₁ and C₂ are two simple curves joining 0 to ,non-intersecting in the finite plane except at 0 and enclosinga domain D which is such that, for all large r, has measure at most 2, where 0 < < .Suppose also that u is a non-constant subharmonic function inthe plane such that u(z) = B(|z|, u) for all large z C₁ C₂.Let A_D(r, u) = inf { u(z):z D and | z | = r }. It is shownthat if A_D(r, u) = O(1) (or A_D(r, u) = o(B(r, u))), then lim_r B(r, u)/r^/2 > 0 (or lim_r log B(r, u)/log r /2).     

A new methodology for the stability analysis of pairwise triangularizable and related switching systems

We consider the asymptotic stability of the time-varying dynamicsystem : = A(t)x, A(t) R^{n x n}, A(t) A= {A₁, ..., A_m}, where A_i is Hurwitz and where a set of non-singularmatrices T_{i j} exist such that any pair of matrices {T_{i j} A_iT_{i j}^–1, T_{i j} A_j T_{i j}^–1}, i, j {1, ..., m}, areupper triangular. Switching systems of this form are referredto as pairwise triangularizable switching systems. It can beestablished that (a) pairwise triangularizability is not sufficientto guarantee the existence of a common quadratic Lyapunov functionfor the linear time-invariant dynamic systems A_i : = A_i x; (b) additional conditions can be specified which guaranteeasymptotic stability of the switching system . In this paperwe also show that pairwise triangularizability is not even sufficientto guarantee asymptotic stability of the switching system .We also show that the method of proof of stability in (b), whichdoes not assume the existence of a common quadratic Lyapunovfunction, can be used to prove the asymptotic stability of moregeneral switching systems (systems that are not pairwise triangularizable).Finally, we show that our results can be used as the basis forthe design of practical control systems; namely, for the designof an automobile speed switched controller with guaranteed stabilityproperties.     

Reduction of CalderON Commutators

Let A₁,..., A_n be Lipschitz functions on R such that A'₁,...,A'_nVMO. We show that on any bounded interval, the Calderóncommutator associated with the kernel (A₁(x)–A₁(y)) ...(A_n(x) – A_n(y))/(x–y) ⁿ¹ is a compact perturbationof , where H is the Hilberttransform. 1991 Mathematics Subject Classification 47B38, 47B47,47G10, 45E99.     

Fields of Definition for Division Algebras

Let A be a finite-dimensional division algebra containing abase field k in its center F. A is defined over a subfield F₀if there exists an F₀-algebra A₀ such that . The following are shown. (i) In many cases A canbe defined over a rational extension of k. (ii) If A has odddegree n 5, then A is defined over a field F₀ of transcendencedegree 1/2(n–1)(n–2) over k. (iii) If A is a Z/mx Z/2-crossed product for some m 2 (and in particular, if Ais any algebra of degree 4) then A is Brauer equivalent to atensor product of two symbol algebras. Consequently, M_m(A) canbe defined over a field F₀ such that trdeg_k(F₀) 4. (iv) IfA has degree 4 then the trace form of A can be defined overa field F₀ of transcendence degree 4. (In (i), (iii) and (iv)it is assumed that the center of A contains certain roots ofunity.)     

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).     

Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

Let A be an algebra over a field K of characteristic zero andlet ₁, ..., _sDer _K(A) be commuting locally nilpotent K-derivationssuch that _i(x_j) equals _ij, the Kronecker delta, for some elementsx₁, ..., x_sA. A set of generators for the algebra is found explicitly and a set of defining relationsfor the algebra A is described. Similarly, let ₁, ..., _s Aut_K(A)be commuting K-automorphisms of the algebra A is given suchthat the maps _i – id_A are locally nilpotent and _i (x_j)= x_j + _ij, for some elements x₁, ..., x_s A. A set of generatorsfor the algebra A: = {a A | ₁(a) = ... = _s(a) = a} is foundexplicitly and a set of defining relations for the algebra Ais described. In general, even for a finitely generated non-commutativealgebra A the algebras of invariants A and A are not finitelygenerated, not (left or right) Noetherian and a minimal numberof defining relations is infinite. However, for a finitely generatedcommutative algebra A the opposite is always true. The derivations(or automorphisms) just described appear often in many differentsituations (possibly) after localization of the algebra A.     

Picard Groups of Irrational Rotation C*-Algebras

Let be an irrational number in [0, 1] and A the correspondingirrational rotation C*-algebra. Let Aut (A) be the group ofall automorphisms of A and Int (A) the normal subgroup of Aut(A) of all inner automorphisms of A. Let Pic (A) be the Picardgroup of A. In the present note we shall show that if is notquadratic, then Pic (A)Aut (A)/Int (A) and that if is quadratic,then Pic (A) is isomorphic to a semidirect product of Aut (A)/Int(A) with Z. Furthermore, in the last section we shall discussPicard groups of certain Cuntz algebras.     

Optimal backward perturbation bounds for the linear least-squares problem with multiple right-hand sides

Let A be an mn matrix, B be an mr matrix, and $$\stackrel{\&macr;}{X}$$be an approximate solution to the problem of minimizing ||AX–B||_F.In this note we consider the following open problem: find anexplicit expression of the optimal backward perturbation bound_F$${\eta }_{F}\left(\tilde{X}\right)$$ dehned by where is a positive number. This problem is solved when $$\tilde{X}$$is of full column rank.