On torsion subgroups of whitehead groups of division algebras

We completely determine all torsion abelian groups that can occur as the torsion subgroup of the Whitehead group of a division algebra of prime index. More precisely, we prove that if D is a division algebra of prime index, then the torsion subgroup of K ₁(D) is locally cyclic. Conversely, if A is a torsion locally cyclic group, then there exists a division algebra D of prime index such that the torsion subgroup of K ₁(D) is isomorphic to A. Our result can be considered as a non-commutative version of May’s Theorem.     

Traces on simple AF C1-algebras

The relations between the set of traces on a simple approximately finite dimensional C¹-algebra A and the algebraic and geometric properties of the Elliott dimension group K₀(A) are studied. It is shown that every metrizable Choquet simplex occurs as the set of normalized traces of a simple unital AF algebra. A simple AF algebra can have both finite and infinite traces, so a finite simple AF algebra need not be algebraically simple. It is shown that a simple AF algebra is algebraically simple if and only if it has no infinite traces, and is stable if and only if it has no finite traces.     

Hyper-Tauberian algebras and weak amenability of Figà-Talamanca-Herz algebras

We study certain commutative regular semisimple Banach algebras which we call hyper-Tauberian algebras. We first show that they form a subclass of weakly amenable Tauberian algebras. Then we investigate the basic and hereditary properties of them. Moreover, we show that if A is a hyper-Tauberian algebra, then the linear space of bounded derivations from A into any Banach A-bimodule is reflexive. We apply these results to the Figà-Talamanca-Herz algebra A_p(G) of a locally compact group G for p∈(1,∞). We show that A_p(G) is hyper-Tauberian if the principal component of G is abelian. Finally, by considering the quantization of these results, we show that for any locally compact group G, A_p(G), equipped with an appropriate operator space structure, is a quantized hyper-Tauberian algebra. This, in particular, implies that A_p(G) is operator weakly amenable.     

The ordinary quivers of Hochschild extension algebras for self-injective Nakayama algebras

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule Hom_K(A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the ?-graded second Hochschild homology group HH₂(A) in the sense of Sköldberg.     

Algebras Whose Coxeter Polynomials are Products of Cyclotomic Polynomials

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is basic connected with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ? _A as the automorphism of the Grothendieck group K ₀(A) induced by the Auslander-Reiten translation τ in the derived category Der(modA) of the module category modA of finite dimensional left A-modules. We say that A is an algebra of cyclotomic type if the characteristic polynomial χ _A of ? _A is a product of cyclotomic polynomials. There are many examples of algebras of cyclotomic type in the representaton theory literature: hereditary algebras of Dynkin and extended Dynkin types, canonical algebras, some supercanonical and extended canonical algebras. Among other results, we show that: (a) algebras satisfying the fractional Calabi-Yau property have periodic Coxeter transformation and are, therefore, of cyclotomic type, and (b) algebras whose homological form h _A is non-negative are of cyclotomic type. For an algebra A of cyclotomic type we describe the shape of the Auslander-Reiten components of Der(modA).     

The Derived Picard Group is a Locally Algebraic Group

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic _K (A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic _K (A) is a locally algebraic group, and its identity component is Out⁰ _K (A). If B is a derived Morita equivalent algebra then DPic _K (A)DPic _K (B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.     

An elementary construction of tilting complexes

Let A be an artin algebra and e∈A an idempotent with add(eA_A)=add(D(_AAe)). Then a projective resolution of Ae_eAe gives rise to tilting complexes for A, where P(l)^• is of term length l+1. In particular, if A is self-injective, then is self-injective and has the same Nakayama permutation as A. In case A is a finite dimensional algebra over a field and eAe is a Nakayama algebra, a projective resolution of eAe over the enveloping algebra of eAe gives rise to two-sided tilting complexes {T(2l)^•}_l?1 for A, where T(2l)^• is of term length 2l+1. In particular, if eAe is of Loewy length two, then we get tilting complexes {T(l)^•}_l?1 for A, where T(l)^• is of term length l+1.     

Factorization properties of subdiagonal algebras

Let M be a von Neumann algebra equipped with a normal finite faithful normalized trace τ, and let A be a tracial subalgebra of M. Let E be a symmetric quasi-Banach space on [0, 1]. We show that A has an L_E(M)-factorization if and only if A is a subdiagonal algebra.     

On the Homological Dimension of Algebras of Differential Operators

Let A be a commutative algebra over a field k, and V_A be the k-subalgebra of End_k(A) generated by End_A(A) = A and all k-derivations of A. A study of the homological properties of V_A was initiated by Hochschild, Kostant, and Rosenberg in [5], and continued by Rinehart [8], [9], Roos [11], Björk [1], Rinehart and Rosenberg [10], and others. It was proved in [5] that, if k is perfect and A is a regular affine algebra of dimension r, then the global dimension of V_A is between r and 2r. Moreover, if k has positive characteristic, then gl.dim V_A = 2r [8]. By a recent celebrated theorem of Roos [11], gl.dim V_A = r if k has characteristic zero and A = k[x₁, …, x_r]; in this case V_A is the so-called “Weyl algebra on 2r variables”.     

Continuous fields of C*-algebras over finite dimensional spaces

Let X be a finite dimensional compact metrizable space. We study a technique which employs semiprojectivity as a tool to produce approximations of C(X)-algebras by C(X)-subalgebras with controlled complexity. The following applications are given. All unital separable continuous fields of C*-algebras over X with fibers isomorphic to a fixed Cuntz algebra On, n∈{2,3,…,∞}, are locally trivial. They are trivial if n=2 or n=∞. For n?3 finite, such a field is trivial if and only if (n−1)[A₁]=0 in K₀(A), where A is the C*-algebra of continuous sections of the field. We give a complete list of the Kirchberg algebras D satisfying the UCT and having finitely generated K-theory groups for which every unital separable continuous field over X with fibers isomorphic to D is automatically locally trivial or trivial. In a more general context, we show that a separable unital continuous field over X with fibers isomorphic to a KK-semiprojective Kirchberg C*-algebra is trivial if and only if it satisfies a K-theoretical Fell type condition.     

Invariant bilinear forms on a vertex algebra

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric. This is a generalization of the result of Li (J. Pure Appl. Algebra 96(3) (1994) 279), who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components.As an application, we introduce a notion of a radical of a vertex algebra. We prove that a radical-free vertex algebra A is non-negatively graded, and its component A₀ of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A₀-linear. We also show that in this case A is simple if and only if A₀ is a field.     

Characterizations of isomorphisms and derivations of some algebras

Let ? be a zero-product preserving bijective bounded linear map from a unital algebra A onto a unital algebra B such that ?(1)=k. We show that if A is a CSL algebra on a Hilbert space or a J-lattice algebra on a Banach space then there exists an isomorphism ψ from A onto B such that ?=kψ. For a nest algebra A in a factor von Neumann algebra, we characterize the linear maps on A such that δ(x)y+xδ(y)=0 for all x,y∈A with xy=0.     

Absolutely indecomposable symmetric matrices

Let A be a symmetric matrix of size n×n with entries in some (commutative) field K. We study the possibility of decomposing A into two blocks by conjugation by an orthogonal matrix T∈Mat_n(K). We say that A is absolutely indecomposable if it is indecomposable over every extension of the base field. If K is formally real then every symmetric matrix A diagonalizes orthogonally over the real closure of K. Assume that K is a not formally real and of level s. We prove that in Mat_n(K) there exist symmetric, absolutely indecomposable matrices iff n is congruent to 0, 1 or −1 modulo 2s.     

Derivations on the algebra of operators in hilbert C*-modules

Let M be a full Hilbert C*-module over a C*-algebra A,and let End*A(M) be the algebra of adjointable operators on M.We show that if A is unital and commutative,then every derivation of End A(M) is an inner derivation,and that if A is σ-unital and commutative,then innerness of derivations on "compact" operators completely decides innerness of derivations on End*A(M).If A is unital(no commutativity is assumed) such that every derivation of A is inner,then it is proved that every derivation of End*A(Ln(A)) is also inner,where Ln(A) denotes the direct sum of n copies of A.In addition,in case A is unital,commutative and there exist x0,y0 ∈ M such that x0,y0 = 1,we characterize the linear A-module homomorphisms on End*A(M) which behave like derivations when acting on zero products.     

An additivity formula for the strict global dimension of C(Ω)

Let A be a unital strict Banach algebra, and let K ₊ be the one-point compactification of a discrete topological space K. Denote by the weak tensor product of the algebra A and C(K ₊), the algebra of continuous functions on K ₊. We prove that if K has sufficiently large cardinality (depending on A), then the strict global dimension is equal to .     

On absolute valued algebras with involution

Let A be an absolute valued algebra with involution, in the sense of Urbanik [K. Urbanik, Absolute valued algebras with an involution, Fund. Math. 49 (1961) 247-258]. We prove that A is finite-dimensional if and only if the algebra obtained by symmetrizing the product of A is simple, if and only if eA_s = A_s, where e denotes the unique nonzero self-adjoint idempotent of A, and A_s stands for the set of all skew elements of A. We determine the idempotents of A, and show that A is the linear hull of the set of its idempotents if and only if A is equal to either McClay’s algebra [A.A. Albert, A note of correction, Bull. Amer. Math. Soc. 55 (1949) 1191], the para-quaternion algebra, or the para-octonion algebra. We also prove that, if A is infinite-dimensional, then it can be enlarged to an absolute valued algebra with involution having a nonzero idempotent different from the unique nonzero self-adjoint idempotent.     

Taylor series and divisibility

LetA be an augmentedK-algebra; defineT:A →A ?k _kA byT(a)=1?a ?a?1,a ∈A. We prove, under some conditions, thatg is in the subalgebraK[f] ofA generated byf if and only ifT(g) is in the principal ideal generated byT(f) inA?k _kA. WhenA=K[[X]],T(f) is a multiple ofT(X) if and only iff belongs to the ringL obtained by localizingK[X] at (X).     

Nonlinear eigenvalue problems with positively convex operators

We consider the equation u = λAu (λ > 0), where A is a forced isotone positively convex operator in a partially ordered normed space with a complete positive cone K. Let Λ be the set of positive λ for which the equation has a solution u?K, and let Λ₀ be the set of positive λ for which a positive solution—necessarily the minimum one—can be obtained by an iteration u_n = λAu_n?1, u₀ = 0. We show that if K is normal, and if Λ is nonempty, then Λ₀ is nonempty, and each set Λ₀, Λ is an interval with $\text{inf}\text{(Λ}{}_0\text{) =}\text{inf}\text{(Λ) = 0}\text{and sup}\text{(Λ}{}_0\text{) =}\text{sup}\text{(Λ) (= λ1,}\text{say}\text{)}$; but we may have λ1 ? Λ₀ and λ1 ? Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ₀ is nonempty. Let u⁰(λ) = lim_n→∞(λA)ⁿ(0) be the minimum positive fixed point corresponding to λ ? Λ₀. Then u⁰(λ) is a continuous isotone convex function of λ on Λ₀.     

Asymptotic expansions of the ordered spectrum of symmetric matrices

In this work, we build on ideas of Torki (2001 [6]) and show that if a symmetric matrix-valued map t?A(t) has a one-sided asymptotic expansion at t=0⁺ of order K then so does t?λ_m(A(t)), where λ_m is the mth largest eigenvalue. We derive formulas for computing the coefficients A₀,A₁,…,A_K in the asymptotic expansion. As an application of the approach we give a new proof of a classical result due to Kato (1976 [3]) about the one-sided analyticity of the ordered spectrum under analytic perturbations. Finally, as a demonstration of the derived formulas, we compute the first three terms in the asymptotic expansion of λ_m(A+tE) for any fixed symmetric matrices A and E.