On the structure of positive homomorphisms on algebras of real-valued continuous functions

In this paper we study the structure of positive homomorphisms on real function algebras. We prove that every positive homomorphism is completely characterized by a family of sets and when the algebra is inverse-closed, by an ultrafilter of zero-sets of functions of the algebra. We show that the known sufficient conditions for every homomorphism of a real function algebra to be countably evaluating or a point evaluation are not necessary. Our results enable us to characterize the countably evaluating algebras as well as the Lindelöf spaces as the spaces in which for every algebra, each countably evaluating homomorphism is a point evaluation.     

On homomorphisms between products of median algebras

Homomorphisms of products of median algebras are studied with particular attention to the case when the codomain is a tree. In particular, we show that all mappings from a product \({\mathbf{A_1} \times\ldots\times {\mathbf{A}_{n}}}\) of median algebras to a median algebra \({\mathbf{B}}\) are essentially unary whenever the codomain \({\mathbf{B}}\) is a tree. In view of this result, we also characterize trees as median algebras and semilattices by relaxing the defining conditions of conservative median algebras.     

On certain homomorphisms of quotients of group algebras

LetA be the algebra of functions on the circle groupT={z: |z|=1} having absolutely convergent Fourier series. For a subsetE ofA, the algebra of restrictions toE of functions inA is denoted byA(E). It is shown that for sets that are “thick” in one of several senses, algebra homomorphisms of theA(E) having norm 1 must arise from point mappings having a certain amount of rigidity.     

On certain homomorphisms of restriction algebras of symmetric sets

LetG be a locally compact abelian group and Γ its dual group. For any closedH ⊆G denote the algebra of restrictions toH of Fourier transforms of functions inL ¹(Γ) byA(H). This paper considers certain Cantor like sets inR and ΠZ _m(j) and gives some necessary algebraic criterion fornatural isomorphisms of their restriction algebras. This work was supported mainly by the U.S. National Science Foundation Graduate Fellowship Program. The author wishes to thank Paul Cohen, Karel de Leeuw, and Yitzhak Katznelson for their counsel.     

Endomorphisms and homomorphisms of Heyting algebras

For a non-trivial Heyting algebraH, the cardinality of its endomorphism monoid is always at least two. If it is exactly two thenH is called 0-endomorphism rigid. It is shown that there exists a proper class of non-isomorphic 0-endomorphism rigid Heyting algebras. This is a consequence of a more general result: the variety of Heyting algebras is 0-universal.The support of NSERC is gratefully acknowledged.Presented by J. Berman.     

On interpolation of Banach algebras and factorization of weakly compact homomorphisms

We show a necessary and sufficient condition on the lattice Γ for the general real method to preserve the Banach-algebra structure. As an application we derive factorization of weakly compact homomorphisms through interpolation properties of weakly compact operators.     

Denominators in cluster algebras of affine type

The Fomin–Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for the denominators of cluster variables in cluster algebras of affine type. The formulas are in terms of the dimensions of spaces of homomorphisms in the corresponding cluster category, and hold for any choice of initial cluster.     

On a category of cluster algebras

We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and provide combinatorial methods for constructing special classes of monomorphisms and epimorphisms. In the case of cluster algebras from surfaces, we describe interactions between this category and the geometry of the surfaces.     

Homology,homomorphisms and presentations of commutative algebras

We obtain sufficient conditions for that a subalgebra of a commutative algebra be a polynomial algebra.     

On Schur algebras and related algebras V: some quasi-hereditary algebras of finite type

In a recent paper, Erdmann determined which of the Schur algebras S(n, r) have finite representation type and described the finite type Schur algebras up to Morita equivalence. The present paper grew out of a desire to see Erdmann's results in the more general context of algebras which are quasi-hereditary in the sense of Cline et al. (1988). Weconsider here the class of quasi-hereditary algebras which have a duality fixing simples. This class includes the “generalized Schur algebras” defined and studied by the first author, and the Schur algebras themselves in particular. In the first part we describe the possible Morita types of the quasi-hereditary algebras of finite representation type over an algebraically closed field with duality fixing simples. This is then applied, in the second part, to give the block theoretic refinement of Erdmann's results.     

Tame concealed algebras and cluster quivers of minimal infinite type

The well-known list of Happel-Vossieck of tame concealed algebras in terms of quivers with relations, and the list of A. Seven of minimal infinite cluster quivers are compared. There is a 1-1 correspondence between the items in these lists, and we explain how an item in one list naturally corresponds to an item in the other list. A central tool for understanding this correspondence is the theory of cluster-tilted algebras.     

On the number of essential arguments of homomorphisms between products of median algebras

In this paper we characterize classes of median-homomorphisms between products of median algebras, that depend on a given number of arguments, by means of necessary and sufficent conditions that rely on the underlying algebraic and on the underlying order structure of median algebras. In particular, we show that a median-homomorphism that take values in a median algebra that does not contain a subalgebra isomorphic to the m-dimensional Boolean algebra as a subalgebra cannot depend on more than \(m-1\) arguments. In view of this result, we also characterize the latter class of median algebras. We also discuss extensions of our framework on homomorphisms over median algebras to wider classes of algebras.     

Decomposition rank and absorbing extensions of type I algebras

We prove the following: Let A and B be separable C^*-algebras. Suppose that B is a type I C^*-algebra such that

(i)

B has only infinite dimensional irreducible *-representations, and

(ii)

B has finite decomposition rank.

If

0→B→C→A→0