Justin R. Peters

Semi-crossed Products of C*-Algebras

Abstract Given a C∗-algebra U and endomorphim α, there is an associated nonselfadjoint operator algebra Z + Xα U , called the semi-crossed product of U with α. If α is an automorphim, Z + Xα U can be identified with a subalgebra of the C∗-crossed product Z + Xα U . If U is commutative and α is an automorphim satisfying certain conditions, Z + Xα U is an operator algebra of the type studied by Arveson and Josephson. Suppose S is a locally compact Hausdorff space, φ: S → S is a continuous and proper map, and α is the endomorphim of U=C0(S) given by α(ƒ) = ƒ ō φ. Necessary and sufficient conditions on the map φ are given to insure that the semi-crossed product Z+XαC0(S) is (i) semiprime; (ii) semisimple; (ii) strongly semisimple.

The ideal structure of certain nonselfadjoint operator algebras

Soit (X, φ) un systeme dynamique localement compact, et Z + Xφ Co(X) la sous-algebre close en norme du produit croise ZXφCo(X) engendre par les puissances non negatives de φ dans le cas φ homeomorphisme. On determine la structure ideale de ces algebres dans le cas ou φ agit librement

MORSE DECOMPOSITION, ATTRACTORS AND CHAIN RECURRENCE

The global behavior of a dynamical system can be described by its Morse decompositions or its attractor and repeller configurations. There is a close relation between these two approaches and also with (maximal) chain recurrent sets that describe the system behavior on finest Morse sets. These sets depend upper semicontinuously on parameters. The connection with ergodic theory is provided through the construction of invariant measures based on chains.

Dynamical systems from function algebras

Soit X compact de Hausdorff, Σ les nombres naturels ou entiers, φ:X→X et {φ k :k∈Σ} un (semi)groupe de fonctions continues de X dans X. Etant donne le systeme dynamique (X,φ,Σ), soit #7B-U une C*-algebre Σ-invariante de fonctions bornees contenant C(X). Il y a une extension naturelle (X^,φ^,Σ) de (X,φ,Σ) ou X^ est le spectre de #7B-U et φ^ est donnee par φ^(x^)f=x^(f○φ)

Semicrossed products generated by two commuting automorphisms

Abstract In this paper, we study the semicrossed product of a finite dimensional C ∗ -algebra for two types of Z + 2 -actions, and identify them with matrix algebras of analytic functions in two variables. We look at the connections with semicrossed by Z + -actions.

Compact operators and nest representations of limit algebras

In this paper we study the nest representations p: A → Alg N of a strongly maximal TAF algebra A, whose ranges contain non-zero compact operators. We introduce a particular class of such representations, the essential nest representations, and we show that their kernels coincide with the completely meet irreducible ideals. From this we deduce that there exist enough contractive nest representations, with non-zero compact operators in their range, to separate the points in A. Using nest representation theory, we also give a coordinate-free description of the fundamental groupoid for strongly maximal TAF algebras. For an arbitrary nest representation p: A → Alg N, we show that the presence of non-zero compact operators in the range of p implies that N is similar to a completely atomic nest. If, in addition, p(A) is closed, then every compact operator in p(A) can be approximated by sums of rank one operators p(A). In the case of N-ordered nest representations, we show that p(A) contains finite rank operators iff ker ρ fails to be a prime ideal.

Meet-irreducible ideals and representations of limit algebras

In this paper we give criteria for an ideal J of a TAF algebra A to be meet-irreducible. We show that J is meet-irreducible if and only if the C∗-envelope of A/J is primitive. In that case, A/J admits a faithful nest representation which extends to a ∗-representation of the C∗-envelope for A/J. We also characterize the meet-irreducible ideals as the kernels of bounded nest representations; this settles the question of whether the n-primitive and meet-irreducible ideals coincide.

Correction to: ``Analytic TAF algebras''

REMARK 1.10. Suppose T = 5MT) is a strongly maximal triangular subalgebra of 91 with diagonal 3). For each n, let %n — C*(9In, 2)). Then there exists a clopen subset %> of 4L such that &{%>) = B„. Note that ^ = U %• L ^ Tn = & H % and 2£ = ¥n \ X. Since T is strongly maximal, we have ^ = Ç U X U ( ^ T 1 Now suppose that for each n, we can define a cocycle cn on !^ such that cn(x, y) > 0 if (x,y) G 2£. Suppose also that for each (jc,_y) G ̂ , there is some m such that (x,y) G ^ and cn(x,y) = cn+\(x,y) for all n > m. Then d(x,y) — limn_00 cn(x, v) exists (as a finite number) for every (x, y) G %^,d satisfies the cocycle condition d(x, z) = d(x, y) + d(y, z) of Definition 1.7, and fP = J 1 [0, oo) since fP = (J (Pn. Conversely, if T=% for some cocycle d, then cn = d\^ is a cocycle on ^ such that cn(x, y) > 0 for (x,y) G 2£, and d(x,y) = cw(x, v) for all n > some m since %, = \J %n. Note that if the cns are given and d is defined by d(x, v) = l i n v ^ c„(x, v), then d satisfies the cocycle condition, but in general d need not be continuous. This leads to a revised Theorem 2.2.

On inductive limits of matrix algebras of holomorphic functions

Let W be a UHF algebra and sQ(D) the disk algebra. If W = [U,ll >1 I ]Jand a is a product-type automorphism of W which leaves each 21,, invariant, then a defines an embedding t1 2>(D) n+ 1 D The inductive limit of this system is a Banach algebra whose maximal ideal space is closely related to that of the disk algebra if the Connes spectrum F(a) is finite.

On traceable factor representations of crossed products

Abstract In this paper we give a complete classification of the traceable factor representations of the C ∗ -algebra which is the crossed product of the gauge group of automorphisms with the fermion algebra. Besides the type I representations, this algebra has an uncountable family of type II ∞ traceable factor representations. Unlike the fermion algebra, it has no finite factor representations. We present a similar analysis of the crossed product of SU (2) with the fermion algebra, where the action is the natural product action.