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We establish the Hao–Ng isomorphism for generalized gauge actions of locally compact abelian groups on product systems over abelian lattice orders and we then use it to explore Takai duality in this context. As an application we generalize some recent work of Schafhauser.

We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis,Inter. Math. Res. Not.2014(2014), 1289–1311 relating to work of Arveson,Acta Math.118(1967), 95–109 from the 1960s, and extends related work of Kakariadis and Katsoulis,J. Noncommut. Geom.8(2014), 771–787.

In an earlier work, the authors proposed a non-selfadjoint approach to the Hao–Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of $\mathrm{C}^*$-correspondences, each one of these conjectures is equivalent to the Hao–Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C$^*$-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of $\mathrm{C}^*$-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana’s injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao–Ng isomorphism for the reduced crossed product and all hyperrigid $\mathrm{C}^*$-correspondences. A culmination of these results is the resolution of the Hao–Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg, and Spielberg.

Let ( G , Σ ) be a (partially) ordered abelian group with Haar measure μ, let ( A , G , α ) be a dynamical system and let A ⋊ α Σ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism A ⋊ α Σ ∼ s ( A ⊗ K ( G , Σ , μ ) ) ⋊ α ⊗ Ad ρ G , where K ( G , Σ , μ ) denotes the compact operators in the CSL algebra Alg L ( G , Σ , μ ) and ρ denotes the right regular representation of G . We also show that there exists a complete lattice isomorphism between the α ˆ -invariant ideals of A ⋊ α Σ and the ( α ⊗ Ad ρ ) -invariant ideals of A ⊗ K ( G , Σ , μ ) . Using Takai duality we also continue our study of the Radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors. Among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra. We also show that the permanence of semisimplicity fails for crossed products by R . A final section of the paper is devoted to the study of radically tight dynamical systems, i.e., dynamical systems ( A , G , α ) for which the identity Rad ( A ⋊ α G ) = ( Rad A ) ⋊ α G persists. A broad class of such dynamical systems is identified.

A higher rank numerical semigroup is a positive cone whose seminormalization is isomorphic to the free abelian semigroup. The corresponding nonselfadjoint semigroup algebras are known to provide examples that answer Arveson's Dilation Problem to the negative. Here we show that these algebras share the polydisc as the character space in a canonical way. We subsequently use this feature in order to identify higher rank numerical semigroups from the corresponding nonselfadjoint algebras., Comment: 10 pages, minor corrections

Let ( G , P ) be an abelian, lattice ordered group and let X be a compactly aligned product system over P with coefficients in A . We show that the C*-envelope of the Nica tensor algebra N T X + coincides with both Sehnem's covariance algebra A × X P and the co-universal C ⁎ -algebra N O X r for injective, gauge-compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of N O X r , thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C ⁎ -envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C ⁎ -algebras of product systems. This generalizes recent results that were obtained by various authors in the case where ( G , P ) = ( Z , N ) .

Given a C ⁎ -correspondence X, we give necessary and sufficient conditions for the tensor algebra T X + to be hyperrigid. In the case where X is coming from a topological graph we obtain a complete characterization.

This paper is an expanded version of the lectures I delivered at the Indian Statistical Institute, Bangalore, during the OTOA 2014 conference.

We study several notions of shift equivalence for C ⁎ -correspondences and the effect that these equivalences have on the corresponding Pimsner dilations. Among others, we prove that non-degenerate, regular, full C ⁎ -correspondences which are shift equivalent have strong Morita equivalent Pimsner dilations. We also establish that the converse may not be true. These results settle open problems in the literature. In the context of C ⁎ -algebras, we prove that if two non-degenerate, regular, full C ⁎ -correspondences are shift equivalent, then their corresponding Cuntz–Pimsner algebras are strong Morita equivalent. This generalizes results of Cuntz and Krieger and Muhly, Tomforde and Pask. As a consequence, if two subshifts of finite type are eventually conjugate, then their Cuntz–Krieger algebras are strong Morita equivalent. Our results suggest a natural analogue of the Shift Equivalence Problem in the context of C ⁎ -correspondences. Even though we do not resolve the general Shift Equivalence Problem, we obtain a positive answer for the class of imprimitivity bimodules.

A triangular limit algebra A is isometrically isomorphic to the tensor algebra of a C*-correspondence if and only if its fundamental relation R(A) is a tree admitting a $Z^+_0$-valued continuous and coherent cocycle. For triangular limit algebras which are isomorphic to tensor algebras, we give a very concrete description for their defining C*-correspondence and we show that it forms a complete invariant for isometric isomorphisms between such algebras. A related class of operator algebras is also classified using a variant of the Aho-Hopcroft-Ullman algorithm from computer aided graph theory., Comment: 25 pages

We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz-Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong $\Delta$-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a $\sigma$-TRO. As an application we show that strong Morita equivalence coincides with strong $\Delta$-equivalence for tensor algebras of aperiodic C*-correspondences., Comment: 20 pages, minor corrections, additional comments in subsection 2.4

Using non-selfadjoint techniques, we establish the Hao-Ng isomorphism for the reduced crossed product and all discrete groups. For the full crossed product an analogous result holds for all discrete groups but the C*-correspondences involved have to be hyperrigid. These results are obtained by calculating the C*-envelope of the reduced crossed product of an operator algebra by a discrete group., Comment: 15 pages, final version to appear in International Mathematics Research Notices

Let A be a unital operator algebra and let α be an automorphism of A that extends to a ⁎-automorphism of its C ⁎ -envelope C env ⁎ ( A ) . We introduce the isometric semicrossed product A × α is Z + and we show that C env ⁎ ( A × α is Z + ) ≃ C env ⁎ ( A ) × α Z . In contrast, the C ⁎ -envelope of the familiar contractive semicrossed product A × α Z + may not equal C env ⁎ ( A ) × α Z . Our main tool for calculating C ⁎ -envelopes for semicrossed products is a new concept, the relative semicrossed product of an operator algebra by an endomorphism. As an application of our theory, we show that if T X + is the tensor algebra of a C ⁎ -correspondence ( X , A ) and α is a ⁎-extendible automorphism of T X + that fixes the diagonal elementwise, then the contractive semicrossed product satisfies C env ⁎ ( T X + × α Z + ) ≃ O X × α Z , where O X denotes the Cuntz–Pimsner algebra of ( X , A ) . This extends the main result of Davidson and Katsoulis (2010) [6] .

In this paper we study the nest representations ρ : A ⟶ Alg ⁡ N \rho : \mathcal {A} \longrightarrow \operatorname {Alg} \mathcal {N} of a strongly maximal TAF algebra A \mathcal {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 \mathcal {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 ρ : A ⟶ Alg ⁡ N \rho : \mathcal {A} \longrightarrow \operatorname {Alg} \mathcal {N} , we show that the presence of non-zero compact operators in the range of ρ \rho implies that N \mathcal {N} is similar to a completely atomic nest. If, in addition, ρ ( A ) \rho (\mathcal {A} ) is closed, then every compact operator in ρ ( A ) \rho (\mathcal {A} ) can be approximated by sums of rank one operators ρ ( A ) \rho (\mathcal {A} ) . In the case of N \mathbb {N} -ordered nest representations, we show that ρ ( A ) \rho ( \mathcal {A}) contains finite rank operators iff ker ⁡ ρ \ker \rho fails to be a prime ideal.

Using representation theory techniques we prove that various spaces of derivations or one-sided multipliers over certain operator algebras are reflexive. A sample result: any bounded local derivation (local left multiplier) on an automorphic semicrossed product is a derivation (resp. left multiplier). In the process we obtain various results of independent interest. In particular, the finite dimensional nest representations of the tensor algebra of a topological graph separate points., Comment: The paper will appear in the Transactions of the AMS. Several typos corrected. 23 pages

We show that the C ∗ -envelope of the tensor algebra of an arbitrary C ∗ -correspondence X coincides with the Cuntz–Pimsner algebra O X , as defined by Katsura [T. Katsura, On C ∗ -algebras associated with C ∗ -correspondences, J. Funct. Anal. 217 (2004) 366–401]. This improves earlier results of Muhly and Solel [P.S. Muhly, B. Solel, Tensor algebras over C ∗ -correspondences: Representations, dilations and C ∗ -envelopes, J. Funct. Anal. 158 (1998) 389–457] and Fowler, Muhly and Raeburn [N. Fowler, P. Muhly, I. Raeburn, Representations of Cuntz–Pimsner algebras, Indiana Univ. Math. J. 52 (2003) 569–605], who came to the same conclusion under the additional hypothesis that X is strict and faithful.

This paper is a comprehensive study of the nest representations for the free semigroupoid algebra ${\mathfrak{L}}_G$ of a countable directed graph $G$ as well as its norm-closed counterpart, the tensor algebra ${\mathcal{T}}^{+}(G)$.We prove that the finite-dimensional nest representations separate the points in ${\mathfrak{L}}_G$, and a fortiori, in ${\mathcal{T}}^{+}(G)$. The irreducible finite-dimensional representations separate the points in ${\mathfrak{L}}_G$ if and only if $G$ is transitive in components (which is equivalent to being semisimple). Also the upper triangular nest representations separate points if and only if for every vertex $x \in {\mathcal{T}}(G)$ supporting a cycle, $x$ also supports at least one loop edge.We also study faithful nest representations. We prove that ${\mathfrak{L}}_G$ (or ${\mathcal{T}}^{+}(G)$) admits a faithful irreducible representation if and only if $G$ is strongly transitive as a directed graph. More generally, we obtain a condition on $G$ which is equivalent to the existence of a faithful nest representation. We also give a condition that determines the existence of a faithful nest representation for a maximal type ${\mathbb{N}}$ nest.

Given an arbitrary countable directed graph G we prove the C*- envelope of the tensor algebra $$\mathcal{T}_ + (G)$$ coincides with the universal Cuntz- Krieger algebra associated with G. Our approach is concrete in nature and does not rely on Hilbert module machinery. We show how our results extend to the case of higher rank graphs, where an analogous result is obtained for the tensor algebra of a row-finite k-graph with no sources.

Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of operators on Hilbert space which are related to the graph in a natural way. This yields a functional calculus determined by quiver algebras and free semigroupoid algebras. We establish a generalization of Coburn's theorem for the C*-algebra of a CKT family, and prove a universality theorem for C*-algebras generated by these families. In both cases, the C*-algebras generated by quiver algebras play the universal role.

We investigate the relationship between the facial structure of the unit ball of an operator algebra A and its algebraic structure, including the hereditary subalgebras and the socle of A. Many questions about the facial structure of A are studied with the aid of representation theory. For that purpose we establish the existence of reduced atomic type representations for certain nonselfadjoint operator algebras. Our results are applicable to C*-algebras, strongly maximal TAF algebras, free semigroup algebras and various semicrossed products.

Given countable directed graphs G and G′, we show that the associated tensor algebras (G) and (G′) are isomorphic as Banach algebras if and only if the graphs G are G′ are isomorphic. For tensor algebras associated with graphs having no sinks or no sources, the graph forms an invariant for algebraic isomorphisms. We also show that given countable directed graphs G, G′, the free semigroupoid algebras and are isomorphic as dual algebras if and only if the graphs G are G′ are isomorphic. In particular, spatially isomorphic free semigroupoid algebras are unitarily isomorphic. For free semigroupoid algebras associated with locally finite directed graphs with no sinks, the graph forms an invariant for algebraic isomorphisms as well.

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.

We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider the consequences of this result. In particular, we give partial solutions to a conjecture and an open problem by Power. As a further consequence, we describe epimorphisms between various classes of limit algebras. In this paper, we study automatic continuity for limit algebras. Automatic continuity involves algebraic conditions on a linear operator from one Banach algebra into another that guarantee the norm continuity of the operator. This is a generalization, via the open mapping theorem, of the uniqueness of norms problem. Recall that a Banach algebra A is said to have a unique (Banach algebra) topology if any two complete algebra norms on A are equivalent, so that the norm topology determined by a Banach algebra is unique. Uniqueness of norms, automatic continuity, and related questions have played an important and long-standing role in the theory of Banach algebras [6, 29, 28, 10, 11, 2]. Limit algebras, whose theory has grown rapidly in recent years, are the nonselfadjoint analogues of UHF and AF C-algebras. We first prove that algebraic isomorphisms between limit algebras are automatically continuous (Theorem 1.4). This proof uses the ideal theory of limit algebras as well as key results from the theory of automatic continuity for Banach algebras. Combining this with [23, Theorem 8.3] verifies Power's conjecture that the C-envelope of a limit algebra is an invariant for purely algebraic isomorphisms, for limits of finite dimensional nest algebras, and in particular, for all triangular limit algebras (Corollary 1.6). In [5], the first two authors studied triangular limit algebras in terms of their lattices of ideals. By combining automatic continuity with this work, we show that within the class of algebras generated by their order preserving normalizers (see below for definitions), algebraically isomorphic algebras are isometrically isomorphic (Theorem 2.5). This shows that the spectrum, or fundamental relation [20], a topological binary relation which provides coordinates for limit algebras and is a useful tool in classifications, is a complete algebraic isomorphism invariant for this class (Corollary 2.6). In recent work, the second two authors studied primitivity for limit algebras [9], showing that a variety of limit algebras are primitive. These results, together with automatic continuity, give descriptions of epimorphisms between various classes of limit algebras, namely lexicographic algebras (Theorem 3.2) and Z-analytic algebras (Theorem 3.3). Received by the editors April 6, 1998 and, in revised form, October 7, 1999. 2000 Mathematics Subject Classification. Primary 47D25, 46K50, 46H40. Research partially supported by an NSF grant. ()2000 American Mathematical Society

The main result of this paper is Theorem 5, which provides a necessary and sufficient condition on a positive operator A A for the existence of an operator B B in the nest algebra A l g N AlgN of a nest N N satisfying A = B B ∗ A=BB^{*} (resp. A = B ∗ B ) A=B^{*}B) . In Section 3 we give a new proof of a result of Power concerning outer factorisation of operators. We also show that a positive operator A A has the property that there exists for every nest N N an operator B N B_N in A l g N AlgN satisfying A = B N B N ∗ A=B_NB_N^{*} (resp. A = B N ∗ B N A=B_N^{*}B_N ) if and only if A A is a Fredholm operator. In Section 4 we show that for a given operator A A in B ( H ) B(H) there exists an operator B B in A l g N AlgN satisfying A A ∗ = B B ∗ AA^{*}=BB^{*} if and only if the range r ( A ) r(A) of A A is equal to the range of some operator in A l g N AlgN . We also determine the algebraic structure of the set of ranges of operators in A l g N AlgN . Let F r ( N ) F_r(N) be the set of positive operators A A for which there exists an operator B B in A l g N AlgN satisfying A = B B ∗ A=BB^{*} . In Section 5 we obtain information about this set. In particular we discuss the following question: Assume A A and B B are positive operators such that A ≤ B A\leq B and A A belongs to F r ( N ) F_r(N) . Which further conditions permit us to conclude that B B belongs to F r ( N ) F_r(N) ?

We examine the strongly extreme point structure of the unit balls of triangular UHF algebras. The semisimple triangular UHF algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal. In contrast to this, for the class of full nest algebras we prove a Krein-Milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points. Results concerning the geometry of the unit ball have a long history both in Banach space theory and in the theory of operator algebras. This geometry can be affiliated with structural and algebraic properties. Moreover, differences in geometric properties can prove useful in classification problems. Two fundamental results are the Russo-Dye Theorem and Kadison’s Theorem on isometries. More recently, there has been interest in the unit balls of nonselfadjoint operator algebras, especially nest algebras [1, 2, 3, 4, 15, 16]. This paper concerns the unit balls of triangular UHF algebras. These and the larger class of triangular AF algebras are nonselfadjoint analogues of the UHF and AF C*-algebras studied by Glimm and Bratteli. Their theory has grown rapidly, cf. [8, 9, 18, 19, 20]. We focus on the extreme point structure, and our results have a different flavor than those for nest algebras. Specifically, we study the strongly extreme points, those boundary points whose “stable character” with respect to approximations makes them behave well under direct limits. Triangular UHF algebras are direct limits of full upper triangular matrix algebras. Unit balls embed into unit balls in the direct limit scheme, and some types of embeddings respect the extreme point structure while others do not. This leads to structural differences in the limit algebras. The geometric structures of the unit balls of different triangular UHF algebras can be very dissimilar. The convex hull of the strongly extreme points, even without closure, always contains the unit ball of the diagonal. Theorem 7 shows that the two coincide if and only if the algebra is semisimple. This is a characterization of a purely geometric property in terms of a purely algebraic one. In contrast to Received by the editors January 11, 1996 and, in revised form, March 28, 1996. 1991 Mathematics Subject Classification. Primary 47D25, 46K50, 46B20.

In this paper we survey our recent work on C*-correspondences and their associated operator algebras; in particular, on adding tails, the Shift Equivalence Problem and Hilbert bimodules.

One of the well-known results in the theory of C* algebras is the Russo-Dye Theorem [19]: given a C* algebra .~r the closed convex hull of the unitary elements in ~r equals the closed unit ball of ~r This result was later refined by Gardner and reached its final form by Kadison and Pedersen; today it is known that every operator in a C* algebra ~r whose norm is less than 1, is the average of unitaries from A. The Russo-Dye Theorem initiated the theory of unitary rank in selfadjoint operator algebras. If ~r is an operator algebra, the unitary rank of an element A E ~r is defined as the smallest number for which there is a convex combination of unitaries from ~r of length u(A) and equaling A. If no such decomposition exists (in particular if liAII > 1) we define u(A) = oo. The literature on unitary rank is vast. The earliest result is due to Murray and yon Neumann who proved that any selfadjoint operator of norm I or less is the mean of two unitary operators ([12] p. 239, 1937). The first systematic study was given by R. Kadison and G. Pedersen [8] in 1984 (previous work in the field included contributions by Popa [15], Robertson [17], Gardner [6] and others). In 1986, C. Olsen and G. Pedersen [14] characterized all elements in a factor von Neumarm algebra with finite unitary rank. In the general case of a C*-algebra, a characterization was obtained by Rordam in his important paper [18]. For more details and further information on the theory of unitary rank we refer to the excellent articles of U. Haagerup [7] and M. Rordam [ 18]. In the first section of the present paper, we prove a Russo-Dye type Theorem for infinite multiplicity nest algebras. The techniques employed in the proof of our result are different from that of Gardner and Kadison-Pedersen. To our knowledge, this is the first result of this type, for non-selfadjoint operator algebras and clearly initiates the unitary rank theory for such algebras.

The complete lattice generated by a totally atomic CSL ℒ and the projection lattice of a von Neumann algebra ℛ, commuting with ℒ, is reflexive. From this it follows that the strongly closed lattice generated by any CSL ℒ and the projection lattice of a properly infinite von Neumann algebra ℛ, commuting with ℒ, is reflexive.

Given a C ∗ C^\ast -algebra A \mathcal {A} and an element A ∈ A A\in \mathcal {A} , we give necessary and sufficient geometric conditions equivalent to the existence of a representation ( ϕ , H ) (\phi ,\mathcal {H}) of A \mathcal {A} so that ϕ ( A ) \phi (A) is a compact or a finite-rank operator. The implications of these criteria on the geometric structure of C ∗ C^\ast -algebras are also discussed.

Given a nest subalgebra of a von Neumann algebra A and vectors f , g we give necessary and sufficient conditions for the existence of an operator A in A so that Af = g . This generalizes a well-known theorem of E. C. Lance and the main result in [Anoussis (1992); Katsoulis et al. (1993)]. We also use the generalized Ringrose criterion to show that the similarity problem for continuous nests in certain crossed product factors of type III has a negative answer.

To a given multivariable C*-dynamical system $(A, \al)$ consisting of *-automorphisms, we associate a family of operator algebras $\alg(A, \al)$, which includes as specific examples the tensor algebra and the semicrossed product. It is shown that if two such operator algebras $\alg(A, \al)$ and $\alg(B, \be)$ are isometrically isomorphic, then the induced dynamical systems $(\hat{A}, \hat{\al})$ and $(\hat{B}, \hat{\be})$ on the Fell spectra are piecewise conjugate, in the sense of Davidson and Katsoulis. In the course of proving the above theorem we obtain several results of independent interest. If $\alg(A, \al)$ and $\alg(B, \be)$ are isometrically isomorphic, then the associated correspondences $X_{(A, \al)}$ and $X_{(B, \be)}$ are unitarily equivalent. In particular, the tensor algebras are isometrically isomorphic if and only if the associated correspondences are unitarily equivalent. Furthermore, isomorphism of semicrossed products implies isomorphism of the associated tensor algebras. In the case of multivariable systems acting on C*-algebras with trivial center, unitary equivalence of the associated correspondences reduces to outer conjugacy of the systems. This provides a complete invariant for isometric isomorphisms between semicrossed products as well., Comment: 16 pages; changes in the Introduction and in Theorem 5.2

In this note we show that several CSL algebras, including Alg Y(200

Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equation Txt = yt, for i = 1, 2,, n. In this article, we continue the investigation of the one-vector interpolation problem for nest algebras that was begun by Lance. In particular, we require the interpolating operator to belong to certain ideals which have proved to be of importance in the study of nest algebras, namely, the compact operators, the radical, Larson's ideal, and certain other ideals. We obtain necessary and sufficient conditions for interpolation in each of these cases.

A family of algebras, which we call topological conjugacy algebras, is associated with each proper continuous map on a locally compact Hausdorff space. Assume that $\eta_i:\X_i\to \X_i$ is a continuous proper map on a locally compact Hausdorff space $\X_i$, for $i = 1,2$. We show that the dynamical systems $(\X_1, \eta_1)$ and $(\X_2, \eta_2)$ are conjugate if and only if some topological conjugacy algebra of $(\X_1, \eta_1)$ is isomorphic as an algebra to some topological conjugacy algebra of $(\X_2, \eta_2)$. This implies as a corollary the complete classification of the semicrossed products $C_0(\X) \times_{\eta} \bbZ^{+}$, which was previously considered by Arveson and Josephson, Peters, Hadwin and Hoover and Power. We also obtain a complete classification of all semicrossed products of the form $A(\bbD) \times_{\eta}\bbZ^{+}$, where $A(\bbD)$ denotes the disc algebra and $\eta: \bbD \to \bbD$ a continuous map which is analytic on the interior. In this case, a surprising dichotomy appears in the classification scheme, which depends on the fixed point set of $\eta$. We also classify more general semicrossed products of uniform algebras., Comment: 25 pages. Accepted for publication in Crelle's Journal

Every operator in the algebra of a continuous nest X can be factored as a product of two operators which belong to certain diagonal disjoint ideals of Alg X . This factorization leads to a new description of nest algebras. The main result of this paper is that every operator in the algebra of a continuous nest can be factored as a product of two operators which belong to certain diagonal disjoint ideals of that algebra. As a consequence, a new description of nest algebras is given, in terms of factorization criteria. Let Y be a nest (i.e., a totally ordered complete set of projections containing 0 and I) on a separable Hilbert space P . We denote by Alg Y the algebra of all operators in B(,i) that leave invariant every element of Y. Also, by Y' we will denote the diagonal of Alg Y, i.e., the set of all operators in Alg Y such that their adjoints also belong to Alg YV. Finally, we call Y" to be the core of Alg IV. Ideals of nest algebras and in particular ideals which are diagonal disjoint have been studied by a number of authors [4, 5, 9]. For example, the Jacobson Radical M. of Alg Y is a diagonal disjoint two-sided ideal of Alg Y and one of the main results of [9] is a characterization of this ideal. Let us give some definitions. We shall call a (not necessarily finite) set {P,a E -X, v C N} of intervals P = M N, M , N, E V, of the nest Y, a partition of Y if the intervals are pairwise orthogonal and the sum E,c P = I in the strong operator topology. Given a nest Y, Larson 's ideal M' is the collection of all operators X in Alg Y for which, given E > 0, there exists a partition {Pk a E -} of Y such that IIP XP 11 < E for all a E v If we restrict all partitions to be finite sets of intervals we will obtain the Jacobson Radical MV of Alg Y [9]. Since, however, infinite partitions can arise in general, M71 will usually contain M. The ideal M70 plays an important role in the theory of nest algebras and their similarities [6]. Recently, J. Orr [1 1] proved that, for an arbitrary nest Y, M70 is the greatest diagonal disjoint ideal of Alg Y. One of the key lemmas in the proof of the above result Received by the editors June 19, 1989. 1980 Mathernatics Subject Classification (1985 Revision). Primary 47D25. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page

Let $(\A, \alpha)$ and $(\B, \beta)$ be C*-dynamical systems and assume that $\A$ is a separable simple C*-algebra and that $\alpha$ and $\beta$ are *-automorphisms. Then the semicrossed products $\A \times_{\alpha} \bbZ^{+}$ and $\B \times_{\beta} \bbZ^{+}$ are isometrically isomorphic if and only if the dynamical systems $(\A, \alpha)$ and $(\B, \beta)$ are outer conjugate., Comment: 12 pages, accepted for publication in Math. Ann

If (χ, ∥ ∥) is a Banach space and S is a subset of the closed unit ball χ1 of χ, then the contractive perturbations of S is the set $$ cp(S) = \left\{ {x \in \chi \left| {\left\| {\chi \pm s} \right\| \leqslant 1,\forall s \in S} \right.} \right\}. $$ The second contractive perturbations of S are defined as the set cp(2)(S)=cp(cp(S)).

We prove that a large class of triangular UHF algebras are primitive. We use two avenues to obtain our results: a direct approach in which we explicitly construct a faithful, algebraically irreducible representation of the algebra on a separable Hilbert space, as well as an indirect, algebraic approach which utilizes the prime ideal structure of the algebra. Using these results, we completely characterize the primitive ideal spaces of all lexicographic algebras: An ideal there is primitive if and only if it is closed prime. Specializing on the algebrasA( Q , ν), we obtain a complete classification of their algebraic isomorphisms and epimorphisms through the use of a new invariant involving the primitive ideal space. Finally, we characterize the primitive ideal spaces of Z -analytic and order-preserving algebras, and obtain information about their epimorphisms.