Quasi-diagonal C1-algebras

We consider perturbations of C¹-algebras by compact operators. We show that if A is a separable liminal algebra of operators on a separable Hilbert space, then it is a subalgebra of a compact perturbation of a block diagonal algebra.     

C1-algebras with finite duals

A forest is a finite partially ordered set F such that for x, y, z?F with x ? z, y ? z one has x ? y or y ? x. In this paper we give a complete characterization of all separable C¹-algebras $\text{A}$ with a finite dual $\text{A}\text{?}$, for which Prim $\text{A}$ is a forest with inclusion as partial order. These results are extended to certain separable C¹-algebras $\text{A}$ with a countable dual$\text{A}\text{?}$. As an example these results are used to characterize completely all separable C¹-algebras $\text{A}$ with a three point dual.     

Semi-crossed products of C1-algebras

Given a C¹-algebra $\text{U}$ and endomorphim α, there is an associated nonselfadjoint operator algebra $\text{Z}$⁺ X_α$\text{U}$, called the semi-crossed product of $\text{U}$ with α. If α is an automorphim, $\text{Z}$⁺ X_α$\text{U}$ can be identified with a subalgebra of the C¹-crossed product $\text{Z}$⁺ X_α$\text{U}$. If $\text{U}$ is commutative and α is an automorphim satisfying certain conditions, $\text{Z}$⁺ X_α$\text{U}$ 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(?) = ? ō φ. Necessary and sufficient conditions on the map φ are given to insure that the semi-crossed product Z⁺X_αC₀(S) is (i) semiprime; (ii) semisimple; (ii) strongly semisimple.     

Derivations of non-separable C1-algebras

The question of which C¹-algebras have only inner derivations has been considered by a number of authors for 25 years. The separable case is completely solved, so this paper deals only with the non-separable case. In particular, we show that the C¹-tensor product of a von Neumann algebra and an abelian C¹-algebra has only inner derivations. Other special types of C¹-algebras are shown to have only inner derivations as well such as the C¹-tensor product of L(H) (all bounded operators on separable Hilbert space) and any separable C¹-algebra having only inner derivations. Derivations from a smaller C¹-algebra into a larger one are also considered, and this concept is generalized to include derivations between C¹-algebras connected by a ¹-homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C¹-algebra which converges to zero (in norm) when restricted to any abelian C¹-subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L(H), but unknown in general.     

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.     

Equivalence and traces on C1-algebras

We introduce an equivalence relation among the positive elements in a C¹ and show that the algebra is (semi-) finite if and only if there is a separating family of (semi-) finite traces. Concentrating on simple, semi-finite C¹-algebras we relate geometrical properties in the cone of equivalence classes to functional analytic properties of the algebra, such as the number of normalized traces and their possible values on a given element. The paper may be considered as an attempt to extend Murray and von Neumann's type and equivalence theory to C¹-algebras.     

Functions and derivations of C1-algebras

It is shown that there is a closed symmetric derivation δ of a C¹-algebra with dense domain $\text{D}$(δ), an element A = A¹ ?$\text{D}$(δ), and a C¹-function $\text{f}$ such that $\text{f}$(A)?$\text{D}$(δ). Some estimates are derived for $\text{∥ δ(¦ A ¦)∥}\text{and}\text{∥ δ(A}{}_{\mathrm{+}}{}^{\mathrm{\alpha }}\text{)∥}$, where 0 < α < 1. It is shown that there exists a family of one-one self-adjoint operators S(t) in $\text{L}$(H) which depends linearly on t, while $\text{¦ S(t)¦}$ is not differentiable. It is also shown that there exists $\text{L}$(H) which is not C¹-self-adjoint even though it satisfies $\text{∥}\text{exp}\text{(itT)∥ ? C(1 + ¦ t ¦)}$ for all t ? $\text{R}$     

Pure states of simple C1-algebras

Pure states of simple C¹-algebras with identity are studied. We prove that pure states of such algebras have a product decomposition property, and that two pure states are unitarily equivalent if and only if they are asymptotically equal.     

Completely contractive factorizations of C1-algebras

It is shown that a C¹-algebra is nuclear if and only if the identity map can be approximated in the point norm topology by complete contractions factoring through matrix algebras.     

A family of simple C1-algebras related to weighted shift operators

In this paper we study a family of C¹-algebras which occurs naturally in the study of C¹-algebras generated by weighted shifts. We show that these algebras are simple modulo the compacts, and while they share many of the properties of uniformly hyperfinite C¹-algebras, they are not approximately finite dimensional.     

Extremal traces on some group-invariant C1-algebras

Let $\text{U}$ be a UHF-algebra of Glimm type n^∞, and {α_g: g?G} a strongly continuous group of ¹-automorphisms of product type on $\text{U}$, for G compact. Let $\text{U}$^α be the C¹-subalgebra of fixed elements of $\text{U}$. We show that any extremal normalized trace on $\text{U}$^α arises as the restriction of a symmetric product state ? on $\text{U}$ of the form ? = ?_k?1 ω. As an example we classify the extremal traces on $\text{U}$^α for the case G = SU(n), α_g = ?_{k ? 1} Ad(g).     

Transformation group C1-algebras with continuous trace

We obtain several results characterizing when transformation group C¹-algebras have continuous trace. These results can be stated most succinctly when ($\text{G, Ω}$) is second countable, and the stability groups are contained in a fixed abelian subgroup. In this case, $\text{C}{}^1\text{(G, Ω)}$ has continuous trace if and only if the stability groups vary continuously on Ω and compact subsets of Ω are wandering in an appropriate sense. In general, we must assume that the stability groups vary continuously, and if ($\text{G, Ω}$) is not second countable, that the natural maps of $\text{G}\text{S}{}_{\mathrm{x}}$ onto G · x are homeomorphisms for each x. Then $\text{C}{}^1\text{(G, Ω)}$ has continuous trace if and only if compact subsets of Ω are wandering and an additional C¹-algebra, constructed from the stability groups and Ω, has continuous trace.     

Scattered C1-algebras with almost finite spectrum

The paper gives a complete classification of all separable C¹-algebras with countable spectrum, for which each point in the spectrum has a finite neighbourhood.     

On tensor products of certain group C1-algebras

If A and B are C¹-algebras there is, in general, a multiplicity of C¹-norms on their algebraic tensor product A ⊙ B, including maximal and minimal norms ν and α, respectively. A is said to be nuclear if α and ν coincide, for arbitrary B. The earliest example, due to Takesaki [11], of a nonnuclear C¹-algebra was $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, the C¹-algebra generated by the left regular representation of the free group on two generators F₂. It is shown here that W¹-algebras, with the exception of certain finite type I's, are nonnuclear.If $\text{C}{}^1\text{(F}{}_2\text{)}$ is the group C¹-algebra of F₂, there is a canonical homomorphism λ_l of $\text{C}{}^1\text{(F}{}_2\text{)}$ onto $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$. The principal result of this paper is that there is a norm ζ on $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, distinct from α, relative to which the homomorphism $\text{λ ⊙ λ}{}_{\mathrm{l}}\text{: C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{) → C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$ is bounded ($\text{C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{)}$ being endowed with the norm α). Thus quotients do not, in general, respect the norm α; a consequence of this is that the set of ideals of the α-tensor product of C¹-algebras A and B may properly contain the set of product ideals {$\text{I ? B + A ? J: I ? A, J ? B}$}.Let A and B be C¹-algebras. If A or B is a W¹-algebra there are on A ⊙ B certain C¹-norms, defined recently by Effros and Lance [3], the definitions of which take account of normality. In the final section of the paper it is shown by example that these norms, with α and ν, can be mutually distinct.     

Noncommutative topological dynamics and compact actions on C1-algebras

The classical notions of topological transitivity and minimality of a topological dynamical system are extended and analyzed in the context of C¹-dynamical systems. These notions are compared with other notions naturally arising in noncommutative ergodic theory. As an application, a C¹-algebra version of a theorem of Araki, Haag, Kastler, and Takesaki (Comm. Math. Phys.53 (1977), 97–134) about the correspondence between a compact automorphism group (here assumed to be abelian) and its fixed-point subalgebra is proved in the presence of a commuting topologically transitive action. A variation of this theorem in the setting of standard W¹-inclusions is also presented.     

Automatic relative boundedness of derivations in C1-algebras

We show that a derivation of a C¹-algebra $\text{A}$ is automatically relative bounded with respect to any closed ¹derivation of $\text{A}$ with smaller domain. We give also some related results on the automatic continuity of derivations in certain Banach algebras.