The Proof of Inseparable Prime AW^＊—Algebras Being Primitive C^＊—Algebras

The relation between the inseparable prime C^*-algebras and primitive C^*-algebras is studied,and we prove that prime AW^*-algebras are all primitive C^*-algebras.     

Normality and Embedding of AW*-Algebras

An AW^*-algebra is a W^*-algebra if and only if it is normal andhas a large W^* corner. Analogous results are proved for therepresentation of AW^*-algebras as operator algebras on AW^*-modules.     

On {sigma}-Normal C*-Algebras

A C*-algebra A is said to be monotone (respectively monotone-) complete if every increasing net (respectively increasingsequence) of elements in the ordered space A_h of all hermitianelements of A has a supremum in A_h. It is straightforward toverify that every monotone complete C*-algebra is an AW*-algebra.For type I AW*-algebras, the converse is known to be true. However,for general AW*-algebras, this question is still open, althoughan impressive attack on the problem was made by E. Christensenand G. K. Pedersen, who showed that properly infinite AW*-algebrasare monotone -complete [4]. 1991 Mathematics Subject Classification46L05, 46L06.     

When does continuity on the center imply continuity?

LetA be a Banach algebra. We give a condition forA which forces a homomorphism fromA into a Banach algebra to be continuous if the closure of its continuity ideal has finite codimension, and if its restriction to the center ofA is continuous. We apply this result to answer the question in the title for centralC ^*-algebras,AW ^*-algebras, andL ¹ (G) for certain [FIA]^?-groupsG.     

Quotient-bounded elements in locally convex algebras. II

Consideration of quotient-bounded elements in a locally convexGB ^*-algebra leads to the study of properGB ^*-algebras viz those that admit nontrivial quotient-bounded elements. The construction and structure of such algebras are discussed. A representation theorem for a properGB ^*-algebra representing it as an algebra of unbounded Hilbert space operators is obtained in a form that unifies the well-known Gelfand-Naimark representation theorem forC ^*-algebra and two other representation theorems forb ^*-algebras (also calledlmc ^*-algebras), one representinga b ^*-algebra as an algebra of quotient bounded operators and the other as a weakly unbounded operator algebra. A number of examples are discussed to illustrate quotient-bounded operators. An algebra of unbounded operators constructed out of noncommutativeL ^p-spaces on a regular probability gauge space and the convolution algebra of periodic distributions are analyzed in detail; whereas unbounded Hilbert algebras andL ^w-integral of a measurable field ofC ^*-algebras are discussed briefly.     

The angular derivative of Fréchet-holomorphic maps in J*-algebras and complex Hilbert spaces

We study the problem of the existence of the angular derivative of Fréchet-holomorphic maps of the generalized right half-planes defined by non-zero partial isometries in infinite dimensional complex Banach spaces called J^*-algebras. These generalized right half-planes and open unit balls (which are bounded symmetric homogeneous domains) are holomorphically equivalent. The results obtained here also hold in C ^*-algebras, JC^*-algebras, B ^*-algebras and ternary algebras, containing non-zero partial isometries, and in complex Hilbert spaces. Some examples are given. The principal tool we use are general results of the Pick-Julia type in J^*-algebras.     

Representations and positive functionals

We discuss the representation theory of both the locally convex and non-locally convex topological^*-algebras. First we discuss the^*-representation of topological^*-algebras by operators on a Hilbert space. Then we study those topological^*-algebras so that every^*-representation of which on a Hilbert space is necessarily continuous. It is well-known that each^*-representation of aB ^*-algebra on a Hilbert space is continuous. We show that this is true for a large class of^*-algebras more general thanB ^*-algebras, including certain non-locally convex^*-algebras. Finally, we investigate the conditions under which a positive functional on a topological^*-algebra is representable.The research of the first-named author was partially supported by an NSERC grant. This work was done by the second-named author when he was a post-doctoral fellow at McMaster University.     

Nuclear C*-algebras and related questions

Results are given characterizing the class of nuclear C^*-algebras from various points of view, and a number of consequences of the nuclearity condition are given, and properties of C^*-algebras and W^*-algebras related to nuclearity are discussed.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 26, pp. 107–126, 1985.     

On the Banach space isomorphism type of AF C*-algebras and their triangular subalgebras

It is shown that all the approximately finite dimensional C^*-algebras which are not of Type I are isomorphic as Banach spaces. This generalizes the matroid case given previously by Arazy. Analogous results are obtained for various families of triangular subalgebras of AF C^*-algebras. In addition the classification of various continua of Type I AF C^*-algebras is discussed.     

Noncommutative differential geometry,quantization, and smooth symmetries of the C*-algebras associated to topological dynamics

Noncommutative differential geometric structures are considered for a class of simple C^*-algebras. This structure is defined in terms of smooth Lie group actions on the C^*-algebra in question together with a certain quantization mapping motivated directly by the known cohomological obstructions for the quantum mechanical quantization correspondence. We show that such a quantization mapping may be constructed for the C^*-algebras associated to antisymmetric bi-characters and for the Cuntz/Cuntz-Krieger C^*-algebras associated to topological dynamics. A certain curvature obstruction is defined in terms of the quantization mapping. It is shown that existence of smooth Lie group actions is determined by the curvature obstruction.     

Tensor Products of Hilbert Modules Over Locally <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

In this paper the tensor products of Hilbert modules over locally C ^*-algebras are defined and their properties are studied. Thus we show that most of the basic properties of the tensor products of Hilbert C ^*-modules are also valid in the context of Hilbert modules over locally C ^*-algebras.     

Multivariable Wiener-Hopf operators II. Spectral topology and solvability

The C^*-algebras generated by Wiener-Hopf Operators on tangible closed cones are solvable: they have finite composition series of ideals with subquotients which are homogeneous C^*-algebras with continuous trace. In particular this refers to cones with piece-wise smooth boundary and to cones which are finite sets of orbits under some linear action.     

Decomposition rank and \mathcal{Z} -stability

We show that separable, simple, nonelementary, unital C^*-algebras with finite decomposition rank absorb the Jiang–Su algebra Z\mathcal{Z} tensorially. This has a number of consequences for Elliott’s program to classify nuclear C^*-algebras by their K-theory data. In particular, it completes the classification of C^*-algebras associated to uniquely ergodic, smooth, minimal dynamical systems by their ordered K-groups.     

TheK-theory of AF embeddings of the rational rotation algebras

Our main result is the construction of an embedding ofC(T²) into an approximately finite-dimensionalC ^*-algebra which induces an injection onK ₀(C(T²)). The existence of this embedding implies that Cech cohomology cannot be extended to a stable, continuous homology theory forC ^*-algebras which admits a well-behaved Chern character. Homotopy properties ofC ^*-algebras are also considered. For example, we show that the second homotopy functor forC ^*-algebras is discontinuous. Similar embeddings are constructed for all the rational rotation algebras, with the consequence that none of the rational rotation algebras satisfies the homotopy property called semiprojectivity.     

On<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-algebras whose Glimm ideals are primitive

Every Glimm ideal of anAW ^*-algebra is a (minimal) primitive ideal.     

The Riccati algebraic equation in a locallyC ⋆-algebra

The stabilizability and complete stabilizability of a pair of elements and the structure of the set of selfadjoint solutions of the algebraic Riccati equation is studied inC ^*-algebras and locallyC ^*-algebras which includes certain algebras of unbounded operators. Under some mild assumptions, selfadjoint solutions are in one-to-one correspondence with a particular set of idempotents.     

NonstableK-Theory for operator algebras

We show that the homotopy groups of the group of quasi-unitaries inC ^*-algebras form a homology theory on the category of allC ^*-algebras which becomes topologicalK-theory when stabilized. We then show how this functorial setting, in particular the half-exactness of the involved functors, helps to calculate the homotopy groups of the group of unitaries in a series ofC ^*-algebras. The calculations include the case of all AbelianC ^*-algebras and allC ^*-algebras of the formAB, whereA is one of the Cuntz algebras O_n n=2, 3, ..., an infinite dimensional simpleAF-algebra, the stable multiplier or corona algebra of a-unitalC ^*-algebra, a properly infinite von Neumann algebra, or one of the projectionless simpleC ^*-algebras constructed by Blackadar.     

Hilbert C*-Modules over Monotone Complete C*-Algebras

The aim of the present paper is to describe self-duality and C*-reflexivity of Hilbert A-modules ?? over monotone complete C*-algebras A by the completeness of the unit ball of ?? with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results ofH. Widom [Duke Math. J. 23, 309-324, MR 17 # 1228] and W. L. Paschke [Trans. Amer. Mat. Soc. 182 , 443-468, MR 50 # 8087, Canadian J. Math. 26, 1272-1280, MR 57 # 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. M. Ozawa [J. Math. Soc. Japan 36, 589-609, MR 85 # 46068]). Especially, one derives that for a C*-algebra A the A-valued inner product of every Hilbert A-module ?? can be continued to an A-valued inner product on it's A-dual Banach A-module ??' turning ??' to a self-dual Hilbert A-module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. Hamana [Internat. J. Math. 3 (1992), 185 - 204]. A classification of countably generated self-dual Hilbert A-modules over monotone complete C*-algebras A is established. The set of all bounded module operators End ′(??) on self-dual Hilbert A-modules ?? over monotone complete C*-algebras A is proved again to be a monotone complete C*-algebra. Applying these results a Weyl-Berg type theorem is proved.     

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.