C *-algebras generated by partial isometries

We study the interconnection between directed graphs and operators on a Hilbert space. The intuition supporting this link is the following feature shared by partial isometries (as operators on a Hilbert space) on the one hand and edges in directed graphs on the other. A partial isometry a is an operator in a Hilbert space H, i.e., a:H→H which maps a (closed) subspace in H isometrically onto a generally different subspace. The respective subspaces are called the initial space and the final space of a. Denoting the corresponding (orthogonal) projections by p _i and p _f , note that a partial isometry a may be thought of as an edge from one vertex to another (which are not necessarily distinct) in a directed graph. And conversely, every directed graph has such a representation. Since neither the partial isometries nor the directed edges in a fixed model allow unrestricted composition, the algebraic construct which is useful is that of a groupoid. In this paper we develop this as a representation theory, and we explore the connection between realizations in the context of C ^*-algebras. The building blocks in our theory are certain matricial C ^*-algebras which we define. We then prove how they serve to localize our global representations.     

Radon-Nikodým theorem for biweights and regular biweights on partial *-algebras

The first purpose of this paper is to investigate Radon-Nikodym theorem for biweights on partial *-algebra. Secondly, we study regularity of biweights on partial *-algebraA and show that a biweightϕ onA is decomposed intoϕ=ϕ _r+ϕ _s, whereϕ _r is a regular biweight onA andϕ _s is a singular biweight onA.     

AW*-algebras Which are Enveloping C*-algebras of JC-algebras

In the given article, enveloping C*-algebras of AJW-algebras are considered. Conditions are given, when the enveloping C*-algebra of an AJW-algebra is an AW*-algebra, and corresponding theorems are proved. In particular, we proved that if $\mathcal{A}$ is a real AW*-algebra, $\mathcal{A}_{sa}$ is the JC-algebra of all self-adjoint elements of $\mathcal{A}$ , $\mathcal{A}+i\mathcal{A}$ is an AW*-algebra and $\mathcal{A}\cap i\mathcal{A} = \{0\}$ then the enveloping C*-algebra $C^*(\mathcal{A}_{sa})$ of the JC-algebra $\mathcal{A}_{sa}$ is an AW*-algebra. Moreover, if $\mathcal{A}+i\mathcal{A}$ does not have nonzero direct summands of type I₂, then $C^*(\mathcal{A}_{sa})$ coincides with the algebra $\mathcal{A}+i\mathcal{A}$ , i.e. $C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}$ .     

A lemma on the approximation of finite-dimensional*-algebras

One proves an analog of Glimm's lemma [Trans. Am. Math. Soc.,95, No. 2, 318–340 (1960)] by replacing the uniform norm by the trace norm A=tr(AA^*)^1/2; if the elements of a finite type factor satisfy (up to ) the relations which have to be satisfied by the matrix units of a finite-dimensional*-algebraA, then in their -neighborhoods there exist operators which satisfy exactly these relations and depends only on the algebraic type ofA and of.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 175–178, 1974.     

Symmetric topological*-algebras

V. Pták's inequality is valid for every hermitian completeQ locallym-convex (:l.m.c.) algebra. Every algebra of the last kind is, in particular, symmetric. Besides, a (Hausdorff) locallyC ^*-algebra (being always symmetric) with the propertyQ is, within a topological algebraic isomorphism, aC ^*-algebra. Furthermore, a type of Raikov's criterion for symmetry is also valid for non-normed topological^*-algebras. Concerning topological tensor products, one gets that symmetry of the-completed tensor product of two unital Fréchet l.m.c.^*-algebrasE, F ( denotes the projective tensorial topology) is always passed toE, F, while the converse occurs when moreover either ofE, F is commutative.     

Haar measures on HopfC *-algebras

In this paper, we first use Markov-Kakutani's fixed point theorem to prove the existence and uniqueness of Haar measures on cocommutative HopfC ^*-algebras. Also we show that in the commutative case, there exists a natural one-to-one correspondence between the Haar measure on a given HopfC ^*-algebra and Haar measures on the associated semigroup. Finally, we show that for HopfC ^*-algebras with Peter-Weyl property, they have Haar measures.Work supported in part by the NSF.     

Positive functionals on BP*-algebras

A new class of locally convex algebras, called BP^*-algebras, is introduced. It is shown that this class properly includes MQ^*-algebras which were introduced and studied by the first author andR. Rigelhof [10]. Among other results, it is proved that each positive functional on a BP^*-algebraA is admissible but not necessarily continuous as shown by an example. However, ifA, in addition, is either (i) a Q-algebra, or (ii) has an identity and is barrelled, or (iii)A is endowed with the inductive limit topology, then each positive functional onA is continuous.This work was supported by an N.R.C. Grant.     

Ring C*-algebras

We associate reduced and full C*-algebras to arbitrary rings and study the inner structure of these ring C*-algebras. As a result, we obtain conditions for them to be purely infinite and simple. We also discuss several examples. Originally, our motivation comes from algebraic number theory.     

Singly generatedC *-algebras

We consider aC ^*-algebraA generated byk self-adjoint elements. We prove that, for , the algebraM _n (A) is singly generated, i.e., generated by one non-self-adjoint element. We present an example of algebraA for which the property thatM _n (A) is singly generated implies the relation . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1136–1141, August, 1999.     

Irreducible * representations of realC*-algebras

In this paper, we show that a topologically irreducible * representation of a realC*-algebra is also algebraically irreducible. Moreover, the properties of pure real states on a realC*-algebra and their left kernels are discussed.Partially supported by the National Natural Science Foundation of China.