{$I\sp{\ast} $}-algebras and their applications

Transactions of the American Mathematical Society, 1970

Bulletin of the American Mathematical Society, 1968

arXiv (Cornell University), 2017

We present a classification theorem for separable amenable simple stably projectionless C *-algebras with generalized tracial rank one whose K 0 vanish on traces which satisfy the Universal Coefficient Theorem. One of C *-algebras in the class is denoted by Z 0 which has a unique tracial state, K 0 (Z 0) = Z and K 1 (Z 0) = {0}. Let A and B be two separable simple C *-algebras satisfying the UCT and have finite nuclear dimension. We show that A ⊗ Z 0 ∼ = B ⊗Z 0 if and only if Ell(A⊗Z 0) = Ell(B ⊗Z 0). A class of simple separable C *-algebras which are approximately sub-homogeneous whose spectra having bounded dimension is shown to exhaust all possible Elliott invariant for C *-algebras of the form A⊗ Z 0 , where A is any finite separable simple amenable C *-algebras. Suppose that A and B are two finite separable simple C *-algebras with finite nuclear dimension satisfying the UCT such that traces vanishe on K 0 (A) and K 0 (B) (but arbitrary K 1). One consequence of the main results in this situation is that A ∼ = B if and only if A and B have the isomorphic Elliott invariant. Recently some sweeping progresses have been made in the Elliott program ([13]), the program of classification of separable amenable C *-algebras by the Elliott invariant (a K-theoretical set of invariant) (see [24], [64] and [17]). These are the results of decades of work by many mathematicians (see also [24], [64] and [17] for the historical discussion there). These progresses could be summarized briefly as the following: Two unital finite separable simple C *-algebras A and B with finite nuclear dimension which satisfy the UCT are isomorphic if and only if their Elliott invariant Ell(A) and Ell(B) are isomorphic. Moreover, all weakly unperforated Elliott invariant can be achieved by a finite separable simple C *-algebras in UCT class with finite nuclear dimension (In fact these can be constructed as so-called ASH-algebras-see [24]). Combining with the previous classification of purely infinite simple C *-algebras, results of Kirchberg and Phillips ([50] and [30]), now all unital separable simple C *-algebras in the UCT class with finite nuclear dimension are classified by the Elliott invariant. This research studies the non-unital cases. Suppose that A is a separable simple C *-algebra. In the case that K 0 (A) + = {0}, then A ⊗ K has a non-zero projection, say p. Then p(A ⊗ K)p is unital. Therefore if A is in the UCT class and has finite nuclear dimension, then p(A ⊗ K)p falls into the class of C *-algebras which has been classified. Therefore isomorphism theorem for these C *-algebras is an immediate consequence of that in [24] (see section 8.4 of [44]) using the stable isomorphism theorem of [4]. Therefore this paper considers the case that K 0 (A) + = {0}. Simple C *-algebras with K 0 (A) + = {0} are stably projectionless in the sense that not only A has no non-zero projections but M n (A) also has no non-zero projections for every integer n ≥ 1. However, as one may see in this paper, K 0 (A) could still exhaust any countable abelian groups as well as any possible K 1 (A). In particular, the results in [24] cannot be applied in the stably projectionless case. It is entirely new situation. If one views C *-algebra theory as the study of non-commutative topological spaces, then unital C *-algebras correspond to the compact spaces and non-unital ones correspond to non-compact spaces. However, stably projectionless simple C *-algebras may be viewed a 1 , 0). Let R(α, 1) = {(f, a) ∈ C([0, 1], Q ⊗ Q) ⊕ Q : f (0) = q α (a) and f (1) = a ⊗ 1 Q }. Note that an element (f, a) is full in R(α, 1) if and only if a = 0 and f (t) = 0 for all t ∈ (0, 1). Let a α = (f, 1) be defined as follows. Let f (t) = (1 − t)(1 ⊗ e α) + t(1 ⊗ 1) for all t ∈ (0, 1). (e 3.16) Note that a α is a strictly positive element of R(α, 1), morover, for any 1/2 > η > 0, f η (a α) is full. C *-algebra R(α, 1) and a α will appear frequently in this paper.

1975

The purpose of this paper is to give some complements to the various extremal decompositions of states on aC*-dynamical system i.e. a pair (A, G) whereA is aC*-algebra andG is a group acting on aA by *-automorphisms. We shall see for instance that the method of decomposition associated with a maximal abelianW*-algebra does not give all the extremal measures in the general case. We also give the explicit form of the greatest lower bound of all the extremal measures and a certain form of continuity of the decomposition. Finally we characterize various systems in the literature (G-abelian algebras, large systems and quasi-large systems) in terms of the equivalence of different notions of ergodicity.

2002

We study the local operator space structure of nuclear $C^*$-algebras. It is shown that a $C^*$-algebra is nuclear if and only if it is an $\OL_{\infty, \la}$ space for some (and actually for every) $\la > 6$. The $\OL_\infty$ constant $\lambda$ provides an interesting invariant \[ \OL_\infty (\A) = \inf\{\la: ~ \A ~{\rm is ~ an} ~ {\OL}_{\infty, \la} ~ {\rm space}\} \] for nuclear $C^*$-algebras. Indeed, if $\A$ is a nuclear $C^*$-algebra, then we have $1\le \OL_\infty (\A) \le 6$, and if $\A$ is a unital nuclear $C^*$-algebra with $\OL_{\infty} (\A) \le (\frac {1+{\sqrt 5}}2)^{\frac 12}$, we show that $\A$ must be stably finite. We also investigate the connection between the rigid $\OL_{\infty, 1^+}$ structure and the rigid complete order $\OL_{\infty, 1^+}$ structure on $C^*$-algebras, where the latter structure has been studied by Blackadar and Kirchberg in their characterization of strong NF $C^*$-algebras. Another main result of this paper is to show that these two local st...

In arXiv:1212.2613, we associated a presheaf \Sigma^A with each unital C*-algebra A. The spectral presheaf \Sigma^A generalises the Gelfand spectrum of an abelian unital C*-algebra. In the present article, we consider one-parameter groups of automorphisms of the spectral presheaf, in particular those arising from one-parameter groups of inner automorphisms of the algebra. We interpret the spectral presheaf as a (generalised) state space for a quantum system and show how one can use flows on the spectral presheaf and on associated structures to describe the time evolution of non-relativistic quantum systems, both in the Schr\"odinger picture and the Heisenberg picture.

Proceedings of the American Mathematical Society, 1996

We obtain some results on the unique extension of (factor) states of C ∗ C^* -algebras which complement various existing results. Our results also lead to a class of C ∗ C^* -algebras whose states are σ \sigma -convex sums of factor states.

Journal of Functional Analysis, 1998

In this article, examples are given to prove that the graded scaled ordered K-group is not the complete invariant for a C*-algebra in the class of unital separable nuclear C*-algebras of real rank zero and stable rank one, even for a C*-algebra in the subclass which consists of those real rank zero, stable rank one C*-algebras being expressed as inductive limits of k n i=1 M [n, i] (C(X n, i)), where X n, i are two-dimensional finite CW complexes and [n, i] are positive integers. (In the case of simple such C*-algebras, it has been proved that the above invariant is the complete invariant by George Elliott and the author.) These examples prove that the classification conjecture of Elliott for the case of non simple real rank zero C*-algebras should be revised one needs extra invariants. The obstruction preventing two such C*-algebras with the same graded scaled ordered K-group from being isomorphic is that they have different unsuspended E-equivalence types (a refinement of KK-equivalence type of C*-algebras due to Connes and Higson). In this article, it is proved that for the above class of inductive limit C*-algebras, the obstruction of unsuspended E-equivalence type is the only obstruction (i.e., if two C*-algebras in the class are unsuspended E-equivalence, then they are isomorphic). It is a surprise that in the case of simple such C*-algebras, or even the case of C*-algebras with finitely many ideals, the obstruction will disappear (see Section 4).

Transactions of the American Mathematical Society, 1976

Reviews in Mathematical Physics, 1999

Let [Formula: see text] be a partial *-algebra endowed with a topology τ that makes it into a locally convex topological vector space [Formula: see text]. Then [Formula: see text] is called a topological partial *-algebra if it satisfies a number of conditions, which all amount to require that the topology τ fits with the multiplier structure of [Formula: see text]. Besides the obvious cases of topological quasi *-algebras and CQ*-algebras, we examine several classes of potential topological partial *-algebras, either function spaces (lattices of Lp spaces on [0, 1] or on ℝ, amalgam spaces), or partial *-algebras of operators (operators on a partial inner product space, O*-algebras).