Active lattices determine AW*-algebras

We prove that operator algebras that have enough projections are completely determined by those projections, their symmetries, and the action of the latter on the former. This includes all von Neumann algebras and all AW*-algebras. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*-algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*-algebras and that of piecewise complete Boolean algebras, and a refinement of the piecewise algebra structure of an AW*-algebra that enables recovering its total structure.     

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. $C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}$ .     

On tracial approximation

Let C be a class of unital C*-algebras. The class TAC of C*-algebras which can be tracially approximated (in the Egorov-like sense first considered by Lin) by the C*-algebras in C is studied (Lin considered the case that C consists of finite-dimensional C*-algebras or the tensor products of such with C([0,1])). In particular, the question is considered whether, for any simple separable A∈TAC, there is a C*-algebra B which is a simple inductive limit of certain basic homogeneous C*-algebras together with C*-algebras in C, such that the Elliott invariant of A is isomorphic to the Elliott invariant of B. An interesting case of this question is answered. In the final part of the paper, the question is also considered which properties of C*-algebras are inherited by tracial approximation. (Results of this kind are obtained which are used in the proof of the main theorem of the paper, and also in the proof of the classification theorem of the second author given in [Z. Niu, A classification of tracially approximately splitting tree algebra, in preparation] and [Z. Niu, A classification of certain tracially approximately subhomogeneous C*-algebras, PhD thesis, University of Toronto, 2005]—which also uses the main result of the present paper.)     

Ideals of factors

Every primitive C*-algebra is prime. An old theorem of Dixmier shows that the converse is true if the algebra is separable. Recently Weaver has shown that without separability this is false. AW* -factors are easily seen to be prime. Are all their ideals primitive? A number of partial results have appeared recently. It turns out that the answer is always positive and that this follows immediately from a theorem of FB Wright proved nearly 50 years ago. His proof was hard and complicated. He works in a more general setting but when his result is specialised to the class of operator algebras investigated here, we are able to give a new, easy proof that the ideals of an AW* -factor form a well-ordered set; from this, it follows swiftly that every proper ideal of an AW*-factor is primitive.     

Universal continuous calculus for Su*-algebras

Universal continuous calculi are defined and it is shown that for every finite tuple of pairwise commuting Hermitian elements of a Su*-algebra (an ordered *-algebra that is symmetric, i.e., “strictly” positive elements are invertible and uniformly complete), such a universal continuous calculus exists. This generalizes the continuous calculus for $C^{\ast }$-algebras to a class of generally unbounded ordered *-algebras. On the way, some results about *-algebras of continuous functions on locally compact spaces are obtained. The approach used throughout is rather elementary and especially avoids any representation theory.     

Lifting Defects for Nonstable K0-theory of Exchange Rings and C*-algebras

The assignment (nonstable K₀-theory), that to a ring R associates the monoid V(?R?) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor:

1. There is no functor Γ, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that V °?Γ?? id.
2. There is no functor Γ, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that V °?Γ?? id.
3. There is a {0,1}³-indexed commutative diagram  ${\vec{D}}$ of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0.

By using categorical tools (larders, lifters, CLL) from a recent book from the author with P. Gillibert, we deduce that there exists a unital exchange ring of cardinality  $\aleph_3$ (resp., an $\aleph_3$ -separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(?R?) is the positive cone of a dimension group but it is not isomorphic to V(?B?) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.     

A note on commutation relations and finite dimensional approximations

In this article we show that the main C*-algebras describing the canonical commutation relations of quantum physics, i.e., the Weyl and resolvent algebras, are in the class of Følner C*-algebras, a class of C*-algebras admitting a kind of finite approximations of Følner type. In particular, we show that the tracial states of the resolvent algebra are uniform locally finite dimensional.     

C~*-代数的实完全等距映射

C*-代数的*-同构一定是(完全)等距映射,反之不然.本文证明了C*-代数的实完全等距映射能够完全决定C*-代数*-同构的结论.     

Noncommutative Positivstellensätze for pairs representation-vector

We study non-commutative real algebraic geometry for a unital associative *-algebra A{\mathcal {A}} viewing the points as pairs (π, v) where π is an unbounded *-representation of A{\mathcal A} on an inner product space which contains the vector v. We first consider the *-algebras of matrices of usual and free real multivariate polynomials with their natural subsets of points. If all points are allowed then we can obtain results for general A{\mathcal {A}}. Finally, we compare our results with their analogues in the usual (i.e. Schmüdgen’s) non-commutative real algebraic geometry where the points are unbounded *-representation of A{\mathcal {A}}.     

Separation properties in the primitive ideal space of a multiplier algebra

We use recent work on spectral synthesis in multiplier algebras to give an intrinsic characterization of the separable C*-algebras A for which Orc(M(A)) = 1, i.e., for which the relation of inseparability on the topological space of primitive ideals of the multiplier algebra M(A) is an equivalence relation. This characterization has applications to the calculation of norms of inner derivations and other elementary operators on A and M(A). For example, we give necessary and sufficient conditions on the ideal structure of a separable C*-algebra A for the norm of every inner derivation to be twice the distance of the implementing element to the centre of M(A).     

Strong NMV-algebras, commutative basic algebras and naBL-algebras

The aim of the paper is to investigate the relationship among NMV-algebras, commutative basic algebras and naBL-algebras (i.e., non-associative BL-algebras). First, we introduce the notion of strong NMV-algebra and prove that

1. a strong NMV-algebra is a residuated l-groupoid (i.e., a bounded integral commutative residuated lattice-ordered groupoid)
2. a residuated l-groupoid is commutative basic algebra if and only if it is a strong NMV-algebra.

Secondly, we introduce the notion of NMV-filter and prove that a residuated l-groupoid is a strong NMV-algebra (commutative basic algebra) if and only if its every filter is an NMV-filter. Finally, we introduce the notion of weak naBL-algebra, and show that any strong NMV-algebra (commutative basic algebra) is weak naBL-algebra and give some counterexamples.     

The Haagerup problem on subadditive weights on W*-algebras. II

In 1975 U. Haagerup has posed the following question: Whether every normal subadditive weight on a W*-algebra is σ-weakly lower semicontinuous? In 2011 the author has positively answered this question in the particular case of abelian W*-algebras and has presented a general form of normal subadditive weights on these algebras. Here we positively answer this question in the case of finite-dimensional W*-algebras. As a corollary, we give a positive answer for subadditive weights with some natural additional condition on atomic W*-algebras. We also obtain the general form of such normal subadditive weights and norms for wide class of normed solid spaces on atomic W*-algebras.     

Crossed products by finite group actions with the Rokhlin property

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (a) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (b) Simple unital AH algebras with slow dimension growth and real rank zero. (c) C*-algebras with real rank zero or stable rank one. (d) Simple C*-algebras for which the order on projections is determined by traces. (e) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. (f) C*-algebras with a unique tracial state. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective.     

A counter-example to a conjecture of Friedland

In 1982, S. Friedland proved that a bounded linear operator A on a Hilbert space is normal if and only if (αI + A + A*)² ≧ AA* − A*A ≧ −(αI + A + A*)² for all real α. And he conjectured the inequality (αI + A + A*)² ≧ AA* − A*A for all real α will imply that A*A − AA* ≧ 0, i.e., A is hyponormal. But his conjecture is incorrect. In this note I’ll give a counter-example for his conjecture.     

Normed algebras with involution

We show that most of the theory of Hermitian Banach algebras can be proved for normed *-algebras without the assumption of completeness. The conditionr(x)≤p(x) for allx (wherep(x)=r(x ^* x) ^1/2 is the Pták function), which is essential in the theory of Hermitian Banach algebras, is replaced for normed *-algebras by the conditionr(x+y)≤p(x)+p(y) for allx, y. In case of Banach *-algebras these conditions are equivalent. The research has been supported by a grant from La Junta de Andalucía and by the Department of Applied Mathematics, University of Seville This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990     

On ultraproducts of operator algebras

Some basic questions on ultraproducts of C~*-algebras and von Neumann al- gebras,including the relation to K-theory of C~*-algebras are considered.More specifically, we prove that under certain conditions,the K-groups of ultraproduct of C~*-algebras are iso- morphic to the ultraproduct of respective K-groups of C~*-algebras.We also show that the ultraproducts of factors of type Ⅱ_1 are prime,i.e.not isomorphic to any non-trivial tensor product.     

On Two-sided Locally Convex H*-algebras

We introduce (left, right, two-sided) locally convex H*-algebras, and we give conditions under which an one-sided locally convex H*-algebra turns to be a two-sided one (actually, a locally convex H*-algebra). We also give an example of a proper right locally convex H*-algebra with a (right) involution, which is not a left involution and an example of a proper two-sided locally convex H*-algebra, which is not a locally convex H*-algebra. Moreover, we connect (via an Arens-Michael decomposition) a two-sided locally m-convex H*-algebra with the classical (Banach) two-sided H*-algebras. Further, we present conditions so that the left, right involutions be continuous, and we see when a twosided locally convex H*-algebra is a dual one. Finally, we present some properties of invariant ideals which play an important rôle in structure theory of two-sided locally convex H*-algebras.