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.     

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.     

bi-strong Smarandache BL-algebras

In this paper, we introduce the notion of bi-Smarandache BL-algebra, bi-weak Smarandache BL-algebra, bi-Q-Smarandache ideal and bi-Q-Smarandache implicative filter, we obtain some related results and construct quotient of bi-Smarandache BL-algebras via MV-algebras (or briefly bi-Smarandache quotient BL-algebra) and prove some theorems. Finally, the notion of bi-strong Smarandache BL-algebra is presented and relationship between bi-strong Smarandache BL-algebra and bi-Smarandache BL-algebra are studied.     

Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras

We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C^∗-algebra, an Exel-Laca algebra, and an ultragraph C^∗-algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C^∗-algebras and Exel-Laca algebras, and that all simple AF-algebras are either graph C^∗-algebras or Exel-Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C^∗-algebra of a row-finite graph with no sinks.     

Construction of Fuzzy σ-algebras using triangular norms

Generalizing the definitions given by the author [Fuzzy Sets and Systems4 (1980), 83–93] we introduce and study T-fuzzy σ-algebras, T being any triangular norm. The main result is that for a large class of triangular norms each T-fuzzy σ-algebra is generated, i.e., consists of all functions μ:X → [0, 1] being measurable with respect to some σ-algebra on X.     

Equational type characterization for σ-complete MV-algebras

In the framework of algebras with infinitary operations, an equational base for the category of σ-complete MV-algebras is given. In this way, we study some particular objects as simple algebras, directly irreducible algebras, injectives, etc. A completeness theorem with respect to the standard MV-algebra, considered as σ-complete MV-algebra, is obtained. Finally, we apply this result to the study of σ-complete Boolean algebras and σ-complete product MV-algebras.     

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.     

Property T＊＊ for C＊-algebras

Extending the notion of property T of finite von Neumann algebras to general von Neumann algebras, we define and study in this paper property T** for (possibly non-unital) C* -algebras. We obtain several results of property T** parallel to those of property T for unital C* -algebras. Moreover, we show that a discrete group Γ has property T if and only if the group C* -algebra Cr* (Γ) (or equivalently, the reduced group C* -algebra Cr* (Γ)) has property T**. We also show that the compact operators K(l2 ) has property T** but c0 does not have property T**.     

On Unital Full Amalgamated Free Products of Quasidiagonal C*-Algebras

In the paper, we consider the question as to whether a unital full amalgamated free product of quasidiagonal C*-algebras is itself quasidiagonal. We give a sufficient condition for a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite-dimensional C*-algebra to be quasidiagonal. By applying this result, we conclude that the unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there exists a faithful tracial state on each of the two AF algebras such that the restrictions of these states to the common subalgebra coincide.     

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.     

Triangular norms and TNF-sigma-algebras

Following the suggestion made by Klement [8], an axiomatic theory of TNF-σ-algebras is given, T being any measurable triangular norm and N any negation. Most of the results about T-fuzzy σ-algebbrs obtained in [8] are extended to the case of TNF-σ-algebras. Some other properties of TNF-σ-algebras are also discussed. Particularly, we point out: (1) there exists a large family of triangular norms, which contains the whole Yager family and almost the whole Sugeno family as subfamilies, such that for any negation N each TNF-σ-algebra is generated, and (2) given a set U with |U|?2 and a measurable triangular norm T, in order that for every negation N each TNF-σ-algebra on U is generated it is necessary that T is Archimedean.     

Chain conditions for C*-algebras coming from Hilbert C*-modules

In this paper we define and study chain conditions for Hilbert C*-modules through their C*-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C*-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C*-algebra with a faithful conditional expectation.     

Some Types of Filters in BE-algebras

In this paper, we introduce some types of filters in BE-algebras and state some theorems which determine the relationship between these filters and other filters of a BE-algebra and by some examples we show that these notions are different.     

The center of approximately finite-dimensional C1-algebras

It is proved that any separable abelian C¹-algebra is the center of a C¹-algebra that is the inductive limit of an increasing sequence of finite-dimensional C¹-algebras.     

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.     

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.     

Crossed products by finite group actions with certain tracial Rokhlin property

We introduce a special tracial Rokhlin property for unital C~*-algebras. Let A be a unital tracial rank zero C~*-algebra(or tracial rank no more than one C~*-algebra). Suppose that α : G → Aut(A) is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~*-algebra. Then, the crossed product C~*-algebra C~*(G, A, α) has tracia rank zero(or has tracial rank no more than one). In fact,we get a more general results.     

Partial isometries and extensions of $${{\rm A}{\mathbb T}}$$-algebras

In this paper, we will define several new isomorphism invariants for C*-algebras by hyponormal partial isometries and discuss the relation between these invariants and K-theory of C*-algebras. This study was in part inspired by the work of H. Lin and H. Su in the context of \({A\mathcal{T}}\)-algebras. An \({{\rm A}\mathcal{T}}\)-algebra often becomes an extension of an \({{\rm A}\mathbb{T}}\)-algebra by an AF-algebra. We show that there is an essential extension of a simple \({{\rm A}\mathbb{T}}\)-algebra which has real rank zero by an AF-algebra such that it has real rank zero and is not an \({A\mathcal{T}}\)-algebra.     

Some examples of homotopy Π-algebras

The homotopy Π-algebra of a pointed topological space, X, consists of the homotopy groups of X together with the additional structure of the primary homotopy operations. We extend two well-known results for homotopy groups to homotopy Π-algebras and look at some examples illustrating the depth of structure on homotopy groups; from graded group to graded Lie ring, to Π-algebra and beyond. We also describe an abstract Π-algebra and give three abstract Π-algebra structures on the homotopy groups of the loop space of X which can be realized as the homotopy Π-algebras of three different spaces.     

Bounded and Unitary Elements in Pro-C*-algebras

A pro-C*-algebra is a (projective) limit of C*-algebras in the category of topological *-algebras. From the perspective of non-commutative geometry, pro-C*-algebras can be seen as non-commutative k-spaces. An element of a pro-C*-algebra is bounded if there is a uniform bound for the norm of its images under any continuous *-homomorphism into a C*-algebra. The *-subalgebra consisting of the bounded elements turns out to be a C*-algebra. In this paper, we investigate pro-C*-algebras from a categorical point of view. We study the functor (−) _b that assigns to a pro-C*-algebra the C*-algebra of its bounded elements, which is the dual of the Stone-Čech-compactification. We show that (−) _b is a coreflector, and it preserves exact sequences. A generalization of the Gelfand duality for commutative unital pro-C*-algebras is also presented.