II1 FACTORS WITH EXOTIC CENTRAL SEQUENCE ALGEBRAS

We provide a class of separable II1 factors $M$ whose central sequence algebra is not the ‘tail’ algebra associated with any decreasing sequence of von Neumann subalgebras of $M$ . This settles a question of McDuff [On residual sequences in a II1 factor, J. Lond. Math. Soc. (2) (1971), 273–280].

Regularity of Villadsen algebras and characters on their central sequence algebras

We show that if $A$ is a simple Villadsen algebra of either the first type with seed space a finite dimensional CW complex, or of the second type, then $A$ absorbs the Jiang-Su algebra tensorially if and only if the central sequence algebra of $A$ does not admit characters.Additionally, in a joint appendix with Joan Bosa (see the following paper), we show that the Villadsen algebra of the second type with infinite stable rank fails the Corona Factorization Property, thus providing the first example of a unital, simple, separable and nuclear $C^\ast $-algebra with a unique tracial state which fails to have this property.

Asymptotic structure of free product von Neumann algebras

Let (M, ϕ) = (M1, ϕ1) * (M2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M1 ⊂ M in arbitrary free product von Neumann algebras.

When central sequence C*-algebras have characters

We investigate C*-algebras whose central sequence algebra has no characters, and we raise the question if such C*-algebras necessarily must absorb the Jiang–Su algebra (provided that they also are separable). We relate this question to a question of Dadarlat and Toms if the Jiang–Su algebra always embeds into the infinite tensor power of any unital C*-algebra without characters. We show that absence of characters of the central sequence algebra implies that the C*-algebra has the so-called strong Corona Factorization Property, and we use this result to exhibit simple nuclear separable unital C*-algebras whose central sequence algebra does admit a character. We show how stronger divisibility properties on the central sequence algebra imply stronger regularity properties of the underlying C*-algebra.

Extreme Version of Projectivity for Normed Modules Over Sequence Algebras

We define and study the so-called extreme version of the notion of a projective normed module. The relevant definition takes into account the exact value of the norm of the module in question, in contrast with the standard known definition that is formulated in terms of normtopology.After the discussion of the case where our normed algebra A is just C, we concentrate on the case of the next degree of complication, where A is a sequence algebra satisfying some natural conditions. The main results give a full characterization of extremely projective objects within the subcategory of the category of non-degenerate normed A-modules, consisting of the so-called homogeneous modules. We consider two cases, ‘non-complete’ and ‘complete’, and the respective answers turn out to be essentially different.In particular, all Banach non-degenerate homogeneous modules consisting of sequences are extremely projective within the category of Banach non-degenerate homogeneous modules. However, neither of them, provided it is infinite-dimensional, is extremely projective within the category of all normed non-degenerate homogeneous modules. On the other hand, submodules of these modules consisting of finite sequences are extremely projective within the latter category.

Some Double Sequence Spaces of Fuzzy Real Numbers of Paranormed Type

We study different properties of convergent, null, and bounded double sequence spaces of fuzzy real numbers like completeness, solidness, sequence algebra, symmetricity, convergence-free, and so forth. We prove some inclusion results too.