In the present paper, we study simple algebras, which do not belong to well-known classes of algebras (associative alternative Lie Jordan etc.). The finite-dimensional over a field characteristic 0 without finite basis identities, constructed by Kislitsin, are such algebras. consider two algebras: seven-dimensional anticommutative algebra \(\mathcal{D}\) and central commutative \(\mathcal{C}\)....

We consider unital simple inductive limits of generalized dimension drop C∗-algebras They are so-called ASH-algebras and include all unital simple AH-algebras and all dimension drop C∗-algebras. Suppose that A is one of these C∗-algebras. We show that A ⊗ Q has tracial rank no more than one, where Q is the rational UHF-algebra. As a consequence, we obtain the following classification result: Le...

since 2005 a new powerful invariant of an algebra has emerged using the earlier work of horvath, hethelyi, kulshammer and murray. the authors studied morita invariance of a sequence of ideals of the center of a nite dimensional algebra over a eld of nite characteristic. it was shown that the sequence of ideals is actually a derived invariant, and most recently a slightly modied version o...

The Jiang–Su algebra Z has come to prominence in the classification program for nuclear C-algebras of late, due primarily to the fact that Elliott’s classification conjecture predicts that all simple, separable, and nuclear C-algebras with unperforated K-theory will absorb Z tensorially (i.e., will be Z-stable). There exist counterexamples which suggest that the conjecture will only hold for si...

Abstract. There has recently been much interest in the C∗-algebras of directed graphs. Here we consider product systems E of directed graphs over semigroups and associated C∗-algebras C∗(E) and T C∗(E) which generalise the higher-rank graph algebras of Kumjian-Pask and their Toeplitz analogues. We study these algebras by constructing from E a product system X(E) of Hilbert bimodules, and applyi...

We investigate the notion of tracial $\mathcal Z$-stability beyond unital C*-algebras, and we prove that this is equivalent to in class separable simple nuclear C*-algebras.

We give substance to the motto “every partial algebra is the colimit of its total subalgebras” by proving it for partial Boolean algebras (including orthomodular lattices), the new notion of partial C*-algebras (including noncommutative C*-algebras), and variations such as partial complete Boolean algebras and partial AW*-algebras. Both pairs of results are related by taking projections. As cor...

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 bound...

Kadison and Kastler introduced a metric on the set of all C∗-algebras on a fixed Hilbert space. In this paper structural properties of C∗-algebras which are close in this metric are examined. Our main result is that the property of having a positive answer to Kadison’s similarity problem transfers to close C∗-algebras. In establishing this result we answer questions about closeness of commutant...