Narrow Results

Let A and B be two maximal abelian *-subalgebras of the n × n complex matrices M_n(ℂ). To study the movement of the automorphisms of M_n(ℂ), we modify the Connes–Størmer relative entropy H(A|B) and the Connes relative entropy H_ϕ(A|B) with respect to a state ϕ, and introduce the two kinds of the constant h(A|B) and h_ϕ(A|B). For the unistochastic matrix b(u) defined by a unitary u with B = uAu*, we show that h(A|B) is the entropy H(b(u)) of b(u). This is obtained by our computation of h_ϕ(A|B). The h(A|B) attains to the maximal value log n if and only if the pair {A, B} is orthogonal in the sense of Popa.

Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set M_n(K) of n × n matrices is either zero, or the set of scalar matrices, or the set sl_n(K) of matrices of trace 0, or all of M_n(K). This conjecture was proved for n = 2 when K is closed under quadratic extensions. In this paper, the conjecture is verified for K = ℝ and n = 2, also for semi-homogeneous polynomials p, with a partial solution for an arbitrary field K.

In this paper we present a new aproach of the noncommutative geometry of the matrix algebra . We define two different differential calculi, and we introduce linear connections on using the framewok of ρ-algebras.

A class of bistochastic maps of three-dimensional matrix algebra which preserves a one-dimensional projector is studied.

We present a new one-parameter family of extremal positive maps on the three-dimensional matrix algebra. The new elements are characterized as mappings that preserve a one-dimensional orthogonal projector.

In this paper, we study Johnson pseudo-contractibility of second dual of some Banach algebras. We show that the semigroup algebra ${\ell }^1{(S)}^{\ast \ast }$ is Johnson pseudo-contractible if and only if $S$ is a finite amenable group, where $S$ is an archimedean semigroup. We also show that the matrix algebra $M_I{(ℂ)}^{\ast \ast }$ is Johnson pseudo-contractible if and only if $I$ is finite. We study Johnson pseudo-contractibility of certain projective tensor product second duals Banach algebras.

In this paper, we introduce a new notion of strong pseudo-amenability for Banach algebras. We study strong pseudo-amenability of some matrix algebras. Using this tool, we characterize strong pseudo-amenability of ${\ell }^1(S)$, provided that $S$ is a uniformly locally finite inverse semigroup. As an application, we show that for a Brandt semigroup $S=M^0(G,I)$, ${\ell }^1(S)$ is strong pseudo-amenable if and only if $G$ is amenable and $I$ is finite. We give some examples to show the differences between strong pseudo-amenability and the other classical notions of amenability.

In this paper, we introduce a new notion of module strong pseudo-amenability for Banach algebras. We study the relation between this new concept to other various notions at this issue, module pseudo-contractibility, module pseudo-amenability and module approximate amenability. For an inverse semigroup $S$ with the set of all idempotents $E$, we show that ${\ell }^1(S)$ is module strong pseudo-amenable as an ${\ell }^1(E)$-module if and only if $S$ is amenable. For specific types of semigroups such as Brandt semigroups and bicyclic semigroups, we investigate the module strong pseudo-amenability of ${\ell }^1(S)$. We show that for every non-empty set $I$, ${𝕄}_I(ℂ)$ under this new notion is forced to have a finite index as an $𝔄$-module, where $𝔄=\{[a_{i,j}]\in {𝕄}_{\mathrm{\Lambda }}(ℂ)|\forall i\ne j,a_{i,j}=0\}$.

In this paper we present some geometric objects (derivations, differential forms, distributions, linear connections, their curvature and their torsion) on matrix algebra using the framework of noncommutative geometry.

We analyze several classes of positive maps in infinite dimensional matrix algebras. Such maps provide basic tools for studying quantum entanglement infinite dimensional Hilbert spaces. Instead of presenting the most general approach to the problem we concentrate on specific examples. We stress that there is no general construction of positive maps. We show how to generalize well known maps in low dimensional algebras: Choi map in M₃(ℂ) and Robertson map in M₄(ℂ).