The article investigates the relationship between the distance of unitary orbits and spectral distance in C*-algebras of generalized tracial rank one. Building on the classical result of H. Weyl, which establishes that for Hermitian matrices x and y, the distance between their unitary orbits dist(U(x), U(y)) coincides with their spectral distance delta(x, y), subsequent extensions have been made: R. Wang proved the equality for simple C*-algebras of tracial rank one, and C.W. Cheong for one-dimensional noncommutative finite CW complexes. In this work, we further extend the result to simple C*-algebras of generalized tracial rank one, showing that for any self-adjoint elements x, y, the equality dist(U(x), U(y)) = D-c(x, y) continues to hold.
In this paper, we consider a system of scaled hypercomplex numbers of Ht for an arbitrarily fixed scale t E R, and construct some functional vector spaces. In particular, we concentrate on a "Hardy-like" space Ht:2[[q] ] in an Ht-variable q, and investigate "Toeplitz-like" operators on Ht:2[[q] ]. Some interesting operator-theoretic and operator-algebraic properties of those operators are characterized.
Let A be a simple non-elementary C & lowast;-algebra, and let alpha : G -> Aut(A) be an action of a finite group G on A which has the weak tracial Rokhlin property. We prove the following results: (1) If A is unital and has property (TM) introduced by Fu and Lin, then A & rtimes;(alpha) G has property (TM). As a corollary, if A is unital, separable, and has stable rank one and strict comparison of positive elements, then A & rtimes;(alpha) G and A alpha have stable rank one. (2) If A has m-comparison, then A & rtimes;(alpha) G and A alpha have m-comparison. (3) If A is n-almost divisible, then A & rtimes;(alpha) G and A alpha are (2n + 1)-almost divisible.
The study of Cowen-Douglas operators involves traditional operator-theoretic and complex-geometric tools. We study the similarity of Cowen-Douglas operators to backward shifts on analytic function spaces whose multiplier algebras equal H-infinity(D). Using the ratio of metrics and a model theorem, we obtain a sufficient condition for similarity. For weighted Dirichlet spaces, whose multiplier algebra is a proper subalgebra of H-infinity(D), we give a similarity condition via the jet bundle. The analysis relies on Muller's model theory for Dirichlet shifts and the corona theorem of Kidane-Trent for weighted Dirichlet spaces.
For r between 0 and 1, let Ar={z∈C:r<|z|<1}. We study the class C1,r={T:T is invertible and ∥T∥,∥rT−1∥⩽1}. The class C1,r is closely related to the \textit{quantum annulus} given by QAr={T:T is invertible and ∥rT∥, ∥rT−1∥⩽1}. McCullough and Pascoe proved that T∈QAr dilates to an operator S satisfying (r−2+r2)I−S∗S−S−1S−∗=0. We extend this dilation result to doubly commuting tuples of operators in quantum annulus as well as in C1,r class. We also provide characterizations and decomposition results for such tuples.
