### A multi-dimensional spectral theory in C*-algebras

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We present two examples. One of an operator T such that ${{T}^{n}(T-I)}_{n=1}^{\infty}$ is precompact in the operator norm and the spectrum of T on the unit circle consists of an infinite number of points accumulating at 1, and the other of an operator T such that ${{T}^{n}(T-I)}_{n=1}^{\infty}$ is convergent to zero but T is not power bounded.

Let A be a locally convex, unital topological algebra whose group of units ${A}^{\times}$ is open and such that inversion $\iota :{A}^{\times}\to {A}^{\times}$ is continuous. Then inversion is analytic, and thus ${A}^{\times}$ is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then ${A}^{\times}$ has a locally diffeomorphic exponential function and multiplication is given locally by the Baker-Campbell-Hausdorff series. In contrast, for suitable non-Mackey complete A, the unit group ${A}^{\times}$ is an analytic Lie group without...

We show that the Banach algebras with continuous involution are the Banach algebras which admit a harmonic functional calculus, while we prove that the hermitian commutative Banach algebras are exactly the involutive commutative Banach algebras that admit a real analytic functional calculus.

We prove that C*-algebras for an equivalent norm are the involutive Banach algebras which admit the continuous functional calculus.

We construct a non-m-convex non-commutative ${B}_{0}$-algebra on which all entire functions operate. Our example is also a Q-algebra and a radical algebra. It follows that some results true in the commutative case fail in general.

We study the continuity of the generalized Drazin inverse for elements of Banach algebras and bounded linear operators on Banach spaces. This work extends the results obtained by the second author on the conventional Drazin inverse.

We give new necessary and sufficient conditions for an element of a C*-algebra to commute with its Moore-Penrose inverse. We then study conditions which ensure that this property is preserved under multiplication. As a special case of our results we recover a recent theorem of Hartwig and Katz on EP matrices.

It is shown that a finite system T of matrices whose real linear combinations have real spectrum satisfies a bound of the form $\left|\right|{e}^{i\u27e8T,\zeta \u27e9}\left|\right|\le C(1+{\left|\zeta \right|)}^{s}{e}^{r\left|\Im \zeta \right|}$. The proof appeals to the monogenic functional calculus.

We investigate the finite-dimensional Lie groups whose points are separated by the continuous homomorphisms into groups of invertible elements of locally convex algebras with continuous inversion that satisfy an appropriate completeness condition. We find that these are precisely the linear Lie groups, that is, the Lie groups which can be faithfully represented as matrix groups. Our method relies on proving that certain finite-dimensional Lie subalgebras of algebras with continuous inversion commute...