PROJECTIVE SPACE OF A C * -MODULE

Mathematische Nachrichten

Corach

Мbekhta³

2005

We study the set S = {(a, b) ∈ A × A : aba = a, bab = b} which pairs the relatively regular elements of a Banach algebra A with their pseudoinverses, and prove that it is an analytic submanifold of A × A. If A is a C * -algebra, inside S lies a copy the set I of partial isometries, we prove that this set is a C ∞ submanifold of S (as well as a submanifold of A). These manifolds carry actions from, respectively, GA × GA and UA × UA, where GA is the group of invertibles of A and UA is the subgroup of unitary elements. These actions define homogeneous reductive structures for S and I (in the differential geometric sense). Certain topological and homotopical properties of these sets are derived. In particular, it is shown that if A is a von Neumann algebra and p is a purely infinite projection of A, then the connected component Ip of p in I is simply connected. If 1 − p is also purely infinite, then Ip is contractible. Gramsch [16] studied the set of Fredholm operators with fixed dimension of the kernel, and proved that it is an analytic homogeneous manifold. He also extended some of these results to the context of Frechet algebras with open group of invertible elements. In [12] there is a geometrical study of certain parts of S. Observe that (a, b) belongs to S if and only if aba = a and bab = b; in particular ab and ba belong to Q, the set of idempotents of A. For a fixed r in Q, consider the set S r = {(a, b) ∈ S : ar = a, rb = b, ba = r}. There is a natural action of G A over S r given by *

Section: Ifmentioning

confidence: 97%

“…In [2] and [4] there is a study of the structure of the set of partial isometries of a C * -algebra with initial space p ∈ A:…”

Section: Proposition 34 the Mapmentioning

confidence: 99%

Section: Proposition 34 the Mapmentioning

confidence: 99%

Section: Ifmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

On the geometry of generalized inverses

Mathematische Nachrichten

Corach

Мbekhta³

2005

Hopf fibration in a C^*‐module^†

Mathematische Nachrichten

Corach

Recht

2023

Self Cite

Let X be a right C*‐module over a unital C*‐algebra A$ {\cal A}$. We study the Hopf fibration of X: frakturh:XP→P1(X)goodbreak=projective space of0.33emboldX,$$\begin{equation*} \mathfrak {h}: {\bf X} _ {\cal P} \rightarrow P1({\bf X}) = \hbox{projective space of } {\bf X} , \end{equation*}$$where the projective space of X is the set of singly generated orthocomplemented submodules of X, boldXscriptP$ {\bf X} _ {\cal P}$ is the set of elements of X, which generate such submodules, and frakturhfalse(boldxfalse)=$\mathfrak {h}({\bf x} )=$module generated by boldx∈XP$ {\bf x} \in {\bf X} _ {\cal P}$. The group of unitary operators of the module X acts on both spaces. We introduce a Finsler metric in boldXscriptP$ {\bf X} _ {\cal P}$, which is invariant under the unitary action. Our main results establish that the map h$\mathfrak {h}$ is distance decreasing (when the projective space of X is considered with its natural unitary invariant metric), and a minimality result in boldXscriptP$ {\bf X} _ {\cal P}$, characterizing metric geodesics in this space.

C*-Modular Vector States

Integr. equ. oper. theory

Varela

2005

Self Cite

Let B be a C * -algebra and X a Hilbert C * B-module. If p ∈ B is a projection, let Sp(X) = {x ∈ X : x, x = p} be the p-sphere of X. For ϕ a state of B with support p in B and x ∈ Sp(X), consider the modular vector state ϕx of LB(X) given byand Sp(X) × {states with support p} → Σp,X = { modular vector states }, (x, ϕ) → ϕx. These fibrations enable us to examine the homotopy type of the sets of modular vector states, and relate it to the homotopy type of unitary groups and spaces of projections. We regard modular vector states as generalizations of pure states to the context of Hilbert C * -modules, and the above fibrations as generalizations of the projective fibration of a Hilbert space. Mathematics Subject Classification (2000). 46L30, 46L05, 46L10, 46L08.