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Sphere and projective space of a C*-algebra with a faithful state

Antunez Andrea C.

Demonstratio Mathematica

2019

Vol. 52, nr 1

410--427

Let A be a unital C*-algebra with a faithful state φ. We study the geometry of the unit sphere Sφ = {x∈A : φ(x*x) = 1} and the projective space Pφ = Sφ/T. These spaces are shown to be smooth manifolds and homogeneous spaces of the group Uφ(A) of isomorphisms acting in A which preserve the inner product induced by φ, which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in Pφ, and prove the existence of minimal geodesics, both with given initial data, and given endpoints.

Crossed product of a C*-algebra by a semigroup of interactions

Kwaśniewski B. K.

Demonstratio Mathematica

2014

Vol. 47, nr 2

352--370

The paper presents a construction of the crossed product of a C*-algebra by a commutative semigroup of bounded positive linear maps generated by partial isometries. In particular, it generalizes Antonevich, Bakhtin, Lebedev’s crossed product by an endomorphism, and is related to Exel’s interactions. One of the main goals is the Isomorphism Theorem established in the case of actions by endomorphisms.

A note on Radon-Nikodym derivatives and similarity for completely bounded maps

Gheondea A., Kavruk A. S.

Opuscula Mathematica

2009

Vol. 29, no. 2

139-145

We point out a relation between the Arveson's Radon-Nikodym derivative and known similarity results for completely bounded maps. We also consider Jordan type decompositions coming out from Wittstock's Decomposition Theorem and illustrate, by an example, the nonuniqueness of these decompositions.

Range projections of idempotents in C*-algebras

Koliha J.

Demonstratio Mathematica

2001

Vol. 34, nr 1

91-103

In this paper we study range projections of idempotents m C*-algebras, and use them to obtain a Schur type decomposition that leads to simple proofs of results on Moore-Penrose inverse and norms of idempotents. We analyze the continuity of range projections, obtain a general result on their approximation, and recover a result of Vidav on two projections in a Hilbert space. Several representations of range projections are given.