Strongly movable C^* algebras
by
Zvonko Čerin
University of Zagreb

We begin with the definition of a ^homomorphism that resembles asymptotic homomorphisms from ch.

Let a and b be C^algebras. Let be a positive real number. A function fAB is a ^homomorphism provided

• f takes the zero element 0A of a into the zero element 0B of b,
• f is nonexpansive, i. e., the relation f(x)-f(y)_B x-y_A holds for all x,  yA, and
• f(x+y)-f(x)-f(y)_B < ( x_A+ y_A) for all x,  yA.
• f(bx)-bf(x)_B < x_A for each xA and each bC.
• f(xy)-f(x)f(y)_B < x_A  y_A for all x,  yA.
• f(x^)-f(x)^_B < x_A for each xA.

Observe that a ^homomorphism is a uniformly continuous function. Moreover, for every real number between 0 and 1 and every C^algebra a the function f from A into itself which takes an xA into the product of and x is an example of a ^homomorphism which is not a homomorphism.

Another basic notion is that of the ^homotopy for nonexpansive functions of C^algebras.

Let > 0. Nonexpansive functions f and g between C^algebras a and b are ^homotopic and we write fg provided there is an ^homomorphism hAC(I; B) with h_0 =  eB0h = f and h_1 =  eB1h = g. We shall also say that h is a ^homotopy which joins f and g or that it realizes the relation or ^homotopy fg.

A C^algebra b is movable provided for every 0 there is a 0 such that for every >0 and every ^homomorphism fAB from any C^algebra a into b is ^homotopic to a ^homomorphism.

A C^algebra b is calm provided there is a 0 such that for every 0 there is a 0 with the property that ^homomorphisms f,  gAB from any C^algebra a into b which are ^homotopic are also ^homotopic.

A C^algebra b is strongly movable provided it is both movable and calm.

We shall prove two theorems about strongly movable C^algebras.

If C^algebras a and b have the same shape in the sense of article cer1 and a is strongly movable, then b is also strongly movable.

Strongly movable C^algebras have finitely generated homotopy groups in the sense of article cer2.

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A. Connes and N. Higson, Deformations, morphismes asymptotiques et K-theorie bivariante, C. R. Acad. Sci. Paris, 313 ( 1990), 163 - 170

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Z. Cerin, Shape theory for arbitrary C^algebras, Proceedings 11th Summer Topology Conference (Gorham), Proc. New York Acad. Sci. 806 (1996), 88-105.

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Z. Cerin, Homotopy groups for C^algebras, Bolyai Society Mathematical Studies, 4 (1995), 29-45.

Date received: July 24, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagt-61.