On the algebra of operators of Hilbert modules over locally C^*- algebras
by
Yu. I. Jurayev
Samarkand State University
Coauthors: F. Saripov (Samarkand State University)

Works of many authors are devoted to the study of the properties of Hilbert modules over C^*-algebra and their applications (see: [1-3]).

In the present talk the idea of Hilbert module over locally C^*-algebra is introduced and some of its properties are studied.

Let A be a locally C^*-algebra(see [4, 5]) and {P_\alpha}_{\alpha in \Delta} be a family of submultiplicative semi-norms defining the topology of A.

Definition 1. A pre-Hilbert A-module is a right A-module X equipped with the map < ., . > :X ×X --> A satisfying:

1. < x, x > >= 0, for each x in X;
2. < x, x > =0 only if x=0;
3. < x₁ + x₂, y > = < x₁, y > + < x₂, y > for each x₁, x₂, y in X;
4. < xa, y > = < x, y > a for each x, y in X, a in A;
5. < x, y > ^*= < y, x > for each x, y in X.

The map < ., . > is called a A-valued inner product on X.

Let X be pre-Hilbert A-module. For each \alpha in \Delta one can define a functional [`P]_\alpha on X by

- P   \alpha  (x):= Ö   P\alpha( < x, x > )   , x in X.

Proposition 1. For each \alpha in \Delta the functional [`P]_\alpha is semi-norm on X and satisfies:

1. [`P]<sub>\alpha</sub>(xa) <= [`P]_\alpha(x)P_\alpha(a) for each x in X, a in A;
2. P_\alpha( < x, y > ) <= [`P]<sub>\alpha</sub>(x)[`P]_\alpha(y) for each x, y in X.

Definition 2. A pre-Hilbert module X over locally C^*-algebra A which is complete locally convex with respect to the family semi-norm {[`P]_\alpha: \alpha in \Delta} is called a Hilbert A-module.

For an locally C^*-algebra A, we denote by A^s the set of elements a in A for which

||a||s:= sup \alpha in \Delta  P\alpha(a) < \infty.

Then, A^s is a C^*-algebra with respect to norm ||.||^s (see [5]). For a Hilbert A-module X we put

Xs:={x in X: ||x||s= sup \alpha in \Delta  - P   \alpha  (x) < \infty}

Note that X^s is a A^s-submodule in X.

Theorem 1. X^s is a Hilbert module over C^*-algebra A^s.

Let I_\alpha=kerP_\alpha, A_\alpha=A/I_\alpha, then A^\alpha is an C^*-algebra with respect to the norm ||a+I_\alpha||:=P_\alpha(a), a in A.

Theorem 2. For each x, y in X we define

< x+ - I   \alpha  , y+ - I   \alpha  > := < x, y > +I\alpha,

where [`I]<sub>\alpha</sub>=ker[`P]_\alpha. Then X_\alpha=X/[`I]_\alpha is a Hilbert C^*-module over A_\alpha.

Definition 3. Let T:X --> X is a A-homomorphism of X. If for each \alpha in \Delta there exists C_\alpha > 0 such that [`P]<sub>\alpha</sub>(Tx) <= C<sub>\alpha</sub>[`P]_\alpha(x), for any x in X, then T is called an bounded A-operator. The set of all bounded A-operators is denoted by End_A(X).

Theorem 3. The set End_A(X) is an complete locally convex algebra whose topology is defined by the family of semi-norm { [^P]_\alpha, \alpha in \Delta} where

^ P   \alpha  (T)= sup [`P]\alpha (x) <= 1  - P   \alpha  (Tx),     T in EndA(X), \alpha in \Delta.

Denote by End_A^*(X) the set of maps T:X --> X, for which there exists an map T^*:X --> X, such that < Tx, y > = < x, T^*y > for any x, y in X.

Theorem 4. The set End_A^*(X) lies in End_A(X) as the closed subalgebra and is an locally C^*-algebra.

References

[1] W. L. Pashke. Inner product modules over B^*-algebras. - Trans. Amer. Math. Soc., 18 (1973), 443-468.

[2] M.A.Rieffel. Induced representations of C^*-algebras. - Adv. Math. 13 (1974), 176-257.

[3] A. S. Mishenko., A. T. Fomenko. The index of elliptic operators over C^*-algebras. - Izv. Acad. Nauk SSSR, ser. Math. 43 (1979), N4, 831-859.

[4] A. Inoue. Locally C^*-algebra. - Memoirs of Fac. Sci. Kyashi Univ. ser.A, vol.25, N2, (1972), 197-235.

[4] C. Apostol. B^*- algebras and their representation. J. London Math. Soc. 33(1971), 30-38.

Date received: May 17, 2000

Copyright © 2000 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 # caeh-11.