1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

4.6. Ountz-Pimsner algebras 137

Let A be a C*-algebra and 'H be a linear space which is a right A-module.

An A-valued inner product on 'His a map

H x 'H :::i (e, rJ) f---+ \e, rJ) EA

satisfying

(1) ( ·, ·) is linear in the second variable,

(2) (e, rJa) = (e, rJ)a for every e, rJ E 'H and a EA,

(3) (e, rJ)* = (rJ, e) for every e, 'fJ E 'H,

(4) (e, e) ?: O; and (e, e) = 0 implies e = 0.

An A-valued semi-inner product is a map which satisfies all the above

conditions except for the second part of (4). Let 'H be a right A-module

with an A-valued inner product. Mimicking Hilbert space, we define llell =

ll(e,e)ll^112. Note that the Cauchy-Schwarz inequality extends to this con-

text. Indeed, the operator inequality

plus a calculation, shows that

o:::; \11e11-^1 e(e, rJ) -11e11'f/, 11e11-^1 e\e, rJ) - llell77) :::; 11e11^2 \77, rJ) - \e, 77)\e, 77).
Hence, II (e, rJ) II :::; llell 117711 for all e, 77 E 'H. It follows that 11e11 = II (e, e) 11112
defines a norm on 'H which satisfies Ilea II :::; llell llall and II (e, 77) II :S llell 117711 ·
We say that 'H is a Hilbert A-module if it is complete with respect to this
norm. Every right A-module with an A-valued semi-inner product can be
promoted to a bona fide Hilbert A-module by separation and completion.
For Hilbert A-modules 'H and /(, we denote by IIB('H, K) the set of ad-
jointable operators from 'H to /(. We simply write IIB('H) for the C -algebra
IIB('H, 'H). For every e, rJ E 'H, we define the "rank-one" operator Br;,,,,,: 'H ----t 1t
by Br;,,,,,(() = e(77, (). Then Br;,,,,, E IIB('H) and we have the following calculus:
Bt;,i+f.2,,,, 1 +,,, 2 = Br;,i,,,, 1 + Br;,i,,,, 2 + Bf.2,,,, 1 +86,,,, 2 , Bt,,, = e,,,,r;, and xBr;,,,,, = Bxr;,,,,, for
every e, 77 E 'Hand x E IIB('H). We define the C-algebra IK('H) of "compact
operators" as the closed linear span of {Br;,,,,, : e, rJ E 'H}. Note that IK('H) is
a closed two-sided ideal of IIB(H).
The simplest Hilbert A-module is A itself with the A-valued inner prod-
uct (a, b) = ab. We often write a for a EA when it is viewed as an element
in the Hilbert A-module A. We note that IK(A) 9i A and IIB(A) 9i M(A),
where M(A) is the multiplier algebra of A. If Hi, i EI, are a set of Hilbert
A-modules, then the algebraic direct sum ffialg 'Hi has an A-valued inner
product: ((ei)iEJ, (77i)iEI) = ~iEJ(ei, 77i)· The completion of E9alg 'Hi is de-
noted by E9 'Hi· We also write 1-(,'<!0n for the n-fold direct sum 'HEB··· EB 'H