Before stating the main results, let us fix the rest of our notation. We assume, unless stated otherwise, throughout this section that
is a unital Banach
-algebra. Also if
is a semi-inner product
-module and
is a
-seminorm on
, we put
, and if
is a positive linear functional on
, we put
. Let
be a finite set of orthogonal elements in
such that
be idempotent, we set
and
.
Dragomir in [8, Lemma 4] shows that in a Hilbert space
, the condition
is equivalent to the condition
where
and
,
. But for semi-inner product
-modules we have the following lemma, which is a generalization of [7, Lemma 1].
Lemma 3.1.
Let
be a semi-inner product
-module and
,
. Then
if and only if
Proof.
Follows from the equalities:
Remark 3.2.
By making use of the previous Lemma 3.1, we may conclude the following statements.
(i)Let
be an inner product
-module and let
be a finite set of orthogonal elements in
such that
are idempotent, then inequality (3.3) implies that
(ii)Let
be an inner product
-module and
be a finite set of orthogonal elements in
such that
are idempotent. If
is a
-seminorm on
then inequality (3.3) implies that
and if
is a positive linear functional on
from inequality (3.3) and [2, Section 37 Lemma 6(iii)], we get
(iii)Let
be a proper
-algebra, let
be an inner product
-module, and let
be a finite set of orthogonal elements in
such that
are idempotent. Since for every
,
inequality (3.3) is valid only if
We are able now to state our first main result.
Theorem 3.3.
Let
be an inner product
-module and let
be a finite set of orthogonal elements in
such that
are idempotent. If
,
,
,
are real numbers and
such that
hold, then one has the inequality
Proof.
By [11, Lemma 2] or, a straightforward calculation shows that for every 
Therefore
Analogously, for every
, we have
The equalities (3.10), (3.13), and (3.14) imply that
Since
therefore the Schwarz's inequality (2.1) holds, that is,
Finally, using the elementary inequality for real numbers
on
we get
Remark 3.4.
-
(i)
Let
be an inner product
-module and let
be a finite set of orthogonal elements in
such that
are idempotent. If
and
are such that
and if we put
, and
, then, by (3.15) and (3.16), we have
These and (3.11) imply that
Therefore, (3.11) is a refinement and a simple formulation of [9, Theorem 4.1.].
-
(ii)
If for
, we set
then similarly (3.11) is a refinement and a simple form of [9, Corollary 4.3].
Corollary 3.5.
Let
be a Banach
-algebra, let
be an inner product
-module, and let
be a finite set of orthogonal elements in
such that
are idempotent. If
,
,
are real numbers and
such that
hold, then one has the inequality
Proof.
Using the schwarz's inequality (2.6), we have
The assumptions (3.26) and the elementary inequality for real numbers (3.19) will provide the desired result (3.27).
Example 3.6.
Let
be a Hermitian Banach
-algebra and let
be the Pták function on
. If
is a semi-inner product
-module and
with the properties that
then we have
That is interesting in its own right.
Corollary 3.7.
Let
be a proper
-algebra, let
be an inner product
-module, and let
be a finite set of orthogonal elements in
such that
are idempotent. If
,
are real numbers and
such that
hold, then one has the inequality
Proof.
Using the strong Schwarz's inequality (2.3), we have
The assumptions (3.31) and the elementary inequality for real numbers (3.19) will provide (3.32).
The following companion of the Grüss inequality for positive linear functionals holds.
Theorem 3.8.
Let
be an inner product
-module, let
be a positive linear functional on
, and let
be a finite set of orthogonal elements in
such that
are idempotent. If
,
,
,
are real numbers and
such that
hold, then one has the inequality
Proof.
By taking
on both sides of (3.12), we have
Analogously
Now, using Acz
l's inequality for real numbers, that is, we recall that
and the Schwarz's inequality for positive linear functionals, that is,
we deduce (3.35).