A proper
-algebra is a complex Banach
-algebra
where the underlying Banach space is a Hilbert space with respect to the inner product
satisfying the properties
and
for all
. A
-algebra is a complex Banach
-algebra
such that
for every
. If
is a proper
-algebra or a
-algebra and
is such that
or
, then
.

For a proper
-algebra
, the trace class associated with
is
. For every positive
there exists the square root of
, that is, a unique positive
such that
, the square root of
is denoted by
. There are a positive linear functional
on
and a norm
on
, related to the norm of A by the equality
for every
.

Let
be a proper
-algebra or a
-algebra. A semi-inner product module over
is a right module
over
together with a generalized semi-inner product, that is with a mapping
on
, which is
-valued if
is a proper
-algebra, or
-valued if
is a
-algebra, having the following properties:

(i)
for all
,

(ii)
for
,
,

(iii)
for all
,

(iv)
for
.

We will say that
is a semi-inner product
-module if
is a proper
-algebra and that
is a semi-inner product
-module if
is a
-algebra.

If, in addition,

(v)
implies
,

then
is called an inner product module over
. The absolute value of
is defined as the square root of
and it is denoted by
.

Let
be a
-algebra. A seminorm
on
is a real-valued function on
such that for
and
:
,
,
. A seminorm
on
is called a
-seminorm if it satisfies the
-condition:
. By Sebestyen's theorem [1, Theorem 38.1] every
-seminorm
on a
-algebra
is submultiplicative, that is,
, and by [2, Section 39, Lemma 2(i)]
. For every
, the spectral radius of
is defined to be
.

The Pták function
on
-algebra
is defined to be
, where
. This function has important roles in Banach
-algebras, for example, on
-algebras,
is equal to the norm and on Hermitian Banach
-algebras
is the greatest
-seminorm. By utilizing properties of the spectral radius and the Pták function, Pták [3] showed in 1970 that an elegant theory for Banach
-algebras arises from the inequality
.

This inequality characterizes Hermitian (and symmetric) Banach
-algebras, and further characterizations of
-algebras follow as a result of Pták theory.

Let

be a

-algebra. We define

by

and call the elements of
positive.

The set
of positive elements is obviously a convex cone (i.e., it is closed under convex combinations and multiplication by positive constants). Hence we call
the positive cone. By definition, zero belongs to
. It is also clear that each positive element is Hermitian.

We recall that a Banach
-algebra
is said to be an
-algebra provided there exists on
a second norm
, not necessarily complete, which is a
-norm. The second norm will be called an auxiliary norm.

Definition 1.1.

Let

be a

-algebra. A semi-inner product

-module (or semi-inner product

-module) is a complex vector space which is also a right

-module

with a sesquilinear semi-inner product

, fulfilling

for

,

. Furthermore, if

satisfies the strict positivity condition

then
is called an inner product
-module (or inner product
-module).

Let
be a seminorm or a positive linear functional on
and
. If
is a seminorm on a semi-inner product
-module
, then
is said to be a semi-Hilbert
-module.

If
is a norm on an inner product
-module
, then
is said to be a pre-Hilbert
-module.

A pre-Hilbert
-module which is complete with respect to its norm is called a Hilbert
-module.

Since
and
are self adjoint, therefore we get the following Corollary.

Corollary 1.2.

If

is a semi-inner product

-module, then the following symmetry condition holds:

Example 1.3.

- (a)

- (b)

- (c)

- (d)

In the present paper, we give an analogue of the Bessel inequality (2.7) and we obtain some further generalization and a simple form for the Grüss type inequalities in inner product modules over
-algebras, proper
-algebras, and unital Banach
-algebras.