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## Über dieses Buch

This book discusses recent developments in semigroup theory and its applications in areas such as operator algebras, operator approximations and category theory. All contributing authors are eminent researchers in their respective fields, from across the world. Their papers, presented at the 2014 International Conference on Semigroups, Algebras and Operator Theory in Cochin, India, focus on recent developments in semigroup theory and operator algebras. They highlight current research activities on the structure theory of semigroups as well as the role of semigroup theoretic approaches to other areas such as rings and algebras. The deliberations and discussions at the conference point to future research directions in these areas. This book presents 16 unpublished, high-quality and peer-reviewed research papers on areas such as structure theory of semigroups, decidability vs. undecidability of word problems, regular von Neumann algebras, operator theory and operator approximations. Interested researchers will find several avenues for exploring the connections between semigroup theory and the theory of operator algebras.

## Inhaltsverzeichnis

### Decidability Versus Undecidability of the Word Problem in Amalgams of Inverse Semigroups

This paper is a survey of some recent results on the word problem for amalgams of inverse semigroups. Some decidability results for special types of amalgams are summarized pointing out where and how the conditions posed on amalgams are used to guarantee the decidability of the word problem. Then a recent result on undecidability is shortly illustrated to show how small is the room between decidability and undecidability of the word problem in amalgams of inverse semigroups.

Alessandra Cherubini, Emanuele Rodaro

### A Nonfinitely Based Semigroup of Triangular Matrices

A new sufficient condition under which a semigroup admits no finite identity basis has been recently suggested in a joint paper by Karl Auinger, Yuzhu Chen, Xun Hu, Yanfeng Luo, and the author. Here we apply this condition to show the absence of a finite identity basis for the semigroup

$$\mathrm {UT}_3(\mathbb {R})$$

UT

3

(

R

)

of all upper triangular real

$$3\times 3$$

3

×

3

-matrices with 0 s and/or 1 s on the main diagonal. The result holds also for the case when

$$\mathrm {UT}_3(\mathbb {R})$$

UT

3

(

R

)

is considered as an involution semigroup under the reflection with respect to the secondary diagonal.

M. V. Volkov

### Regular Elements in von Neumann Algebras

The semigroup of all linear maps on a vector space is regular, but the semigroup of

continuous

linear maps on a Hilbert space is not, in general, regular; nor is the product of two regular elements regular. In this chapter, we show that in those types of von Neumann algebras of operators in which the lattice of projections is modular, the set of regular elements do form a (necessarily regular) semigroup. This is done using the construction of a regular biordered set (as defined in Nambooripad, Mem. Am. Math. Soc. 22:224, 1979, [

9

]) from a complemented modular lattice (as in Patijn, Semigroup Forum 21:205–220, 1980, [

11

]).

### LeftRight Clifford Semigroups

Clifford semigroups are certain interesting class semigroups and looking for regular semigroups close to this is natural. Here we discuss the leftright Clifford semigroups.

M. K. Sen

### Certain Categories Derived from Normal Categories

Normal categories are essentially the category of principal left(right) ideals of a regular semigroup which are used in describing the structure of regular semigroups. Several associated categories can be derived from normal categories which are of interest.

A. R. Rajan

### Semigroup Ideals and Permuting 3-Derivations in Prime Near Rings

Let

N

be a near ring. A 3-additive map

$$\Delta : N\times N\times N\longrightarrow N$$

Δ

:

N

×

N

×

N

N

is called a 3-derivation if the relations

$$\Delta (x_1x_2, y,z)=\Delta (x_1,y,z)x_2+x_1\Delta (x_2, y,z)$$

Δ

(

x

1

x

2

,

y

,

z

)

=

Δ

(

x

1

,

y

,

z

)

x

2

+

x

1

Δ

(

x

2

,

y

,

z

)

,

$$\Delta (x, y_1y_2, z)=\Delta (x, y_1, z)y_2+y_1\Delta (x, y_2, z)$$

Δ

(

x

,

y

1

y

2

,

z

)

=

Δ

(

x

,

y

1

,

z

)

y

2

+

y

1

Δ

(

x

,

y

2

,

z

)

, and

$$\Delta (x, y, z_1z_2)=\Delta (x,y,z_1)z_2+z_1\Delta (x,y,z_2)$$

Δ

(

x

,

y

,

z

1

z

2

)

=

Δ

(

x

,

y

,

z

1

)

z

2

+

z

1

Δ

(

x

,

y

,

z

2

)

are fulfilled, for all

$$x,y,z,x_i,y_i,z_i\in N, i=1,2$$

x

,

y

,

z

,

x

i

,

y

i

,

z

i

N

,

i

=

1

,

2

. The purpose of the present paper is to prove some commutativity theorems in the setting of a semigroup ideal of a 3-prime near ring admitting a permuting 3-derivation, thereby extending some known results of biderivations and permuting 3-derivations.

Asma Ali, Clauss Haetinger, Phool Miyan, Farhat Ali

### Biordered Sets and Regular Rings

Biordered sets were introduced in [

3

] to describe the structure of regular semigroups. In [

1

] it is shown that the ideals of a regular ring forms a complemented modular lattices. Here we describe the biordered set of such a regular ring.

P. G. Romeo

### Topological Rees Matrix Semigroups

An important problem in the theory of topological semigroups is to formulate a suitable definition of continuity for the choice of generalized inverses. In this paper, we will show that under certain natural conditions, a topology can be defined on a Rees matrix semigroup, which turns it into a topological semigroup, and in which a canonical continuous choice of inverses is possible. As an example, we show that this construction applied to the semigroup of operators of rank less than or equal to 1 on a Hilbert space gives a topology which is stronger than the norm topology, under which this semigroup is a topological semigroup and the assignment of every operator to its Moore-Penrose inverse is continuous.

E. Krishnan, V. Sherly

### Prime Fuzzy Ideals, Completely Prime Fuzzy Ideals of Po- $$\Gamma$$ Γ -Semigroups Based on Fuzzy Points

Using fuzzy points the notions of prime fuzzy ideals, weakly prime fuzzy ideals, completely prime fuzzy ideals, and weakly completely prime fuzzy ideals of a po-

$$\Gamma$$

Γ

-semigroup have been introduced. Some important properties and characterizations of these ideals have been obtained. The relations among various types of primeness have also been investigated.

Pavel Pal, Sujit Kumar Sardar, Rajlaxmi Mukherjee Pal

### Radicals and Ideals of Affine Near-Semirings Over Brandt Semigroups

This work obtains all the right ideals, radicals, congruences, and ideals of the affine near-semirings over Brandt semigroups.

Jitender Kumar, K. V. Krishna

### Operator Approximation

We present an introduction to operator approximation theory. Let

T

be a bounded linear operator on a Banach space

X

over

$${\mathbb C}$$

C

. In order to find approximate solutions of (i) the operator equation

$$z\,x-Tx=y$$

z

x

-

T

x

=

y

, where

$$z\in {\mathbb C}$$

z

C

and

$$y\in X$$

y

X

are given, and (ii) the eigenvalue problem

$$T\varphi =\lambda \varphi$$

T

φ

=

λ

φ

, where

$$\lambda \in {\mathbb C}$$

λ

C

and

$$0\ne \varphi \in X$$

0

φ

X

, one approximates the operator

T

by a sequence

$$(T_n)$$

(

T

n

)

of bounded linear operators on

X

. We consider pointwise convergence, norm convergence, and nu convergence of

$$(T_n)$$

(

T

n

)

to

T

. We give several examples to illustrate possible scenarios. In most classical methods of approximation, each

$$T_n$$

T

n

is of finite rank. We give a canonical procedure for reducing problems involving finite rank operators to problems involving matrix computations.

Balmohan V. Limaye

### The Nullity Theorem, Its Generalization and Applications

The Nullity theorem says that certain pairs of submatrices of a square invertible matrix and its inverse (known as complementary submatrices) have the same nullity. Though this theorem has been around for quite some time and also has found several applications, some how it is not that widely known. We give a brief account of the Nullity Theorem, consider its generalization to infinite dimensional spaces, called the Null Space Theorem and discuss some applications.

S. H. Kulkarni

### Role of Hilbert Scales in Regularization Theory

Hilbert scales, which are generalizations of Sobolev scales, play crucial roles in the regularization theory. In this paper, it is intended to discuss some important properties of Hilbert scales with illustrations through examples constructed using the concept of Gelfand triples, and using them to describe source conditions and for deriving error estimates in the regularized solutions of ill-posed operator equations. We discuss the above with special emphasis on some of the recent work of the author.

M. T. Nair

### On Three-Space Problems for Certain Classes of $$C^*$$ C ∗ -algebras

It is shown that being a GCR algebra is a three-space property for

$$C^*$$

C

-algebras using the structure of composition series of ideals present in GCR algebras. A procedure is presented to construct a composition series for a

$$C^*$$

C

-algebra from the unique composition series for any GCR ideal and the corresponding GCR quotient being a

$$C^*$$

C

-algebra. We deduce as a consequence that, a GCR algebra is a three-space property. While noting that being a CCR algebra is not a three-space property for

$$C^*$$

C

-algebras, sufficient additional conditions required on a

$$C^*$$

C

-algebra for the CCR property to be a three-space property are also presented. Relevant examples are also presented.

A. K. Vijayarajan

### Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

Normal categories are essentially those arising as the category of principal left [right] ideals of a regular semigroup. These categories have been used in describing the structure of regular semigroups. The structure theory in this context is known as cross connection theory. Several associated categories can be derived from a normal category which are also of interest in the structure theory of regular semigroups. The subcategory of inclusions, the subcategory of retractons, the groupoid of isomorphisms etc. are some of the associated categories.

K. Kumar

### On k-Minimal and k-Maximal Operator Space Structures

Let

X

be a Banach space and

k

be a positive integer. Suppose that we have matrix norms on

$$M_2(X)$$

M

2

(

X

)

,

$$M_3(X)$$

M

3

(

X

)

,...,

$$M_k(X)$$

M

k

(

X

)

that satisfy Ruan’s axioms. Then it is always possible to define matrix norms on

$$M_{k+1}(X)$$

M

k

+

1

(

X

)

,

$$M_{k+2}(X),\ldots ,$$

M

k

+

2

(

X

)

,

,

such that

X

becomes an operator space. As in the case of minimal and maximal operator spaces, here also we have a minimal and a maximal way to complete the sequence of matrix norms on

X

k

-minimal

and

k

-maximal

operator space structures on

X

. These spaces were first noticed by Junge [

10

] and more generally studied by Lehner [

11

]. Recently, the relationship of

k

-minimal and

k

-maximal operator space structures to norms that have been used in quantum information theory have been investigated by Johnston et al. [

9

]. We discuss some properties of these operator space structures.

P. Vinod Kumar, M. S. Balasubramani

### Erratum to: Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

K. Kumar
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